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Compressed Sensing Magnetic Resonance Imaging UsingFourier and Non-Fourier
Based Bunched Phase Encoding
This dissertation is submitted for the degree of Doctor of Philosophy
KAZI RAFIQUL ISLAM
Principal Supervisor:Jingxin Zhang
Co-supervisor:Cishen Zhang
School of Software and Electrical Engineering
Faculty of Science Engineering & Technology
Swinburne University of Technology, John St, Hawthorn, VIC-3122, Australia
July 2020
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Declaration
I hereby declare that the contents of this thesis are original and have not been submitted in
whole or in part for the award of any other degree, qualification or any other university
except where specific reference is made to the work of others.
Kazi Rafiqul Islam
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Acknowledgements
PhD study has been one of the valuable parts of my education life. During this period, I have
acknowledged help and encouragement from many people and institute. Here, I would like
to remember their name.
First of all, I would like to praise my creator and sustainer Allah Ta’ala for his bounty and
Rahmah. Then, I would like to express my wholehearted thanks to my supervisor Prof.
Jingxin Zhang. Without his proper supervision, I could not finish my PhD study. During my
PhD study, He has always been helpful, resourceful and patient. His continuous
encouragement, kindness and sympathetic helped me a lot throughout my research. Also, I
would like to thanks Prof. Cishen Zhang for his guidance and advice towards me.
IwouldalsoliketothankSwinburneUniversityofTechnologyforprovidingscholarships, research facilities to me so that I could enjoy a peaceful life, and concentrate my time on research. Staff from this institute are always willing to help. I am grateful for their kind support and willingattitude.
I would like to thank my colleagues from Swinburne University and Technology and
Monash University, who helped and cooperate me during my research, especially Kamlesh
Powar.
Finally and most importantly, I would like to thank my family, who encouraged and supported me to obtain my PhD degree. I could not have achieved my degree without their support.
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Abstract
Magnetic resonance imaging (MRI) technology is one of the most significant biomedical
innovations. However, the MRI process speed is still too slow to meet the demand of
biomedical research and clinical diagnosis. Its applications, affordability, and accessibility
remain limited by the time required to sample adequate information of k-space for the
desired field of view (FOV), resolution, and signal-to-noise ratio (SNR). A large number of
methods have been developed over the years to accelerate MR imaging speed.
However, the ultimate targeted acquisition speed with quality is still a big issue that has
inspired active research in recent years on accelerated MRI. Compressed sensing (CS) is an
emerging technique to accelerate conventional MRI by reducing the number of acquired
data. Bunched phase encoding (BPE) is a technique that uses oscillating trajectories to
minimise phase encoding steps, and hence to accelerate data acquisition. These two
techniques have proven their respective advantages standalone, and their combination may
result in further acceleration of MRI data acquisition, but has never been exploited in the
literature.
This research work presents a novel method that combines CS with BPEusing Fourier
encoding to further accelerate data acquisition and preserve image quality. Physically BPE
trajectories are never the same as the theoretically calculated ones due to gradient
imprecision and field inhomogeneity, which results in poor image quality. A fix to this
problem is to measure the coordinates of the trajectories and use them in BPE image
reconstruction. Such measurements generally require prescan calibration that complicates the
process and increases operation cost.
To overcome this difficulty, we present a simple and effective method to estimate the
coordinates of BPE trajectories from normal scan dataand demonstrate its efficacy in image
reconstruction of in vivo scan data acquired from ZIGZAG trajectory.To smooth and remove
the aliasing from the image, a modulation map estimation method has been incorporated in
compressed sensing part of our proposed method.
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Non-Fourier encoding like spread spectrum instead of Fourier encoding in CS increases the
level of incoherence between the sparsifying transform matrix and the measurement
(sensing) matrix and hence the quality of reconstruction image. Therefore, a non-
Fourierencoding like Chirp modulation encoding method is used in this workto further
improve the quality of reconstructed image. It is realized by controlling the complex
orthogonality of the quadratic phase encoding. The experiments on the simulated and real-
life MRI data demonstrate that the proposed methods outperform the existing CS MRI and
BPE MRI methods in the literature.
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Table of Contents Chapter 1 Introduction .......................................................................................................... 1
1.1 Overview ................................................................................................................ 1
1.2 The Time Limitation of MRI .................................................................................. 3
1.3 Research Objective and Technical Approaches ....................................................... 7
1.4 Main Contributions ................................................................................................. 8
1.5 Thesis Outline ........................................................................................................ 9
Chapter 2 Fundamentals of MRI Process ............................................................................ 11
2.1 Introduction .......................................................................................................... 11
2.2 Theory of MRI ..................................................................................................... 11
2.2.1 Physics of NMR ............................................................................................ 11
2.2.2 Physical Behaviour of Tissue Particle ............................................................ 16
2.3 NMR Signal Excitation......................................................................................... 17
2.3.1 Slice Selection and Gradient Selection ........................................................... 17
2.3.2 Frequency and Phase Encoding ..................................................................... 18
2.3.3 Image Formulation ........................................................................................ 19
2.3.4 The K-space Signal ........................................................................................ 21
2.3.5 2D Spatial Encoding ...................................................................................... 22
2.3.6 Field of View (FOV) and Spatial Resolution ................................................. 23
2.3.7 MR Imaging Pulse Sequence ......................................................................... 25
2.4 Image Reconstruction Technique of MRI ............................................................. 26
2.4.1 Reconstruction Complexity ........................................................................... 26
2.4.2 Cartesian Trajectory ...................................................................................... 27
2.4.3 Non-Cartesian Trajectory .............................................................................. 28
2.4.4 Noise and System Uncertainty in Image Reconstruction ................................ 29
2.5 Image Reconstruction from Subsampled Data in Cartesian Coordinates................ 31
2.5.1 SENSE .......................................................................................................... 31
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2.5.2 Extensions of SENSE .................................................................................... 34
2.5.3 GRAPPA ....................................................................................................... 35
2.5.4 Extensions of GRAPPA ................................................................................. 37
2.5.5 Compressive Sensing based Image Reconstruction Method ........................... 40
2.5.6 Extension of CS-based Image Reconstruction Method ................................... 42
2.6 Novel Sequence Design ........................................................................................ 43
2.6.1 Bunched Phase Encoding .............................................................................. 44
2.6.2 Rotating RF Coil ........................................................................................... 45
2.6.3 Wave-CAIPI ................................................................................................. 45
2.7 Some Limitations of the Present Research ............................................................ 45
2.8 Conclusion ........................................................................................................... 46
Chapter 3 Fundamentals of Sparse Sampling ...................................................................... 48
3.1 Introduction .......................................................................................................... 48
3.2 Theory of Compressive Sensing ........................................................................... 49
3.3 Restricted Isometry in Compression Detection...................................................... 52
3.4 Incoherence in Compressive Sensing .................................................................... 53
3.5 Compressed Sensing in Magnetic Resonance Imaging .......................................... 54
3.6 Application of Compressive Sensing .................................................................... 55
3.6.1 Sparse Error Correction ................................................................................. 55
3.6.2 Linear Regression and Model Selection ......................................................... 56
3.6.3 Group Testing and Data Stream Algorithms .................................................. 57
3.6.4 Coronary Heart Imaging ................................................................................ 58
3.6.5 Brain Imaging ............................................................................................... 58
3.6.6 Rapid 3-D Angiography ................................................................................ 58
3.7 Some Limitations of CSMRI ................................................................................ 59
3.8 Conclusion ........................................................................................................... 60
Chapter 4 Compressive Sensing MRI using Fourier Based Bunched Phase Encoding ......... 61
4.1 Introduction .......................................................................................................... 61
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4.2 BPE and CS Techniques ....................................................................................... 64
4.3 Multichannel CSBPFE ......................................................................................... 74
4.4 Aliasing Coefficient Correction ............................................................................ 76
4.5 Smoothness Enhancing ......................................................................................... 80
4.6 Methods ............................................................................................................... 81
4.7 Simulations and Experimental Results .................................................................. 91
4.7.1 Multichannel CSBPE Simulation ................................................................... 91
4.7.2 Multichannel CSBPE Experiment .................................................................. 97
4.7.3 Phantom Scan Data Experiment................................................................... 100
4.7.4 Brain in vivo Scan Data Experiment ............................................................ 103
4.8 Computation Complexity of CSBPE as Compared with CSMRI. ........................ 103
4.9 Conclusion ......................................................................................................... 104
Chapter 5 Compressive Sensing MRI using Non Fourier Based Bunched Phase Encoding 105
5.1 Introduction ........................................................................................................ 105
5.2 Chirp Modulation based CSBP Encoding ........................................................... 106
5.3 Chirp Modulation of Multichannel CSBPE ......................................................... 110
5.4 Chirp Modulated CSBPE Sampling Pattern ........................................................ 111
5.5 Simulation and Experimental Results ................................................................. 112
5.5.1 Simulation Results ....................................................................................... 114
5.5.2 Experimental Results ................................................................................... 115
5.6 Conclusion ......................................................................................................... 122
Chapter 6 Conclusion and Future work ............................................................................. 123
6.1 Conclusion ......................................................................................................... 123
6.2 Future Work ....................................................................................................... 124
REFERENCES .................................................................................................................. 126
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List of Figures Figure 2.1: Nuclear spin in different condition (a) Spins without magnetic field (b) Spins with the magnetic field ....................................................................................................... 12
Figure 2.2: Net magnetization as a function of time ............................................................ 14
Figure 2.3: Transverse relaxation Mxy as a function of time ................................................ 15
Figure 2.4: FID process in the x-y plan after 900RF pulse ................................................... 16
Figure 2.5: The process of the slice selection by the gradient pulse ..................................... 18
Figure 2.6: The image resolution and FOV ......................................................................... 24
Figure 2.7: The sample spacing ∆kx in k-space and the extent kx,max, relate to the FOV, FOVx and voxel size ∆x, respectively, of the reconstructed image in the x-direction. Similarly, ∆ky and ky,max are connected to FOVy and ∆y. ........................................................................... 25
Figure 2.8: Typical pulse sequence for MRI data acquisition .............................................. 26
Figure 2.9: the trajectory pulse sequence in the non-Cartesian coordinates .......................... 29
Figure 2.10: Comparison between the original and the subsampling images ........................ 32
Figure 2.11: Schematic diagram of GRAPPA reconstruction technique .............................. 36
Figure 2.12: The image reconstruction method of TGRAPPA ............................................. 38
Figure 2.13:The image rebuilding method of k-t GRAPPA ................................................. 39
Figure 2.14:schematic diagram of gradient and BPE encoding technique ............................ 44
Figure 4.1: Sequential steps of CSBPE image reconstruction process.................................. 64
Figure 4.2: Schematic diagram of k-space trajectories of bunched phase encoding. (a) Rectilinear sampling data. (b) zigzag sampling data in the BPE scheme. ............................. 65
Figure 4.3: Generalized 1D bunched phase encoding technique in k-space.......................... 67
Figure 4.4: Schematic diagram of compressive sensing BPE ............................................... 73
Figure 4.5: Baseline data and shifted data in the Zigzag trajectory of Bunched phase encoding. The image from all PE baseline data on left and rest of all images from shifted PE line data and RO line data for all four sets of image. The whole 256 × 1028 BPE data image is showing on below. .......................................................................................................... 78
Figure 4.6: K-space lines on or near the k-space centre for eight individual k-space volume of the matrix represents the same image with the entire k-space shifted by the step size dk(n) and n = [0:7], top: n = 0 and bottom: n = 7. ......................................................................... 80
Figure 4.7: A sample pulse sequence design of bunched phase encoding; (Gse) Gradient in slice direction; (Gpe) Gradient in phase encoding direction; (Gro) Gradient in readout direction. A zigzag gradient is incorporated in the phase encoding direction during readout. The amplitude and period of the oscillatory gradient in this figure are not to scale. ............. 82
Figure 4.8:(a) Inverse Fourier transform of the acquired all 32 channel data gives the aliased image; (b) Inverse Fourier transform of the acquired channel 4 data gives the aliased image; (c) k-space of channel 4. ..................................................................................................... 83
Figure 4.9: Phase encoded data sampling technique; (a) Random sampling; (b) Variable density Random sampling. .................................................................................................. 83
Figure 4.10: BPE k-space data acquisition. Ccalculated (simulated) trajectory which shown in red marks (----); Original (measured) trajectory where data positions are marked as (×’s,
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×’s, ×’s, ×’s), baseline (zero shift) data points are marked as ×’s, measured shifted data points are marked as ×’s, ×’s, ×’s. ....................................................................................... 85
Figure 4.11: The sequential steps of Aliasing Coefficient Correction by cross correlation technique ............................................................................................................................ 87
Figure 4.12: Calculated (red) and measured (blue) BPE k-space positions. The data were shifted to be symmetrical around zero PE shift.................................................................... 88
Figure 4.13: BPE image reconstruction (channel 4), phase images, using the synthetic BPE k-space PE shifts, Ks (left) and using the measured and corrected BPE k-space PE shifts, Kc (right). ................................................................................................................................ 89
Figure 4.14: BPE image reconstruction, all channels combined, R = 2, matrix 128 x 128, using the synthetic BPE k-space PE shifts, Ks (left), and using the measured and corrected BPE k-space PE shifts, Kc (right). ....................................................................................... 90
Figure 4.15: The coil Sensitivity maps used in CSBPE simulation. The sensitivity maps are estimated from data acquired on MR scanner ...................................................................... 92
Figure 4.16: The mean relative error versus the acceleration factor in CS-MRI, CSBPE for simulation data. .................................................................................................................. 93
Figure 4.17 and Figure 4.18 show the simulation results of Fourier based compressed sensing reconstruction using BPE data. Figure 4.17 is magnitude image and Figure 4.18 is phase image respectively. The BPE acquired data images are all aliased images which has obtained before the CS reconstruction and the middle column show the aliased images after CS reconstruction at reduction factor R = 4, 6, 8 and 10 respectively. Right column shows the error images of Fourier based CS reconstruction for acceleration factors of 4, 6, 8 and 10 respectively; The result show the CS reconstructed images from BPE data can preserve image resolution. ................................................................................................................ 93
Figure 4.17: Simulation results of Fourier based compressed sensing reconstruction using BPE data; (a)-(d) (left two column) comparing the BPE acquired data image and CS reconstructed images at reduction factor R = 4, 6, 8 and 10 respectively; (a)-(d) (right column) error images of Fourier based CS reconstruction for acceleration factors of 4, 6, 8 and 10 respectively; The result show the CS reconstructed images from BPE data can preserve image resolution. .................................................................................................. 94
Figure 4.18: Simulation results (phase image)of Fourier based compressed sensing reconstruction using BPE data; (a)-(d) (left two column) comparing the BPE acquired data phase image and CS reconstructed phase images at reduction factor R = 4, 6, 8 and 10 respectively; (a)-(d) (right column) error images of Fourier based CS reconstruction for acceleration factors of 4, 6, 8 and 10 respectively; The result show the CS reconstructed images from BPE data can preserve image resolution. ........................................................ 95
Figure 4.19: Simulation results for comparing the conventional CS and Fourier based CSBP encoding technique. (a) reference image; (b)-(d) reconstructed images of Fourier based CS for acceleration factors of 4, 6, and 8 respectively; (e)-(g) error images of Fourier based CS for acceleration factors of 4, 6, and 8 respectively; (h)-(j) reconstructed images of Fourier based CSBP encoding for acceleration factors of 4, 6, and 8 respectively; (k)-(m) error images of Fourier based CSBP encoding for acceleration factors of 4, 6, and 8 respectively; The result of reconstructed images of CSBPE outperforms the CS for preserving image resolution. ........................................................................................................................... 96
Figure 4.20: Comparison between with and without aliasing coefficient correction of reconstructed MRI images from BPE acquired in vivo k-space data; (a) reference image; (b) Reconstructed MRI image using calculated (simulated) trajectories; (c) Reconstructed MRI
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image after aliased coefficient correction using cross-correlation technique; (d) and (e) are error images. The maximum error in the calculated trajectory is 0.2095 and in corrected trajectory is 0.0423. ............................................................................................................ 98
Figure 4.21 : Comparison between CSBPE image reconstruction using estimated modulation maps and non-estimated modulation maps; (a) Original image (b) CSBPE image reconstruction without an estimated modulation map at reduction factor R = 4, (c) CSBPE image reconstruction using the estimated modulation map at reduction factor R = 4. (d) and (e) are error images. ............................................................................................................ 99
Figure 4.22: Experimental results of Fourier based CSBPE phantom data. The left column top image represents the reference image and Fig. Rest of the left column shows reconstruction images using Fourier based CSBPE for down sampling R = 2, 4, 6, 8 respectively and the right column show the error images. The error for the reduction factor 2, 4, 6 and 8 are 0, 0.021, 0.3637, and 0.3867 respectively. ................................................... 101
Figure 4.23: Experimental results of Fourier based CSBP encoding. The left column represents the reference image and Fig. (a) show image reconstruction using Fourier based CSBPE for down sampling R = 4 and Fig. (b) – (c) show reconstruction images of Fourier based CSBP encoding for acceleration factors of R = 6, 8 respectively and Fig.(d) - (f) (right Column): show the error images. The error for the reduction factor 4, 6 and 8 are 0.021, 0.478 and 0.523 respectively. ............................................................................................ 102
Figure 5.1: Sequential steps of Chirp modulated CSBPE image reconstruction process ..... 107
Figure 5.2: (a) Fourier modulated k-space (Mesh view), (b) Chirp modulated k-space (Mesh view), (c) Chirp modulated k-space .................................................................................. 109
Figure 5.3: (a) shows that the Fourier encoded variable density sampling pattern and (b) shows the chirp modulated random sampling pattern which energy is spread along the phase direction. .......................................................................................................................... 112
Figure 5.4 The mean relative error versus the acceleration factor in CS-MRI, CSBPE Fourier encoding and CSBPE chirp encoding for simulation data. ................................................. 113
Figure 5.10 shows the comparing real phantom results between the Fourier based CSBPE and chirp modulation based CSBP encoding schemes. Figure 5.10 (b)-(d) show Chirp modulated Fourier based CSBPE reconstructed images at different acceleration factors. Figure 5.10 (e)-(g) show error images with Chirp modulated Fourier based CSBPE for acceleration factors of 4, 6, and 8 respectively. Figure 5.10 (h)-(j) show images reconstructed with Fourier encoding based CSBPE using for acceleration factors of 4, 6, and 8 respectively. Figure 5.10 (k)-(m): show error images with Fourier encoded based CSBPE for acceleration factors of 4, 6, and 8 respectively. But, for the real phantom data case as shown in Figure 5.10, The Chirp modulated CSBPE results are not significantly high quality compare to Fourier based CSBPE methods. ...................................................................... 115
Figure 5.5: Simulation results for performance evaluation of chirp based compressed sensing reconstruction using BPE data; (a)-(d) (left two column) comparing the chirp based BPE acquired data image and CS reconstructed images at reduction factor R = 4, 6, 8 and 10 respectively; (a)-(d) (right column) error images of Fourier based CS reconstruction for acceleration factors of 4, 6, 8 and 10 respectively; The result show the CS reconstructed images from chirp based BPE data can preserve image resolution. .................................... 116
Figure 5.6: Simulation results for performance evaluation of chirp based compressed sensing reconstruction using BPE data; (a)-(d) (left two column) comparing the chirp based BPE acquired data image and CS reconstructed images at reduction factor R = 4, 6, 8 and 10 respectively; (a)-(d) (right column) error images of Fourier based CS reconstruction for
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acceleration factors of 4, 6, 8 and 10 respectively; The result show the CS reconstructed images from chirp based BPE data can preserve image resolution. .................................... 117
Figure 5.7: Simulation results for Chirp modulated Fourier encoding based CSBPE. Left column represents the reference image 256 × 256 (up/down: phase encodes, left/right: frequency encode), Fig. (a) - (d) (middle column): shows reconstructed images by Chirp modulated Fourier basedCSBPE with random down sampling patterns for acceleration factors of R= 4, 5.7, 6, 8 respectively and Fig.(a)-(d) (right Column): shows the error images. ............................................................................................................................. 118
Figure 5.8: Comparing results between the Fourier based CSBPE and chirp modulation based CSBPE encoding schemes (up/down: phase encodes, left/right: frequency encode). (b)-(d): show images reconstructed with Chirp modulated based CSBPE for acceleration factors of 4, 6, and 8 respectively; (e)-(g): show error images with Chirp modulated based CSBPE for acceleration factors of 4, 6, and 8 respectively; (h)-(j): show images reconstructed with Fourier encoding based CSBPE using for acceleration factors of 4, 6, and 8 respectively; (k)-(m): show error images with Fourier encoded based CSBPE for acceleration factors of 4, 6, and 8 respectively; ................................................................. 119
Figure 5.9: Comparing results between the CS-MRI and chirp modulation based CSBPE encoding schemes (up/down: phase encodes, left/right: frequency encode). (b)-(d): show images reconstructed with Fourier encoding based CSBPE using for acceleration factors of 4, 6, and 8 respectively; (e)-(g): show error images with Fourier encoded based CSBPE for acceleration factors of 4, 6, and 8 respectively; (h)-(j): show images reconstructed with Chirp modulated based CSBPE for acceleration factors of 4, 6, and 8 respectively; (k)-(m): show error images with Chirp modulated based CSBPE for acceleration factors of 4, 6, and 8 respectively; .................................................................................................................. 120
Figure 5.10: Compare the results between the Fourier based CSBPE and chirp modulation based CSBPE encoding schemes (up/down: phase encodes, left/right: frequency encode). (b)-(d): show images reconstructed with Chirp modulated based CSBPE for acceleration factors of 4, 6, and 8 respectively; (e)-(g): show error images with Chirp modulated based CSBPE for acceleration factors of 4, 6, and 8 respectively; (h)-(j): show images reconstructed with Fourier encoding based CSBPE using for acceleration factors of 4, 6, and 8 respectively; (k)-(m): show error images with Fourier encoded based CSBPE for acceleration factors of 4, 6, and 8 respectively; ................................................................. 121
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List of Tables Table 1: Calculated trajectory vs corrected trajectory of all 32 channel. .............................. 88
Table 2: Comparison of the artifact power (AP) between proposed CSBPE-MRI methods and the reference CS-MRI in a simulated brain image at different acceleration factor (R). .. 93
Table 3: Compare the simulation results of Fourier based CSBPE and chirp modulated CSBPE for single channel data. ........................................................................................ 115
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Nomenclature
B radius of weak-Apball
B0 external magnetic field
B1 magnetic field caused by the RF pulseaconstant
C complex set
d negative gradient of objective function
E energy
∆E energy differencefrequency
f(x) regularized functiongradient of objectivefunctionGramMatrix
Gx x component of magnetic gradient
Gy y component of magnetic gradient
Gz z component of magnetic gradient
Gs slice selectiongradient
GΦ phase encoding gradient
H hard thresholdingoperation
h the Planck’sconstant
I image/identity matrix
i index
j index
K sparsity of the originalsignal
k Boltzmannconstant
A00norm
A11norm
A22norm
Ap pnorm
A∞infinitynorm
M length of the measureddata
M0 net magnetization
Mx x component of the netmagnetization
My y component of the netmagnetization
Mxy transverse component of the net magnetization due to T2
M∗xytransversecomponentofthenetmagnetizationduetoT2
∗
Mxy0 Mxyat the start of the dephasingprocess
x
Mz z component of the netmagnetization
N length of the originalsignal
Nc number ofcoils
[N] a set (1,2,….. ,N)
card(N) cardinality of set[N ]
PD protondensity
p aconstant
r residual
R reductionfactor
Rnet net reduction factor
Rnom nominal reduction factor
R realset
s transformed compressiblesignal
si(j) acquired black data points in coil i at locationj
s4,ACS the gray data in the ACS line of coil4
s(kx,ky) acquired signal in Fourierdomain
S NMRsignal
t time
tp a timeduration
T1 spin-lattice relaxationtime
T2 spin-spin relaxationtime
T2,inhomorelaxation time caused byinhomogeneity
T2∗ combined relaxation time
TE echotime
TR repetitiontime
T measurementmatrix
TF partial Fourier encodingoperator
T̃F fullFourierencodingoperator
T̂F Fourierencodingoperatorwithwavelettransform
TN partial noiselet encodingoperator
T̃N fullnoiseletencodingoperator
T̂N noiseletencodingoperatorwithwavelettransform
Ts selected encoding technique with sub-samplingapplied
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T̂s wholesub-samplingandencodingprocess
U transformmatrix
W wavelet transformoperator
wi(j) GRAPPAweightsincoiliatlocationj x adirection
x original signal orimage
xi the ithentry ofx
x̂1 overlappedpixel
x̂2 overlappedpixel
x(1) pixel in aimage
x(2) pixel in aimage
x̂ reconstructedsignalorimage
y adirection
y measureddata
yi sub-sampled data from coili
z adirection
α aconstant
β aconstant
Γ1(1),Γ1(2) sensitivity in the first coil
Γ2(1),Γ2(2) sensitivity in the second coil
Γi sensitivity profile of coili
Γ̂i transformedΓi
γ the gyromagneticratio
δ ratio of M toN
δK,δ2K RIPconstant
s parameters with relationship tonoise
θ degree of RFpulse
Λ supportset
λ regularizationparameter
µ coherencevalue
ν frequency ofproton
ν0 Larmorfrequency
ξ positive smoothingparameter
π aconstant
ρ ratio of K toM
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ρ(x,y) spin density at location (x, y)
σ stopcriterion
σmin minimum singularvalues
σmax maximum singularvalues
τ parameters with relationship tonoise
Φ sensingmatrix
Φ̂ asquaresensingmatrix
φi ithrow or column of sensing matrix Φ
ΦK a matrix consisting of K columns fromΦ
Φsub(K) sub-matrix formed from K distinct columns ofΦ
χ(x) mother bases function of noiselet
Ψ sparsifyingmatrix
ψi ithrow or column of sparsifying matrixΨ
ψi(x) ithentry of transformed vector x under Ψ
Ψ̃ Ψwitheachcolumnnormalized
Ψ̂ dictionary
ψ̂i ithcolumnofΨ̂
ω frequency of sinewave
∅ emptyset
|·| magnitude
||·||0 number of non-zeroelements
||· ||1 sumofthemagnitudeofallelements
||·||2 sumofthesquareofallelements
||·||p pthrootofsumofpthpowerofthemagnitudeofallelements
||·||∞ maximum magnitude of avector
<·,·> innerproduct
∞ infinity
Ⓢ element-wisemultiplication
Chapter 1 Introduction
Introduction
1.1 Overview
Visual observation is one of the essential keys of almost all human activities. Hence,
the most useful information source, such as video and image, plays an indispensable
role in our daily lives. Along with science and technology, a significant number of
imaging techniques have been developed and applied in almost all of our life events.
The impact of imaging in medicine on clinical diagnosis and research has been
unprecedented. Among all of the clinical imaging techniques, magnetic resonance
imaging (MRI) is the most popular and advanced imaging technique.
The main concept of MRI is based upon the practical application of Nuclear Magnetic
Resonance (NMR) which was first explained independently by Bloch and Purcell, in
1946, who was awarded Nobel Prize in Physics in 1952 [1]. After establishing the
theory, scientists started to realise that imaging could be a great application of NMR.
For the first time imaging by NMR was used for a military purpose. At the time of cold
war, aircraft navigation in the water project was established using NMR signals in the
former USSR in the 1960s. During this project, Lieutenant Vladislav Ivanov noted that
this technique could be used in the human body because living species are made up of
water [2].
However, his proposal was denied due to political reasons. A couple of years later, the
real era of MRI had literally begun. Researcher Lauterbur published a paper unfolding a
new technique to form images of biological structure using NMR signal of protons[3].
After establish this concept, Mansfield published a series of paper that introduced the
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fundamental concept of spatial encoding in MR imaging. Due to the major
contributionin medicine, Researcher Lauterbur and Mansfield were awarded the Nobel
Prizein Medicine in 2003[4].
Now, MRI [5] is a widely used medical imaging modality in contemporary biomedical
research and diagnosis.The importance of MRI is grown dramatically to radiologists,
clinicians and researchers for its competency to produce high-quality images without
the side effects of harmful radiation since its invention in the 1970s.MRI can construct
an image from soft tissue throughout the whole body [6] and can efficiently differentiate
grey and white matters in imaging the brain. It provides excellent contrast in imaging
the different parts of the body such as heart, muscles, blood and cancer cell compared
with other modalities such as CT or X-rays. It can distinguish blood flow and can be
used to measure necessary quantities diagnostically such as cortical thickness [7-9].
In addition, MRI has an extra feature to generate various contrast–weighted images such
as T1, T2, and T2*. This flexibility in image contrast is advantageous in clinical diagnosis
and medical research. Furthermore, thenon-invasive technique has made MRI harmless
for the patient. Hence, MRI has excellent versatility and numerous applications in both
medical research and clinical diagnostic and preoperative surgical imaging.
MRI is suitable for application to any part of the body, such as the heart, brain, neck,
spine, kidney, abdomen and feet. Besides that, there are many applications of MRI for
different types of diagnosis and research. For example, functional MRI (fMRI) can be
used to measure the signal activity of neuron in the brain; dynamic contrast-enhanced
MRI (DCE-MRI) is useful to evaluate specific contrast agents of the muscle blood flow;
3D MRI can evaluate 3D volumetric images of different body parts, etc. These are some
applications of the MRI. It is understood that with further development in the software
and hardware, MRI will be even more broadly applied in different medical and non-
medical fields.
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1.2 The Time Limitation of MRI
Despite the ever increasing importance and wide application of MRI in biomedical
research and clinical diagnosis, it still suffers from some major limitations.For example,
under the normal imaging conditions, a conventional MR imaging of a brain may take
around ten minutes, and inother cases, it mayprolonguptoan hour. This constraint is a
hardship for some people, such as children,aged patients, and patients suffering from
chronic pain. Moreover, most MRI machine bores are narrow bounded spaces, which
may cause claustrophobia for some patients.Also, multiple images are typically required
for various applications. This may extend ascan sessionto more than one hour in
duration, resulting in increasedexpenses and reducedaccessibility of MRI scanner [10].
MRI data acquisition speed is restricted by biological limitations associated with the
properties of spatially fluctuating magnetic fields in the body. Varying applied magnetic
fields can induce fluxes at high rates indifferent parts of the nervous system and may
stimulate the nerves causing discomfort andeven damage in the subject [11].As the
electromagnetic fields used to encode the data for Fourier coefficients, this constraint
limits the rate that can collect MRI data.Hence, in some time-critical applications, such
as fMRI (functional MRI), the spatial image resolution has to be compromised for
higher temporal resolution to accelerate MRI process, and researchershave focused on
modifying the sampling pattern and acquiring multiple samples concurrently.
Different kinds of sampling patterns or multiple simultaneous samples have some
advantages and disadvantages. Modifying the data sampling pattern may
reduceimageresolution, lose phase informationand also may require more complicated
reconstruction methods [12, 13]. Moreover, fast MR image acquisition methods can use
multiple echoes to acquire an image in shorter time while decreasing contrast or
increasing susceptibility to field inhomogeneity [14-16].
Several image processing approaches have been developed for accelerating MRIoverthe
last two decades, e.g., parallel MR imaging process and post-processing technique to
4
recover images from fewer data samples. Most popular parallel imaging methods are
SENSE (SENSitivity Encoding), GRAPPA (Generalized Auto Calibrating Partially
Parallel Acquisition) [17-20]. These techniques use multiple receive channels with
different non-uniform magnetic sensitivities, whereas conventional MRI has no such
capacity to accelerate image processing due to using a single channel with uniform
sensitivity to magnetisation. Parallel MR imaging (pMRI) techniques are able to
accelerate different types of imaging which have already been implemented in
commercial scanners. However, those pMRI schemes alone are insufficient to achieve
the level of acceleration we would like to achieve due to the number of difficulties
associated with pMRI.These difficulties are responsible for degrading the quality of the
reconstructed image if these are not appropriately addressed and solved during the
reconstruction process.
These limitations are summarized below:
Inadequate sampling data for reconstruction: For example, keyhole imaging
technique is applied to reconstruct the full-size image using only a subset of the
acquired k-space data.
Noise in acquired data: It is nearly impossible to acquire data without noise in
all MRI systems[21-23]. Moreover, it has more effects on accelerated imaging
because fewer amounts of data are acquired in k-space. Hence, we have
observed significant SNR loss in accelerated imaging [19].
Process Noise: The procedure of encoding is responsible for imposing noise
during the acquisition, and it affects the reconstruction results, especially when
the others condition are not fulfilled.
Anonymous imaging scheme:This kind of problem can arise from unknown
sensitivity functions of receiver coils. As the imaging procedure is unidentified,
it requires to be estimated. Butestimation process may create many difficulties,
such as process estimation error and system structural constraints.
5
Process uncertainty: The main reasons for the uncertainty in the imaging
technique are process noise and process estimation error. It can also affect image
quality, such as field’s inhomogeneity, improper pulse sequence design and
some other machine related issues.
Anewly invented methodof image reconstruction from undersampled k-space data is
known as compressed sensing (CS) [24-27].CS has a unique feature of constructability
to recover data from fewer randomly selected measurements than the original size of the
signal by using a suitable recovery technique. However, two prerequisites of CS
theoryshould be fulfilled in order to obtain the maximum benefit from CS as mention
below:
(1) Sparsity:The image signal should be sparse or sparse after a specific
transformation.
(2) Incoherency and RIP: The acquired signal should be sensed randomly.
Moreover, the random sensing matrix should satisfy a particularcondition
such as the restricted isometric property (RIP)[24, 28, 29] and
incoherence[30, 31]to ensure successful reconstruction of the image signal.
So, MRI is one of the appropriate candidates for CS because MRI data are generally
sparse in the wavelet transformed domain, and can be acquired with some random
sampling patterns[32-36]. Hence, MRI has the ability to fulfil the condition of RIP and
coherency to reconstruct the images from sub-sampled k-space data successfully.
Over the last two decades, the researchers have endeavoured to achieve the highest goal
of the MRI process time by combining different methods. The combination of
compressed sensing with some other existed popular pMRI methods such as SENSE,
and GRAPPA was already established in recent years [19, 20]. Some of the researchers
havetried to use different encoding techniques such as Fourier, Noiselet, and chirp
modulation. Some other researchers have also developed different encoding trajectories
such as Cartesian and non-Cartesian to enhance the processing speed.
6
The various under-sampled data acquisition techniquesand compressive sensing
reconstruction methods have the capability to reduce the total scan time and running
costto some extent. This can give the opportunityformore patients or allow more
imaging to be completed per unit time. That is very important for systemic metabolic
and genetic diseases, where multiple organ systems are involved. Additionally, other
scan types can be made feasible. However, there are still some limitations in the present
work as listed below:
Problem 1.Parallel MRI techniques such as SENSE, GRAPPA require a large
number of coils when few number of data are acquire to reconstruct the
image.Also due to inconsistency in coil geometries and sensitivities, severe
residual aliasing artifacts and amplified noise may occur when high reduction
factor is chosen.
Problem 2.As a non-array coil method like BPE also has some problem when
reduction factor is increased. That means signal to noise ratio of BPE approach
becomes considerably reduced as the reduction factor is increased and we observe
that if reduction factor is greater than two, the noise of reconstructed image is
gradually increased even through aliasing artefact are not exist.
Problem 3. Similarly, the reduction factor is also crucial for high-quality
reconstruction in compressed sensing even though it is used with pMRI. That
means if it is increased to a certain level, the artifacts of the image and SNR also
increased simultaneously.
Problem 4. In the conventional CS-MRI, Fourier matrix as a sensing matrix and
Wavelet matrix as a sparsifying transform matrix respectively are not optimally
incoherent. Moreover, Fourier encoding usually concentrates energy in the centre
of the k-space known as low frequencies region. This imposes restriction on the
subsampling pattern to adequately sample the low frequency region and
insufficiently sample the high frequency region at high acceleration factors,
7
resulting in degraded image quality. Such limitation can cause a considerable loss
in image resolution.
The purpose of the research is to address above mentioned limitations to accelerate
imaging speed and improve the imaging quality within the constraints of present
commercial MRI scanners. Currently, the scan time of MRI is still too slow to meet the
demand of clinical diagnosis and biomedical research. Therefore, the accelerated
acquisition and reconstruction techniques need to be employed to reduce encoding and
hence scan time. Inthis research, we develop a method to improve and accelerate the
conventional signal acquisition and high-resolution image reconstruction using a hybrid
method.These methods will provide economic benefits for patients and doctors.
1.3 Research Objective and Technical Approaches
The objective of this research is to advance further the present state of reconstruction
speed with quality in MRI. The fundamental concept of this research is to find a new
optimal way to reconstructhigh-quality images using less amount of acquired data by
implementing different approaches. These Novel techniques offer optimal solutions in
accelerated imaging and hence, increase the image quality. In this thesis, the enhanced
imaging methods are modelled, analysed and solved from a system point view by using
the technique extensively from the theories of system modelling and parameter
estimation, system inversion and system optimization.
The technical approaches of the research work areto achievea higher acceleration factor
with a quality image using both compress sensing and Fourier and non-Fourier
encoding of non-Cartesian zigzag trajectories. Non-Cartesian trajectory such as
Bunched Phase encoding (BPE) can be directly combined with the compressed sensing
reconstruction framework. Some existing methods such as Sparse SENSE [27] and L1
SPIRiT [37] follow this approach, yielding a sparsity promoting standardized
reconstruction technique that can reconstruct high-quality images from moderate
accelerations with random under sampling.Initially, general simulations will be used to
validate these ideas. After validation, the MR pulse sequence will beused to acquire data
8
from the MRI scanner for physical experiment and verification of the proposed
methods.
The novel results of this thesis are two new accelerated MR imaging techniques. The
first one is a technique to reduce scan time in 2D static imaging and is called
compressed sensing MRI using Fourier based bunched phase encoding.This technique
accelerates the conventional MR signal acquisition and improves high-resolution image
reconstruction using a hybrid method which combines multiple coils compressed
sensing MRI with Fourier or non-Fourier based bunched phase encoding.
The second one is the non-Fourier chirp modulatedBPE to further reduce scan time in
MR imaging and is called compressed sensing MRI using non-Fourier based bunched
phase encoding. The motivation of this work is that chirpmodulated encoding schemes
outperformFourier encoding in compressed sensing reconstruction because it spreads
signal energy in the measurement domain that improves sampling incoherence.
1.4 Main Contributions
The major contributions of this thesis are summarized as follows:
1. In order to improve the quality of image reconstruction of compressive sensing
for high reduction factor, this thesis has proposed the first compressive sensing
bunched phase encoding (CSBPE) technique. The simulation results of Fourier
based CSBPE have shown that the quality of the reconstructed image is higher
than that of the conventional compressive sensing at the same reduction factor.
These results are presented in Chapter 4.
2. Due to magnetic field inhomogeneity and imprecise field gradients, the phase
of the zigzag trajectories is different from theoretically calculated ones, which
lowers image quality. To overcome this problem, we have introduced a new
technique called cross-correlation to remove the phase deviation from the
zigzag data position. The experiment results show that the CSBPE with zigzag
9
data-position corrected by cross-correlation outperforms the conventional BPE
without the correction at high reduction factor. This part of the study is
presented in Chapter 4.
3. To smooth and remove the aliasing from the image, a modulation map
estimation method has been incorporated in compress sensing part of our
proposed method.This part of the study is presented in Chapter 4.
4. In order to further improve the image quality of CSBPE, we have introduced a
non-Fourier encoding scheme called chirp modulated Fourier encoding. The
simulation result shows that the chirp modulated CSBPE outperform the
Fourier method. This research result is given in chapter 5.
1.5 Thesis Outline
The thesis is organized as follows:
• Chapter 1: Presents the limitation of CSMRI, motivation and objectives of this
thesis. A general background of the study is explained to highlight the
importance of the research. The objectives and technical approaches are then
provided. Finally, the thesis outline is given.
• Chapter2:Presents the fundamentals of the MRI process. First, the theory of
NMR is reviewed to introduce the physics of NMR, It then explains the NMR
signal excitation, image reconstruction technique of MRI, the non-cartesian
trajectory, different types of sub-sampling and reconstruction in the Cartesian
coordinates such as SENSE, and GRAPPA. Finally,it explains some novel
sequence designs, including bunched phase encoding, rotating RF coil, Wave–
CAIPI.
10
• Chapter 3: Summarizes the fundamental of sparse sampling. First, the theory
of compressed sensing is reviewed to introduce restricted isometry
propertyand, incoherence in compressive sensing. It then explains the
application of CS in magnetic resonance imaging and other fields.
• Chapter 4: Discusses compressive sensing MRI using bunched phase
encoding scheme. It first explains the theory of CSMRI using BPFE and
multichannel CSBPE, then explainshow to correct the reconstruction error of
the reconstruction techniques. It further explains the method of CSBPE and
describes the simulation and experimental results. Finally, the discussion and
conclusion of this chapter are presented.
• Chapter 5:Introduces compressive sensing MRI using bunched phase non-
Fourier encoding scheme. It first describes the theory of CSBPE with chirp
modulated Fourier encoding and explains the multichannel CSBPE with chirp
encoding. It then presents the simulation results with qualitative and
quantitative analysis. Finally, the discussion and conclusion of this chapter is
presented.
• Chapter 6:Concludes the thesis with discussions of the presented results and
recommendations for future study.
11
Chapter 2 Fundamentals of MRI Process
Fundamentals of MRI Process
2.1 Introduction
One of the purposes of this chapter is to identify the limitationsof present work in the
literature review. At first, the fundamentals of MRI will be discussed then the open
problems in the existing literature will be identified.
2.2 Theory of MRI
Magnetic resonance imaging (MRI) concept works by using the principle of nuclear
magnetic resonance. The spatial gradient of the applied magnetic field provides spatial
information to create images. In MRI, the spatial frequency domain is known as k-
space. It transforms the time-domain signal to the spatial frequency domain. The
sampling trajectory of MRI in the k-space domain is determined by the time of the
applied magnetic field gradient. The appropriately designing the magnetic field is the
most important engineering aspect of MRI to induce nuclear magnetic resonance to
capture the spatial distribution of the hydrogen protons in the body.
2.2.1 Physics of NMR
The physical basis of Nuclear Magnetic Resonance (NMR) phenomenonis the concept
of Nuclear Spin, which is first invented by Bloch and Purcell in 1946[38-40]. NMR
signals can generate by flipping the nuclear protons from high energy state to low
energy state. Nuclear spin is a natural form of intrinsic angular momentum carried by
atomic and sub-atomic and other elementary particles. It is a hypothesis that protons of
12
Hydrogen Atom get aligned in parallel or anti-parallel in the direction of the magnetic
field [41].Same directional spin particle repeal but opposite directional particle make
pair with each other.
In NMR, only unpaired spins are played most of the role. The rotate nuclei act as a
magnetic dipole with random orientations. However, in the natural environment, the net
magnetization of nuclei which donated as M0 is untraceable due to random orientations
and the magnetic effect iscancelled by each other.
When a proton spins on its axis, it generates a magnetic field. Hence, the nucleus can be
considered to be a tiny bar,and these small bar magnets are randomly oriented in space.
However, in the presence of a magnetic field B0, they are oriented with or against this
applied field. Therefore, the difference in numbers of dipoles orientation with and
without this field leads to the NMR signal.
Figure 2.1: Nuclear spin in different condition (a) Spins without magnetic field (b) Spins with the magnetic field
The energy difference between the two states isminimal,and it is defined as,
∆E = hγ|𝐁𝐁𝟎𝟎|2π
(2.1)
13
Where,ℎ, 𝛾𝛾, |𝐁𝐁𝟎𝟎|is the Planck’s constant, gyromagnetic ratio, and magnitude of the static
magnetic field.Spins can switch between the two energy states at a particular frequency
by absorbing a photon. The energy of the photon E is related to its frequency f, therefore
𝐸𝐸 = ℎ𝑓𝑓 (2.2)
The photon only can move from one state to another state, when the energy difference
between the two energy states is the same as the photon energy, which means
𝐸𝐸 = ∆𝐸𝐸 (2.3)
Hence, we can rewrite the equation (2.1) as
ℎ𝑓𝑓 = hγ|𝐁𝐁𝟎𝟎|2π
(2.4)
Therefore the frequency of the photon is then given as
𝑓𝑓 = γ|𝐁𝐁𝟎𝟎|2π
(2.5)
The frequency fis called Larmor frequency,and the magnetization is resonant at this
frequency.
2.2.1.1 Spin-Lattice Relaxation The net magnetization of nucleiM0reaches an equilibrium state after placed in a
magnetic fieldB0 for a specific time, and the direction of M0isparallel to the magnetic
field B0. In the coordinate system, the direction of the field B0is generally considered
parallel to as the z-direction, whereas perpendicular to the x-direction and y-direction.
At the equilibrium state, the net magnetization M0 is same as the z component
magnetization Mz, while x and y component of the magnetization is zero. The linear
condition of the net magnetization with B0 can be changed by a sequence at Larmor
Frequency. The net magnetization can only reach its equilibrium condition after time
and the time constant can explain how it reaches its equilibrium state is called the spin-
lattice relaxation time or longitudinal constant which is denoted as T1time constant. The
net magnetization comes to its equilibrium state can be express by the following
equation,
14
𝑀𝑀𝑧𝑧 = 𝑀𝑀0 �1− 𝑒𝑒−𝑡𝑡𝑇𝑇1�. (2.6)
Figure 2.2 shows that the net magnetization Mz as a function of time t when M0= 1 and
T1 = 256. Due to natural phenomenon, spin tends to come low energy state, when the
energy of the magnetization comes to its equilibrium state, the energy of the spin goes
to the lowenergy state. The energy of the spin is deteriorated as a transfer of heat
through the electromagnetic interaction, collisions and rotations.
Figure 2.2: Net magnetization as a function of time
2.2.1.2 Spin-spin Relaxation When a radiofrequency (RF) is applied to the magnetization field, it is switch into the
transverse plane which is perpendicular to the z-direction. The applied RF pulse should
be equal to Larmor frequency due to modifying the net magnetization. The modified
magnetic field due to RF pulse known as B1 magnetic field. After applying RF pulse,
the flip angle α will be
𝛼𝛼 = 𝑓𝑓𝑡𝑡𝑝𝑝 = 𝛾𝛾𝐵𝐵1𝑡𝑡𝑝𝑝. (2.7)
15
Wheretp is the time duration, f is the RF frequency,and B1 is the modified magnetic
field. The flip angle can be varied from 0 to 1800. However, when the flip angle is 1800,
the net magnetization is the same as B0 but opposite direction. At any other angle except
1800, the rotating magnetization appears around z-axis which is known as precession.
The rotation speed fallows the Larmor frequency. After the external RF pulse cut off,
the net magnetization of the z component will experience T1 relaxation, and on the other
hand, the net magnetization Mxy will start, which expose a different relaxation process.
Suppose after 900RF pulse excitation, the transverse component of the magnetization
can be represented as
𝑑𝑑𝑀𝑀𝑥𝑥𝑥𝑥
𝑑𝑑𝑑𝑑= −𝑀𝑀𝑥𝑥𝑥𝑥
𝑇𝑇2 (2.8)
Since, after a 900excitation 𝑀𝑀𝑥𝑥𝑥𝑥(0) = 𝑀𝑀0, so
𝑀𝑀𝑥𝑥𝑥𝑥 = 𝑀𝑀0𝑒𝑒(−𝑑𝑑/𝑇𝑇2) (2.9)
Figure 2.3: Transverse relaxation Mxy as afunction of time
The characteristics of the decay of transverse magnetization are called Transverse
relaxation or spin-spin relaxation,and the time constant T2 is called spin-spin time
constant. Due to the intermolecular interaction of the spins,the transverse magnetization
Mxy dephases in the transverse plane resulting inthe time constant T2 decay. The T2
decay curve is shown in Figure 2.3. However, transverse relaxation process also
16
depends on the z-component magnetic field oscillation. The longitudinal fluctuation
dominates transverse relaxation process, therefore T2 < T1. Due to the domination of the
z-component fluctuations,a new relaxation is evolved that is called T2*. In the presence
of transverse component oscillation, the z-component israpidlydephased, which is
characterized as T2*decay, which also called free induction decay (FID).Figure2.4 is
indicated in the FID process.
Figure 2.4:FID process in the x-y plan after 900RF pulse
2.2.2 Physical Behaviour of Tissue Particle One of the common element in the human and animal bodyiswater and hydrocarbons,
which are based on the hydrogen (1H) atom. This hydrogen (1H) atom is highly
sensitive to applied magnetic fields. When the gradient magnetic field is applied to the
body, the hydrogen atom of the body tissue gets excited to spin in one direction. This
spin happens because hydrogen Atom has only one proton, and it could be aligned
easily with MRI magnet.
The proton possesses the property of spin, which causes the nucleus to generate an
NMR signal that further catches by the radio frequency (RF) coils to produce k-space
data. Since bones have not had any water; it does not generate any data for the image.
17
Bones leave a blank area in the images; that is why MRI scanners are best for scanning
soft tissues. The Radio Frequency coils detect the NMR signals generated by the
nuclear spin.
2.3 NMR Signal Excitation
2.3.1 Slice Selection and Gradient Selection The core magnetic field B0 of the MRI scanner is considered as homogeneous. When an
RF signal is applied to excite, the net magnetization is switched into the transverse
plane,and the FID will be recorded from the spins in the magnetic field since both the
spins and RF pulse have the Larmor frequency. A specific magnetic field gradient helps
us to select a particular position. The gradient field is a variation of the magnetic field
familiarized locations,and this gradient field is generated by gradient coils in an MRI
scanner.
The MRI scanner has three types of gradient coils,and those gradient coils generate
fluctuating magnetic field so that spins at different location precess at frequencies
unique to their location, allowing us to reconstruct 2D or 3D images. The x, y, and z
directional gradient known as Gx, Gy, and Gz, respectively.
MR images are drawn a slice of the object with a specific thickness. A one dimensional
(1D) selected magnetic gradient pulse is applied along z-direction to accomplish slice
selection during the period of the RF signal. During processing, the applied exciting
magnetic field B1 with a linear gradient field Gz tips the net magnetization B0 into the
transverse plane in a specified period. Spins of the net magnetization field outside the
slice selection gradient are not influenced. After thatacquired signal from the selected
slice and both the RF pulse and slice selection gradient pulse are turned off, and the
main magnetic field B0back to the initial homogeneous state. Figure 2.5 shows a
schematic diagram of the slice selection process.
18
Figure 2.5: The process of the slice selection by the gradient pulse
2.3.2 Frequency and Phase Encoding During the image data acquisition period, the slice selection gradient along the z-
direction is used to select a specific slice with thickness.The frequency encoding
gradient and the phase encoding gradient are used to define the x and y directional value
respectively to select a specific location in the transverse plane. The midpoint of the
transverse plane, the magnetic field is homogeneousB0 and frequency is Larmor
frequencyf which point is called isocenter. Anx-directionallinear magnetic gradient is
applied to fluctuate linearly as a function of position. The magnetic field along this
direction can be represented as
𝐵𝐵(𝑥𝑥) = 𝐵𝐵0 + 𝑥𝑥𝐺𝐺𝑥𝑥 . (2.10)
And the corresponding Larmor frequency is
𝑓𝑓(𝑥𝑥) = 𝛾𝛾𝐵𝐵(𝑥𝑥) = 𝛾𝛾𝐵𝐵0 + 𝛾𝛾𝑥𝑥𝐺𝐺𝑥𝑥 = 𝑓𝑓 + 𝛾𝛾𝑥𝑥𝐺𝐺𝑥𝑥 (2.11)
The equation (2.11) shows that the Larmor frequency f also varies linearly as a function
of gradient position. Therefore, the encoded NMR signal has a different frequency at a
19
different location. We can determine the x-directional location by measuring those
frequencies of the acquired MRI signal.This is known as frequency encoding technique.
For the 2D image, frequency encoding can determine the x-directional data position that
is called frequency encoding direction. By the same way, to determine y-directional
position value, phase encoding technique is used. A one-dimensional linear magnetic
field gradient along the phase encoding direction is used to accomplish phase encoding.
The equation of the magnetic field along the y-direction represent as
B(y) = 𝐵𝐵0 + yG𝑥𝑥. (2.12)
And the corresponding Larmor frequency is
𝑓𝑓(𝑦𝑦) = 𝛾𝛾𝐵𝐵(𝑦𝑦) = 𝛾𝛾𝐵𝐵0 + 𝛾𝛾𝑥𝑥𝐺𝐺𝑥𝑥 = 𝑓𝑓 + 𝛾𝛾𝑦𝑦𝐺𝐺𝑥𝑥 (2.13)
Therefore, we can define a three-dimensional gradient where slice excitation is one of
the first dimension,and frequency encoding provides the second dimension. The third
dimension is provided by phase encoding, which is done by applying a gradient in the
y-direction parallel to both slice selection and frequency encoding. From equation
(2.13), we conclude that the phase can be different for the same frequency, and the
phase encoding direction can be identified by measuring the phase.
2.3.3 Image Formulation The time-varying magnetization due to free relaxation is called the free induction decay.
The process of relaxation needs to transform into voltage or current in order to be
detected and further processed.The transformation of FID into voltage signal can be
done by using Faraday’s law of electromagnetic induction,the equation is given by
𝑉𝑉(𝑡𝑡) = 𝜕𝜕𝜕𝜕𝑥𝑥𝛷𝛷(𝑡𝑡) = 𝜕𝜕
𝜕𝜕𝑥𝑥 ∫ 𝐶𝐶(𝑟𝑟).𝑀𝑀��⃗ (𝑟𝑟, 𝑡𝑡)𝑑𝑑𝑟𝑟,𝑉𝑉𝑉𝑉𝑉𝑉 (2.14)
20
where Φ(𝑡𝑡) denoted as the magnetic flux of the RF coil, VOI represent as volume of
interests of the subject being imaged, and 𝐶𝐶(𝑟𝑟) denoted as the reciprocity of the coil
calculated by the Biot–Savart Law
𝐶𝐶(𝑟𝑟) = 𝜇𝜇4𝜋𝜋 ∮
𝑑𝑑𝑑𝑑×𝑅𝑅�⃗
𝑅𝑅3, (2.15)
where 𝜇𝜇 represent as the material permeability, 𝑙𝑙 denoted as the path of the coil, 𝑅𝑅�⃗
represent the vector between 𝑑𝑑𝑙𝑙 and the spatial position𝑟𝑟. Equation (2.15) represents the
spatial sensitivity of the receive coil. Visually assuming, the applied transverse RF field
𝐵𝐵�⃗ 𝑟𝑟𝑟𝑟 is uniformly applied to sample, the image will show brighter where the coil’s
reciprocity is strong and vice versa. The coil’s reciprocity 𝐶𝐶(𝑟𝑟)value in single receive
coil scanners is uniform compare to multireceiver coil scanner. In multireceiver coil
scanners, a set of 𝐶𝐶(𝑟𝑟)s is used, each coil value can be designed to have a different and
localized sensitivity region for the purpose of acceleration imaging.
The Eq. (2.14) can be written for the multiple receiver coils as
𝑉𝑉(𝑡𝑡) = ∫ �𝐶𝐶𝑥𝑥(𝑟𝑟) 𝜕𝜕𝑀𝑀𝑥𝑥(𝑟𝑟,𝑑𝑑)𝜕𝜕𝑑𝑑
+ 𝐶𝐶𝑥𝑥(𝑟𝑟) 𝜕𝜕𝑀𝑀𝑥𝑥(𝑟𝑟,𝑑𝑑)𝜕𝜕𝑑𝑑
+ 𝐶𝐶𝑧𝑧(𝑟𝑟) 𝜕𝜕𝑀𝑀𝑧𝑧(𝑟𝑟,𝑑𝑑)𝜕𝜕𝑑𝑑
� 𝑑𝑑𝑟𝑟.𝑉𝑉𝑉𝑉𝑉𝑉 (2.16)
Here, 𝜕𝜕𝑀𝑀𝑧𝑧(𝑟𝑟,𝑑𝑑)𝜕𝜕𝑑𝑑
is considered as negligible component compare to other stronger
transverse components, so the MR signal will be
𝑠𝑠(𝑡𝑡) = ∫ �𝐶𝐶𝑥𝑥(𝑟𝑟) 𝜕𝜕𝑀𝑀𝑥𝑥(𝑟𝑟,𝑑𝑑)𝜕𝜕𝑑𝑑
+ 𝐶𝐶𝑥𝑥(𝑟𝑟) 𝜕𝜕𝑀𝑀𝑥𝑥(𝑟𝑟,𝑑𝑑)𝜕𝜕𝑑𝑑
� 𝑑𝑑𝑟𝑟.𝑉𝑉𝑉𝑉𝑉𝑉 (2.17)
Finally, we can represent MR signal in the following,
𝑠𝑠(𝑡𝑡) = 𝜔𝜔0 ∫ 𝑒𝑒−𝑡𝑡
𝑇𝑇2(𝑟𝑟��⃗ ) 𝐶𝐶𝑥𝑥,𝑥𝑥(𝑟𝑟)𝑀𝑀0(𝑟𝑟, 0) 𝑒𝑒−𝑗𝑗(∆𝜔𝜔(𝑟𝑟)𝑑𝑑+∅(𝑟𝑟,𝑑𝑑)−𝜗𝜗(𝑟𝑟) 𝑑𝑑𝑟𝑟,𝑉𝑉𝑉𝑉𝑉𝑉 (2.18)
Where 𝑠𝑠(𝑡𝑡) is the MR signal, ∆𝜔𝜔(𝑟𝑟) is denoted as demodulated frequency, ∅(𝑟𝑟, 𝑡𝑡) is
the accumulated phase of the procession up to time t and 𝜗𝜗(𝑟𝑟) is represented as the
21
receive field directional angle with x-axis. However, the global signal 𝑠𝑠(𝑡𝑡) cannot be
used to produce image signal from the spatial distribution of hydrogen nuclei. It needs
to establish a new technique called spatial encoding to obtain the image successfully.
2.3.4 The K-space Signal The MRI encoding system generates and acquires data in k-space by applying varying
magnetic field gradients, and this system is called spatial encoding of MRI. The
spatially encoded MRI signal are used to reconstruct the digital image through spatial
algorithm process. To understand the whole MRI encoding idea, the mathematical
explanation of k-space is first introduced.
We can represent the Equation (2.18) to obtain the MR signal in terms of the transform
of spatially weighted spin density function
𝑠𝑠(𝑡𝑡) = ∫ 𝐶𝐶 (𝑟𝑟) 𝑃𝑃(𝑟𝑟) 𝑒𝑒−𝑗𝑗(∆𝜔𝜔(𝑟𝑟)𝑑𝑑+∅(𝑟𝑟,𝑑𝑑)) 𝑑𝑑𝑟𝑟,𝑉𝑉𝑉𝑉𝑉𝑉 (2.19)
Where all the constants and the relaxation terms are dropped and 𝐶𝐶 (𝑟𝑟)is represent in
place of 𝐶𝐶𝑥𝑥,𝑥𝑥(𝑟𝑟) and 𝑒𝑒−𝑗𝑗𝜗𝜗(𝑟𝑟) is absorbed into 𝑃𝑃(𝑟𝑟) which is called
𝑃𝑃(𝑟𝑟) ≔ 𝑃𝑃0(𝑟𝑟)𝛾𝛾2ℏ2𝐵𝐵04𝑘𝑘𝑇𝑇
𝑒𝑒−𝑗𝑗𝜗𝜗(𝑟𝑟). (2.20)
Further, Introducing ∅𝐺𝐺(𝑟𝑟, 𝑡𝑡) = ∆𝜔𝜔(𝑟𝑟)𝑡𝑡 + ∅(𝑟𝑟, 𝑡𝑡), the Eq. (2.19) can be simplified to
𝑠𝑠(𝑡𝑡) = ∫ 𝐶𝐶 (𝑟𝑟) 𝑃𝑃(𝑟𝑟) 𝑒𝑒−𝑗𝑗∅𝐺𝐺(𝑟𝑟,𝑑𝑑)) 𝑑𝑑𝑟𝑟,𝑉𝑉𝑉𝑉𝑉𝑉 (2.21)
Now, ∅𝐺𝐺(𝑟𝑟, 𝑡𝑡) can be replaced by an encoding field gradient �⃗�𝐺 (𝑡𝑡) to give
𝑠𝑠(𝑡𝑡) = ∫ 𝐶𝐶 (𝑟𝑟) 𝑃𝑃(𝑟𝑟) 𝑒𝑒𝑗𝑗 𝛾𝛾 𝑟𝑟 ���⃗ ∫ 𝑑𝑑�́�𝑑𝑡𝑡0 �⃗�𝐺(�́�𝑑) 𝑑𝑑𝑟𝑟,𝑉𝑉𝑉𝑉𝑉𝑉 (2.22)
Now, the k-space can be represented as follows
22
𝑘𝑘�⃗ = 12𝜋𝜋𝛾𝛾 ∫ �⃗�𝐺(�́�𝑡)𝑑𝑑
0 𝑑𝑑�́�𝑡 (2.23)
Therefore, the encoded k-space signal can be represented as
𝑠𝑠�𝑘𝑘�⃗ � = ∫ 𝐶𝐶 (𝑟𝑟) 𝑃𝑃(𝑟𝑟) 𝑒𝑒𝑗𝑗 2𝜋𝜋 𝑘𝑘�⃗ 𝑟𝑟 ���⃗ 𝑑𝑑𝑟𝑟,𝑉𝑉𝑉𝑉𝑉𝑉 (2.24)
2.3.5 2D Spatial Encoding
The 2D spatial encoding method is the process of encoding to generate the k-space
signal𝑠𝑠�𝑘𝑘�⃗ �. As observed from Eq. (2.23), the full k-space signal can be produced by
changing the encoding gradient or the time duration. In 2D slice selective encoding, az-
directional gradient set as a readout gradient to produce
𝑃𝑃(𝑥𝑥, 𝑦𝑦) = ∫ 𝑃𝑃(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) 𝑑𝑑𝑧𝑧𝑎𝑎2− 𝑎𝑎2
(2.25)
Where 𝑎𝑎 is denoted as slice selection thickness. 2D encoding techniques can be applied
after a slice is selected. Here, The Fourier encoding technique is explained to
demonstrate the principle of spatial 2D encoding.
𝑘𝑘𝑥𝑥 = 𝛾𝛾𝐺𝐺𝑥𝑥(𝑡𝑡 − 𝑇𝑇𝑒𝑒) (2.26)
𝑘𝑘𝑥𝑥 = 𝛾𝛾𝛾𝛾∆𝐺𝐺𝑥𝑥𝑇𝑇𝑝𝑝𝑒𝑒 (2.27)
Where 𝐺𝐺𝑥𝑥 is denoted as frequency gradient, 𝑇𝑇𝑒𝑒 is frequency encoding time, ∆𝐺𝐺𝑥𝑥 is thei-
th phase encoding cycle and 𝑇𝑇𝑝𝑝𝑒𝑒 represent the phase encoding time. Hence, x-direction
is used as frequency encoding,whereas y-direction performs as phase encoding. Thus,
the frequency and phase-encodedk-space can be illustrated as
23
𝑠𝑠�𝑘𝑘𝑥𝑥 ,𝑘𝑘𝑥𝑥� = ∬ 𝐶𝐶 (𝑥𝑥, 𝑦𝑦) 𝑃𝑃(𝑥𝑥, 𝑦𝑦) 𝑒𝑒𝑗𝑗 2𝜋𝜋 (𝑘𝑘𝑥𝑥𝑥𝑥+𝑘𝑘𝑥𝑥𝑥𝑥) 𝑑𝑑𝑥𝑥 𝑑𝑑𝑦𝑦,𝐹𝐹𝑉𝑉𝑉𝑉 (2.28)
Where FOV denoted as the 2D field of view of the receiver coil. In the single-channel
receiver coil, the area of FOV has to be equal to the region of interest (ROI). However,
when multiple channels are used for imaging, the encoded k-space equation will be
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ 𝑠𝑠0�𝑘𝑘𝑥𝑥 ,𝑘𝑘𝑥𝑥� = ∬ 𝐶𝐶0 (𝑥𝑥, 𝑦𝑦) 𝑃𝑃(𝑥𝑥, 𝑦𝑦) 𝑒𝑒𝑗𝑗 2𝜋𝜋 �𝑘𝑘𝑥𝑥𝑥𝑥+𝑘𝑘𝑥𝑥𝑥𝑥�𝑑𝑑𝑥𝑥 𝑑𝑑𝑦𝑦,
𝐹𝐹𝑉𝑉𝑉𝑉
⋮
𝑠𝑠𝑑𝑑�𝑘𝑘𝑥𝑥 ,𝑘𝑘𝑥𝑥� = ∬ 𝐶𝐶𝑑𝑑 (𝑥𝑥, 𝑦𝑦) 𝑃𝑃(𝑥𝑥, 𝑦𝑦) 𝑒𝑒𝑗𝑗 2𝜋𝜋 �𝑘𝑘𝑥𝑥𝑥𝑥+𝑘𝑘𝑥𝑥𝑥𝑥�𝑑𝑑𝑥𝑥 𝑑𝑑𝑦𝑦,𝐹𝐹𝑉𝑉𝑉𝑉
⋮
𝑠𝑠𝐿𝐿−1�𝑘𝑘𝑥𝑥 ,𝑘𝑘𝑥𝑥� = ∬ 𝐶𝐶𝐿𝐿−1 (𝑥𝑥, 𝑦𝑦) 𝑃𝑃(𝑥𝑥,𝑦𝑦) 𝑒𝑒𝑗𝑗 2𝜋𝜋 (𝑘𝑘𝑥𝑥𝑥𝑥+𝑘𝑘𝑥𝑥𝑥𝑥) 𝑑𝑑𝑥𝑥 𝑑𝑑𝑦𝑦,𝐹𝐹𝑉𝑉𝑉𝑉
(2.29)
Where L is the number of receiver channel, 𝑠𝑠𝑑𝑑�𝑘𝑘𝑥𝑥 ,𝑘𝑘𝑥𝑥� is the received signal at the l-th
channel in the array, and 𝐶𝐶𝑑𝑑 (𝑥𝑥, 𝑦𝑦) is the sensitivity function of the l-th channel and this
value are not same for all channel. This multiple coils system are used in parallel MRI
to accelerate image processing.
2.3.6 Field of View (FOV) and Spatial Resolution
The resolution and the FOV of the final image are determined by the highest spatial
frequency sampled (kmax) and sample rate (Δk)respectively,which shown in Figure
2.6.The sampling rate in the frequency encodes direction (x) is determined by the ADC
receiver, where the aliasing problem rarely exists. But, the sampling rate in the phase
encoding direction (y) is determined by the magnitude of the k-space shift executed by
the phase encoding gradient. The phase encoding process takes a long time to acquire
phase directional data. To minimize the number of steps, a fixed extent of k-space are
used to reduce the scan time. There is two way to reduce the scanning time; we have to
take as large step as possible or minimize the FOV.
24
Figure 2.6: The image resolution and FOV
However, if the FOV has no ability to contain the whole image, it is represented as an
aliasing image. And using different reconstruction technique, the full-size image can be
reconstructed. New decade are emphasising on the area to accelerate the MRI
processing.
The appropriate selection of the field of view (FOV) and the spatial resolution is very
important in the MRI data acquisition. The FOV should be sufficiently larger than the
dedicated size of the image to remove the aliasing artifacts as well as the voxel size
should be enough small to find the smallest features we are intent to observe.
If we consider the frequency domain spacing is Δk between two samples in k-space,
then the FOV will be FOV = 1/Δk and the voxel size (denoted as Δ) is inversely
proportional to the range of k-space (-kmax to kmax) of the sampled object. That mean ∆ =
1/(2kmax). The physical illustration of those parameters are presented in Figure 2.7. We
need more scan line and length to improve the quality of image because a larger FOV or
smaller voxels in the phase encode direction require more scan lines, and smaller voxels
in the frequency encode direction increases the length of those scan lines.
Also, image quality is affected by the signal-to-noise ratio (SNR), if SNR is too low,
tissue contrast and abnormal regions are troublesome to differentiate from observation
noise. SNR is roughly proportional to acquisition time; therefore, reducing the
25
acquisition time is in the midst of the same reduction in SNR. The degradation in SNR
is going to be a significant concern in reconstructing quality pictures from accelerated
MRI information.
Figure 2.7: The sample spacing ∆kxin k-space and the extent kx,max, relate to the FOV, FOVxand voxel size ∆x, respectively, of the reconstructed image in the x-direction. Similarly,∆ky and ky,maxare connected to FOVyand ∆y.
2.3.7 MR Imaging Pulse Sequence Figure 2.8shows a typical timing diagram of gradient-echo (GRE) MRI pulse sequence
to generatek-space data from objects. This timing diagram design with an RF pulse, a
slice selection gradient signal, a phase gradient pulse, a frequency gradient pulse and, an
output signal. At first slice selection, z-gradient is applied in a specific part of the
objectto create a linear variation of the field along the z-direction in the presence of the
uniform magnetic field B0,and at the same time, RF pulse also applied. And this applied
RF pulse which is usually 0 to 900flips the net magnetization of a particularslice in the
transverse (x-y) plane.
After applying RF pulse, a y-directional gradient pulse is applied for some time to
generate phase variation along the y-direction. In this stage,the frequency encoding
signal is applied which is called a readout signal. As a result, the net magnetization
precesses along x-direction with different frequency. The total procedure is repeated
with different phase gradient pulse each time,and k-space is filled up correspondingly
26
through the analogue to digital converter (ADC). For example, 256 phase encoding line
and 256 sampling points acquired during the readout process for a 256×256 dimensional
image.
Figure 2.8: Typical pulse sequence for MRI data acquisition
2.4 Image Reconstruction Technique of MRI
2.4.1 Reconstruction Complexity MRI is one of the leading age technology in biomedical diagnosis field. It has many
advantages,such as high resolution, radiation-free , flexible slice orientation [5].
However, MRI has somecritical problem; long data acquisition time is one of those.
Suppose, If the length of TRcycle is one second,then a total of 256 seconds or 4 minutes
and 16 seconds require for 256 phase encoding steps. Some application use repeated
phase encoding steps to improve the signal to noise ratio (SNR). Therefore, the total
scanning time will be double, when each phase encoding steps are repeated twice.
Besides, in multi-slice or volume imaging, the multi-sliceacquisition is essential to
acquire k-space data which extend the total scanning time up to one hour.Generally,
27
contemporarytime reducing scanning methods of multi-coil scheme take advantages in
modern MRI process.
The advanced MRI scanner is equipped with phased array coils,and these phased array
coils can collectdata from multiple coils simultaneously.Each phase array coils has an
individual sensitivity profile which is used to emphasizes some part of the proton
density. A full image from all coils can obtain by the sum of square (SOS) of individual
channel image,and the SOS of the image is displayed the proton density with uniform
distribution weights. There are several methods have been proposed to reduce the data
acquisition time such as data sub-sampling of Cartesian trajectory, using non-Cartesian
trajectory, novel pulse sequence, and combinations of those methods.
2.4.2 Cartesian Trajectory The Cartesiantrajectoryis the spatial transform of the magnetization by acquiring a
Cartesian grid of samples. In this technique, the frequency encoding lines are acquired
one by one that means parallel to each other. Also, the distance between the linesis the
same. However, the phase encoding line is perpendicular to the frequency encoding
line. This method is widespread for generating images from the object, and many pulse
sequences techniquesare now present to implement Cartesian trajectory. The critical
part of the Cartesian and other Fourier sampling methods is the use of spatial gradient
fields during the relaxation of excited spins. As we describe above, the spacing and
extent of the Cartesian grid of samples both affect the acquisition time and the field of
view and a voxel size of the resulting image.
In the Cartesian coordinates, the brightness and the main structure of the image are
defined by the centre part of the frequency encoding lines of the k-space, whereas, the
details of the images are described by the peripheral of the k-space of the frequency
encoding line. In a nutshell, the different frequency encoding line in the k-space has a
different importanceto reconstruct the image. The quality of the reconstructed image
will not be profoundly affected due to the dropped out of some frequency encoding
lines.However, the image quality will be significantly changed due to discarding the
centre part of the frequency encoding lines of the k-space.
28
The image reconstruction technique can be implemented easily and fast by applying 2D
Fourier transform. However, apparently, the Cartesian coordinates of the MRI data
acquisition techniquesuffer from long scan time for multiple TR and the phase encoding
steps. To increase the data acquisition speed, some frequency encoding steps are
dropped out deliberately.
2.4.3 Non-Cartesian Trajectory
The systems that assemble data in the Cartesian coordinates are Multi-slice imaging,
The Gradient Echo Sequenceand Volume Imaging. The frequency encoding lines are
parallel to each other, and the distances between adjacent frequencies encoding line are
identical. The phase encoding direction is perpendicular to the frequency encoding
direction. Images are reconstructed using the 2D Fourier transform from data acquired
with the trajectory in the Cartesian coordinate, and this is fast and not very hard to
implement.
However, trajectories within the Cartesian coordinates experience the lengthy scan time
equal to the multiplication of the repetition time (TR) and the total phase encoding steps.
The frequencyencoding lines at the middle of the Fourier domain decides the main
structure and brightness of the image, while the frequency encoding lines at the
periphery of the Fourier domain describes the facts of the image. In other words, the
importance of each frequency encoding lines is different in the k-space. Therefore, the
significance of varying frequency encoding lines across the Fourier transform is not
identical.
For example, certain frequency encoding lines are discarded at the periphery of the
Fourier transform, the resolution of the image will not be heavily influenced. However,
if some frequency encoding lines are removed at the centre part of k-space, the quality
of the image will be severely affected. Figure 2.9 below indicates a radial trajectory of a
signal structure in the non-Cartesian coordinates.
29
Even though the time used for the non-Cartesian trajectories is not very high, the signal
reconstruction procedure is not direct. Supplementary stages are essential toreconstruct
images from acquired data using trajectories in non-Cartesian coordinates. Therefore,
the use of trajectories in non-Cartesian coordinates is constrained to certain processes. A
large chunk of scans is accomplished using trajectories in the Cartesian coordinates [42-
45]
Figure 2.9: the trajectory pulse sequence in the non-Cartesian coordinates
2.4.4 Noise and System Uncertainty in Image Reconstruction Noise in reconstructed image in MRI has been analysed in different works [21-23]. The
noise in the acquired data sample is generally exhibited as white additive noise with
variance
𝓋𝓋2 ∝ 4𝑘𝑘𝑇𝑇 × 𝑅𝑅 × 𝐵𝐵𝐵𝐵 (2.30)
Where 𝓋𝓋 is denoted as noise, R is the resistance loaded into the coil, coil resistance and
other electronic resistance. The BW is the bandwidth of the given imaging scheme
directed by the imaging parameters.
The discrepancy between mathematical estimated model and reality are described as
imaging scheme uncertainty. It is denoted as∆𝒯𝒯. For example, in the conventional two
dimensional Fourier encoding have many possible imperfect imaging conditions that
30
can deviate the real encoding process 𝒯𝒯 away from the perfect Fourier basis. There are
some reasons behind this imperfection, such as field inhomogeneity of B0, the motion of
the imaged substance and selective excitation offset. In the model, these issues need to
resolve precisely in order to obtain the perfect reconstruction.
In exercise, these type of effects can reduce significantly by correctly choosing pulse
sequence in the current state of the art single receive coil scanners. Generally, the
inverse discrete Fourier transform (DFT) and Singular Value Decomposition (SVD) can
be used for image reconstruction without significantly affecting image quality.
However, due to the unknown coil sensitivity in pMRI, it needs to be estimated. And,
the estimated error is generally significant compare to a single receive coil scanners.
Therefore, in the applied parallel MRI system, both the noise and the system uncertainty
have to analyse explicitly. The right imaging function is thus given by
𝒯𝒯 = 𝒯𝒯0 + ∆𝒯𝒯, (2.31)
Where 𝒯𝒯0 is the estimated function, and the image reconstruction function is specified
by
𝒥𝒥 = 𝒯𝒯0−1. (2.32)
The encoded data measurement can be written as
𝒮𝒮 = 𝒯𝒯{𝑃𝑃} + 𝓋𝓋
= 𝒯𝒯0{P} + ∆𝒯𝒯{P} + 𝓋𝓋. (2.33)
Therefore, the reconstructed image function is given by
P� = 𝒥𝒥{𝒮𝒮}
= 𝒥𝒥𝒯𝒯0{P} + 𝒥𝒥∆𝒯𝒯{P} + 𝒥𝒥𝓋𝓋.
31
= P + 𝒥𝒥[𝓋𝓋 + ∆𝒯𝒯P], (2.34)
Also, the corresponding reconstruction error is given by
ℰ = 𝒥𝒥[𝓋𝓋 + ∆𝒯𝒯P]. (2.35)
As can be observed from Equation (2.35),the error ℰ is not only dependent on the size
of noise 𝓋𝓋 and uncertainty function ∆𝒯𝒯 but also the gain of the reconstruction system𝒥𝒥.
The errorℰ can be significant if the gain of reconstruction system 𝒥𝒥 is substantial.
However, in the single receiver case, the error ℰ is not an important issue because the
hardware setup for both 𝓋𝓋 and ∆𝒯𝒯 are minimal, and the gain of the reconstruction
system is 1. Hence, there is no amplification.
2.5 Image Reconstruction from Subsampled Data in
Cartesian Coordinates
There is about 128 or 256 phase encoding line normally utilized to form an image in the
Cartesian coordinates. In order to decrease the time for acquiring the image signal, some
frequency encodinglines are intentionally escaped. The factor by which the number of
frequency encodinglines is reduced is referred to as the reduction factor R, which is
defined by dividingthe number of total frequency encoding lines by the number of
actual frequencyencoding lines in the actual scan.Reconstructedimages will have
artifacts by directly applying two-dimensional inverse Fourier transform to the under-
sampled k-space data. Specifically designed image reconstruction procedures are
required to reconstruct images with satisfactoryquality from under sampled k-space
data. SENSE [19], GRAPPA [20], CS-based reconstruction[35] and their some
extensions are the main focus for image reconstruction methods in MRI.
2.5.1 SENSE SENSitivity Encoding is a concept of reconstructingfull FOV images from sub-sampled
k-space data of multiple coils by using the sensitivity profile of coils[19]. If the k-space
32
signal is under-sampled, the reconstructed signal has aliasing. For example, the images
are shown in Figure 2.10 (a) is initially transformed to k-space, and the k-space data are
under-sampled by assembling one for every two-frequency encoding lines. The image
are reconstructed by using 2-Dimensional inverse Fourier transform is depicted in
Figure 2.10 (b). The two bright pixels in the original image are vertically
symmetrical. The reconstructed image in (b) portrays an image that shows that these
two pixels have overlapto the point that they can no longer be differentiated. Actually,
the top and bottom halves are combined to form a 1/2 FOV image.
(a) The original image (b) The reconstructed image
Figure 2.10: Comparison between the original and the subsampling images
SENSE has two steps to unfold the reconstructed image with aliasing. In the first step of
this technique, the aliased image of each coil is reconstructed by applying 2-
Dimensional inverse Fourier transform to the under-sample k-space data. The second
step involves the reconstruction of the whole FOV image from this intermediate aliasing
images.
The SENSEprocess is demonstrated in the following example. Assume that the two
bright pixels in the figure above are represented by x(1) and x(2). Let’s also assume that
the scanning process is done using two coils for which their sensitivities in these bright
regions are r1(1), r1(2) and r2(1), r2(2) corresponding to the first and second coil
respectively. Also, suppose that 𝑥𝑥�1and 𝑥𝑥� 2represents the overlapped pixel in the two
33
intermediate aliased images. The process of aliasing is illustrated using the equation
below:
r1(1)x(1)+ r1(2)x(2)= 𝑥𝑥�1, (2.36)
r2(1)x(1)+ r2(2)x(2) = 𝑥𝑥�2. (2.37)
In the matrix format, the above expressions are represented as
�𝑟𝑟1(1) 𝑟𝑟1(2)𝑟𝑟2(1) 𝑟𝑟2(2)� �
𝑥𝑥(1)𝑥𝑥(2)� = �𝑥𝑥�1
𝑥𝑥�2� (2.38)
In the above matrix expression, two linear expressions and two unknown variables
exist. This issue can be easily solved by multiplying the inverse matrix of the first
matrix expression on the LHS to both sides of the equation. The number of the coils, Nc,
must not be lower than R (Reduction factor). IfNc ≥ R, there are an equal number of
equations or more exists as the unknown variable. Using a pseudo-inverse methodology,
the unknown variables can be obtained. However, ifNc< R, the image reconstruction
issue may not be certain. To solve the equation, the pseudo-inverse-matrix or inverse
matrix is required to be present.
The precondition of the SENSE technique for successfulimage reconstruction process is
the precise sensitivity profile of coils. Assume that the sensitivity profiles for the coils
are not known, the process inversion cannot be carried out. A prescan can be exploited
to establish the sensitivity profiles of coils such that signals with a lower FOV are
obtained[19]. But then, in the course of the real scan, sensitivity profiles could be
dissimilar from the prescan to some extent due to many factors, such as the body
movements of the patients. Hence, the author suggestscreating a sensitivity profile
during the actual scan, by adopting variable density sub-sampling pattern, which means
collecting more frequency encoding lines the low frequency part of k-space, instead of
sub-sampling them the same way as in the high frequency part[46]. Sincethe calibration
data are acquired simultaneously with the data to be reconstructed,errors due to
sensitivity miscalibration are eliminated.
34
Although these two methods differ in the time when the scanning for
sensitivityinformation is conducted, both share a similar sensitivity estimation
procedure. First,a low resolution image for each coil is reconstructed from k-space data
obtained bythe prescan or the low frequency part. Then the reference image is obtained
by theSOS of the sensitivity encoded images from all coils. The sensitivity profile of
eachcoil is calculated by dividing the sensitivity encoded image by the reference
image.An interpolation step is finally adopted to increase the size of the sensitivity
profiles.
One of the successful application of SENSE is real-time imaging of cardiac
diagnosis[47] and also, some other clinical practice[48, 49]. However, the sensitivity
profile is difficult to estimate in some application to fulfil the requirement of accurate
sensitivity.
2.5.2 Extensions of SENSE A lot of extensions have been utilized with the conceptualization of the SENSE
technique. Some of these extensions are introduced in this thesis.
2.5.2.1 2-Dimensional SENSE In [50], the authors propose to extend SENSing Encoding to 2-Dimensional under-
sampling, called 2D SENSE, which is an under-sampling technique used in 2-
Dimensions.Since there are two phase encoding directions in volume imaging, under-
sampling can be done in either direction. Hence, aliasing happens in both directions.
There are two successive steps in 2-Dimensional SENSE. And each of these steps
unfolds unidirectional aliasing[50].
2.5.2.2 Auto-SENSE Auto-SENSE was proposed in [51] and has been applied in cardiac imaging, which is
known as dynamic MRI. Sequential frames for dynamic MRI are acquired in a time
35
series.A frequency encoding line is shifted when under-sampling k-space in adjacent
even or odd frames. Just like noted before, a higher frequency region of the k-space
decides the details of the image while the lower frequency region of the k-space is the
deciding factor of the main shape of the image. In dynamic MRI, it is understood that
the modifications between two neighbouring frames are significantly low.
Consequently, the lower frequency regions of k-space from neighbouring frames are
thought to be the same or very similar, while only the high-frequency region of Fourier
transform data alter seriously[51]. After estimated sensitivity profile of the coil, the
SENSE reconstruction technique is applied in both the odd and even frame to generate
images. Auto-SENSE can save data acquisition time for avoiding prescan, and
sensitivityprofiles can be dynamically updated, which leads to fewer artifacts and less
noise.
2.5.2.3 Tikhonov Regularization This is an image reconstruction algorithm that effectively decreases the loss of Signal-
to-Noise-Ratio (SNR) as a result of certain geometric connections in the spatial
information from the coil elements. In this algorithm, a reference image is utilized as
preceding information about the reconstruction image in order to provide regularized
approximates for image reconstruction. Pre-scanning is a method of getting this
reference image. It is frequently understood that this technique can decrease g-factors
and raise image resolutions in SENSE image reconstruction. In a characteristic SENSE
method, precise sensitivity profiles should be supplied before the image reconstruction
occurs[52].
2.5.3 GRAPPA GeneRalized Autocalibrating Partially Parallel Acquisitions (GRAPPA) is a widely
used image reconstructionmethods from the under-sampled k-space data [20]. Contrary
to SENSing Encoding, GRAPPA does not need precise data on sensitivity profiles. As a
substitute, GRAPPA needs that the lower frequency region of the k-space is fully
sampled. An additional milestone that differentiates the SENSE image reconstruction
36
scheme and the GRAPPA method is that SENSE reconstruction is majorly completed in
the image domain while GRAPPA is majorly finished in the Fourier domain.
The process of reconstruction in GRAPPA is demonstrated below. Phase encoding
direction is shown by the horizontal region while the various coils are shown in the
vertical region. Perpendicular to the plane is the frequency encoding direction. Those
circles with holes in them denote the frequency encoding lines that were unsampled.
Those circles with a totally black solid grey or black colour represent sampled
frequency encoding lines. The solid black circle follows the same sampling module
used in SENSE meaning that an R line is obtained. Those circles that are grey represent
the additional calibration signal lines (ACS) which are deliberately obtained in
GRAPPA.
Figure 2.11: Schematic diagram of GRAPPA reconstruction technique
On a broad basis, GRAPPA image reconstructionmethod has two vital steps. The
GRAPPA weights are defined by an interpolation procedure in the very first step. Just
as showed in the diagram above, the solid black circles obtained from all coils in the
neighbouring phase encoding steps are utilized to interpolate the grey circles in the ACS
line of coil 4.The method of interpolation is represented by
∑ ∑ 𝑠𝑠𝑖𝑖4𝑗𝑗=𝑑𝑑
4𝑖𝑖=𝑑𝑑 (𝑗𝑗)𝑤𝑤𝑖𝑖(𝑗𝑗) = 𝑠𝑠4,𝐴𝐴𝐴𝐴𝐴𝐴, (2.39)
37
i and j respectively denote the coil and phase encoding steps. si(j) represents the points
in the i of j domain obtained black data in the coil and 𝑠𝑠4,𝐴𝐴𝐴𝐴𝐴𝐴 ,stands for the grey data in
the ACS line of coil 4 and 𝑤𝑤i(j)stands for the weights of GRAPPA.
Each of the solid black circle data points is assigned a weight to stabilize the right- and
left-hand part of the equation of interpolation. For the scenario, a total of about 16
weights are to be determined, but just one equation is needed. There are about 15 free
variables that mean the equation has endless solutions. The course of interpolation is
iterated for all data points in the frequency encoding direction. Suppose there are 256
data points in the frequency encoding direction, meaning that there are 256 equations
but only 16 unknown variables, which makes the problem of finding GRAPPA weights
over-determined. In GRAPPA, it is assumed that the GRAPPA weights across the k-
space are the same. In addition to moving along the frequency encoding direction, the
same interpolation process is repeated along the phase encoding direction in the fully-
sampled low-frequency part of k-space. Since there are more equations than unknown
variables, the GRAPPA weights canbe obtained by a pseudo-inverse step, as in SENSE.
With the same procedure, theGRAPPA weights for interpolating the gray data points in
the other 3 coils areobtained.
After when the interpolation step has been finished, the points where the data are
missing are rebuilt. Those points are kept on the right part of the equation (2.17) in
place of the ACS lines, which contains the grey data points. Because the weights and
black data points are recognized, the points where the data are missing are simply
computed by using simple and straight forward mathematical process. Finally, 2-
Dimensionalinverse Fourier transform is applied to transform the reconstructed image
using the data in the k-space. The greatest benefit of GRAPPA in contrast to SENSE is
that GRAPPA does not require the coil’s precise sensitivity profiles. Rather, it uses a
variable density under-sampling module that acquires the ACS lines in the lower
frequency region of k-space. It is advised that in circumstances where it is possible to
approximate the sensitivity profiles, then it is also advised first to consider SENSE.
2.5.4 Extensions of GRAPPA
38
From the time when GRAPPA was introduced, a lot of other extensions for
reconstruction images have also been introduced to make the ease of image
reconstruction a tangible reality. This thesis also strives to point out those extensions.
TGRAPPA[53] is an example of such an extension. This extension was introduced for
the purpose of applying GRAPPA in dynamic Magnetic Resonance Imaging. K-space
data are obtained at different times in dynamic MRI, which is normally being obtained
in ahigher dimensional k-t space. There are no extra ACS lines obtained in TGRAPPA.
The combination of the obtained frequency encoding lines in the lower frequency region
of the k-space in every R frames is one way of obtaining the ACS lines. Upon the
scanning of a new frame, the oldest frame is discarded. Hence, as time advances, the
ACS lines are upgraded. When the ACS lines are upgraded, GRAPPA weights can be
estimated and applied in every frame. The reconstruction principle of TGRAPPA is
demonstrated in the diagram below (Figure 2.12).
Figure 2.12: The image reconstruction method of TGRAPPA
The extension, k-t GRAPPA is introduced in a journal [54]. In contrast to the
TGRAPPA technique, k-t GRAPPA obtains ACS lines in each frame. This method
utilizes data from neighbouring time frames, in conjunction with data from
neighbouring blocks, for the interpolation of GRAPPA weights. The image
reconstruction technique of k-t GRAPPA is demonstrated in Figure 2.13.
39
Figure 2.13:The image rebuilding method of k-t GRAPPA
The extensions k-t GRAPPA and TGRAPPA may be applied in either multiple or one
coil because since they also explore information in time-space. For the GRAPPA
extension, data obtained in neighbouring phase encoding steps in all coils are employed
to interpolate one data point in the ACS line of a coil. In a journal [55], they suggest not
only obtained data in neighbouring phase encoding steps of every coil are to be utilized,
but they also suggest that data obtained in the same frequency encoding lines are also to
be utilized, because close to the ACS line is the data point. By bringing in additional
data points during the course of interpolation, GRAPPA weights become increasingly
precise. The weights in the GRAPPA extension are understood to be uniform at every
region. When the weights are estimated, they are used across the whole k-space. In a
journal [56]titled “[55],” Magnetic Resonance in Medicine, the writers assert that for
higher and lower frequency regions of k-space, the differences should be reflected upon,
meaning that a plethora of varying GRAPPA weights is to be estimated and used in
these areas.
In order to get a better image reconstruction outcome, there should be a segmentation of
ACS lines along the frequency encoding direction. The GRAPPA weights in different
regions are to be computed individually. Furthermore, the Fourier transform data are be
sampled out of the ACS area, using varying factors of reduction in accordance with
locations.
According to the journal [57]titled “2D-GRAPPA-Operator for faster 3D parallel
MRI,” Magnetic Resonance in Medicine”, authored by M. Blaimer et al and published
in 2006, GRAPPA is used in 3-Dimensional Fourier transform data, and called 2D
40
GRAPPA Operator or 2-Dimensional GRAPPA, in relation to the way of reconstruction
image from under-sampledFourier transform data in two-phase encoding directions. For
3-Dimensional Fourier transform data, asnoted before, under-sampling in two-phase
encoding directions is utilized to advance the gross rate of reduction. Just like the first
GRAPPA extension, 2-Dimensional GRAPPA weights are obtained using the method of
interpolating the data points in ACS lines using the data obtained in every direction.
This procedure is called 2-Dimensional GRAPPA because it is fairlyrelated to the
reconstruction process from under-sampled unidirectional Fourier transform.On the
other hand, the reconstruction process is divided into two individual successive 1-
Dimensional GRAPPA operations. For each of the GRAPPA operation, it reforms the
data missing in a unidirectional fashion. This is known as 2-Dimensional GRAPPA
Operator.
So far, no extension arising from GRAPPA has probed into this issue of noise.
Realistically data obtained by scanning, are usually damaged by noise, leading to
incorrect GRAPPAweights. In order to determine the off points, the authors of the
journal “Robust GRAPPA reconstruction and its evaluation with the perceptual
difference model,” devised a methodology named RobustGRAPPA[58]. Using this
methodology, the above points are multiplied by zero or a small constant so the weights
can then be estimated with higher precision because GRAPPA weights are supposed to
be uniform in the whole k-space, it is feasible to approximate GRAPPA weights from
reconstructed lines that are missing. GRAPPA weights can be calculated with
theconsideration of reconstructed missing data. GRAPPA weights are initially estimated
from a small number of ACS lines, and then the missing frequency encoding lines are
reconstructed[59].
2.5.5 Compressive Sensing based Image Reconstruction
Method In accordance with the current proposition of the compressive sensing theory, a signal
can be reconstructed with an overwhelming probabilityfrom data which are randomly
sampled if the original signal is sparse. Images obtained by the method of MRI are
intrinsicallysparse, such as angiography, or compactable following some types of
41
change, such as brain images. Hence, it is straightforward to use compressive sensing in
the problem of image reconstruction from under-sampled k-space data in Magnetic
Resonance Imaging.
We can reconstruct images by minimizing the ℓ1 norm of the transformed image, subject
to data reliability restraints. In the compressive sensing based image reconstruction
process, the k-space data is under-sampled by a variable density under-sampling
procedure[32]. The equation for the image reconstruction problem can be represented as
𝑚𝑚𝛾𝛾𝑚𝑚||Ψx||1 s.t.||TFx−𝑦𝑦||2< ε (2.40)
x represents the objective image, while Ψ represents the wavelet transform operator. TF
is the partial Fourier transform operator obtainedby sub-sampling the fulldiscrete
Fourier matrix using variable density under-sampling method, ε is that factor that relates
to noise and 𝑦𝑦 is the measured data.In order to adequately address the equation above, it
is paramount first to restrain the problem to the unrestrained form:
𝑚𝑚𝛾𝛾𝑚𝑚λ||Ψx||1 + || TFx−𝑦𝑦||22 (2.41)
where λ is the factor in ascertaining the exchange between data sparsity and data
fidelity. λ is ascertained by working out the equation above with different numbers, and
then selecting λ so that the reliability of the data in Eq. (2.18) is addressed. The authors
propose to solve Eq. (2.19) by using a conjugate gradient descent algorithm with
backtracking line search. The function f(x) is characterized as in the equation above.
The conjugate gradient ∇f (x) is
∇f (x) = λ∇||Ψx||1+ 2𝑇𝑇𝐹𝐹∗(TFx−𝑦𝑦) (2.42)
The ℓ1 norm is described as the sum of absolute values of entries, not smooth for every
x value. Hence, the ℓ1 norm is estimated using a smooth function expressed as |𝑥𝑥| =
�𝑥𝑥 ∗ 𝑥𝑥 + ξ. ξ is termed the positive smoothing parameter. Using this estimation, the
conjugate gradient of the ℓ1 norm can be written as
42
∇||x||1 = 𝑥𝑥�𝑥𝑥∗𝑥𝑥+ξ
(2.43)
Hence, ∇f (x) of function x can be represented as
∇f (x)= λ Ψ∗Ψx�(Ψx)∗(Ψx)+ξ
+ 2𝑇𝑇𝐹𝐹∗(TFx−𝑦𝑦) (2.44)
In [32], it is demonstrated that the enhanced spatial resolution and accelerated
acquisition for multi-slice fast spin echo brain imaging and 3D contrast enhanced
angiography are attained. The conjugate gradient descent algorithm with backtracking
line search is briefly described in Algorithm 3 [32].
2.5.6 Extension of CS-based Image Reconstruction Method A lot of extensions for the Compressive Sensing based image reconstruction scheme
have been reported over the decade. In[35], the authors explain the requirements for
successful CS reconstruction.They describe the natural fit of CS to MRI, and give an
intuitive understanding ofCS reconstruction by describing it as a process of interference
cancellation. Some example and applications of Compressive Sensing in MRI and
certain parameters influencing the use of Compressive Sensing in MRI are introduced
here. Some of the restrictions experienced here are forced by a malfunctioning of the
MR hardware, the characteristics of different types of images, and clinical concerns.
Similar to GRAPPA and SENSE, it is upfront to use Compressive Sensing in dynamic
MRI. Dynamic MRI provides additional sparsity than traditional 2-Dimensional
imaging schemes, by making use of the k-space data in the temporal dimension, if only
that part of FOV gets altered at temporal rates which are high, while other regions stay
immobile or alter gradually. Random under-sampling for Compressive Sensing can be
appreciated by arbitrarily dodging frequency encoding lines in every dynamic
frame[60].
43
On a broader basis, CS-based image reconstructionmethod from under-sampled k-space
data in MRI is computed by efficient algorithms. A primary issue with these algorithms
is the expensive and lengthy computational time[61]. In[62], the use of multicore CPUs
can resolve this computational cost and efficiency by forwarding multichannel data to
multicore CPUs. Also, GPUs can use to accelerate computation[63]. Parallel computing
is a promising method to reduce the reconstruction time of CS based reconstruction
methods when multiple coils are used in MRI.
2.6 Novel Sequence Design
This section introduces some novel sequence designs that are different from the
conventional Fourier transform scheme of data encoding.
44
2.6.1 Bunched Phase Encoding In conventional 2-Dimensional sequences such as GRE, methodically to form an image
of size 256×256, a total of about 256 phase encoding steps are implemented. Also, 256
data points are acquired in the frequency encoding direction. In [63], the authors
propose a high rated data acquisition technique in the frequency encoding direction and
along a zigzag trajectory. Basically, data are acquired in a bunch of phase encoding
steps simultaneously, and hence, it is called BPE (bunched phase encoding)[63]. The
difference between an oscillating frequency encoding line in BPE and a straight
frequency encoding line in GRE is shown in the figure below. It is observed that the
solid black circles in this figure stand for individual data points in its place of frequency
encoding lines.
Figure 2.14: Schematic diagram of gradient and BPE encoding technique
In bunched phase encoding, additional data are acquired in the frequency encoding
direction while TR is unchanged. Hence, phase encoding steps is decreased, leading to
reduce data acquisition time. Even though BPE needs just a coil, it may combine with a
phased array of coils to provide an even higher factor of reduction. A combination of
BPE plus SENSE takes advantage of many coils in the system[64]. In the initial step,
the aliased image is reformed for each coil by BPE reconstruction. Then the aliased free
image is formed by SENSE by making use of the sensitivity profiles of the coils. In
BPE, tailored sequences and quickly varying gradients are needed to obtain bunched
data. These necessities restrict its use in regular MR scanners. It has been suggested by
some authors that the GRAPPA operator gridding (GROG) technique to create a cluster
45
of data, that may be achieved using MR scanners[65]. More details explanation about
BPE are given in chapter 4.
2.6.2 Rotating RF Coil The RF pulse that is required to flip the net magnetization to the transverse surface is
formed by an RF coil, which is generally stationary in the scanner. The authors
recommend a rotating RF coil system in papers [66-68]. The RF coil spins around the
object being scanned for spin excitation and signal acquisition. The rotating RF coil
system may prevent the requirement for multiple channels and complex RF decoupling
of several coils[66-68]. The signal reconstruction from data with a rotating RF coil is
similar to the images reconstruction from data with the conventional RF system.
2.6.3 Wave-CAIPI A high reduction factor is hard to attain in conventionalpMRI. According to the journal
[69], partition encoding gradients and sinusoidal phase encoding occur concurrently in
wave- CAIPI during the readout of frequency encoding lines. Hence, inter-shifts are
formed by modifying the partition encoding and the phase encodingstrategy. By using
this scheme, highly efficient k-space data sampling can be achieved, which extends the
aliasing consistently in all three dimensions. Highly accelerated volume imaging with
low artifacts and lowSNRpenalties is gainedbecause wave-CAIPI completely utilizes
the spatial variation in coil sensitivities.
2.7 Some Limitations of the Present Research The use of undersampled data acquisition has the potential to reduce the total scan time
and cost of MR acquisitions and to reduce the burden of imaging to patients. This can
permit either greater patient throughput, or permit more imaging to be completed per
unit time. However still there are some limitations in the present workas listed below:
46
o Parallel MRI, such as SENSE and GRAPPA, requires large number of coils to
achieve a high reduction factor. Due to inconsistency in coil geometries and
sensitivities, severe residual aliasing artefacts and amplified noise may occur when
high reduction factor is chosen.
o The non-array coil method such as BPE also has some crucial problem when
reduction factor is increased. The signal to noise ratio of BPE may become
considerably reduced as the reduction factor is increased.It has been observed that
asreduction factor R>2, the noise of reconstructed image gradually increases even if
the aliasing artefactis absent.
2.8 Conclusion The MRI scheme is an extensively imaging scheme in the context of recent medical
practice. It is proficient in revealing images of the human body with no invasive
procedure being undergone. This procedure has a challenge that is comparatively
extended data obtaining time as a result of physical restraints. Hence as to decrease the
time for obtaining data, under-sampling is used in the phase encoding direction.
SENSE, GRAPPA, and Compressive Sensing have been productively used to the MRI
image reconstruction from under-sampled k-space data. In traditional MRI with Fourier
encoding, the majority of the energy of the calculated data is situated in the lower
frequency region of k-space.
Hence, variable density under-sampling is usually employed in CS-based
reconstruction, which gets more data from the lower frequency region of k-space and
fewer data from the high-frequency region of k-space. 2-Dimensional image
reconstruction outcomes show that a reformed image from erratically under-sampled
BPE data is of higher resolution than that of variable densely under-sampled Fourier
transform-encoded data. This thesis has addressed the aspects of MRI analysis and CS-
based MRI image reconstruction scheme so as to expand in the perception of the feat of
47
Compressive Sensing-based MRI image reconstruction applying under-sampled data
from Fourier encoding and BPE.
In this chapter, the fundamentals of MRI process has reviewed briefly. Here, we have
explained the MR physics from the generation of bulk magnetization to the observable
MR signal in the time domain. In addition, the spatial domain data is introduced, which
is known as k-space data of MRI. The sampling of k-space and data acquisition
procedure is also discussed where we have mentioned the limitation for image
acceleration. In addition, different kind of system uncertainty and noise related issue in
MRI processing has been discussed here.
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Chapter 3 Fundamentals of Sparse Sampling
Fundamentals of Sparse Sampling
3.1 Introduction
The purposes of this chapter are to identify the limitations in the current research work
of popular compressive sensing MRI schemes. At first, the fundamentals of
compressive sensing will be discussed, then the open problems in the existing literature
are analysed results and identified.
Compressive sensing (or compressed sensing, sparse sampling, or compression
sampling) (CS) is a relatively new methodology that has been used in many research
areas where MR imaging is one of them. One of the major advantages of CS
techniquethat permits the faithful reconstruction of the signal from the data acquired
below the Nyquist sampling rateif the signal domains are sparse also it can accelerate
the speed of data acquisition in MRI.
Magnetic Resonance Imaging (MRI) is one of the most useful imaging modalities that
have been extensively used in medical science for the acquisition of biomedical data
over the years[70]. MRI has capability to analyse the detail structure of a human body
as well as the metabolic processes in the body. So, the application of CSMRI
significantly reduces the time of image reconstruction as well as improved the quality of
images.Compressive sensing has been applied to MRI to accelerate the acquisition
process and has been demonstrated in various MRI procedures. CSMRI has been
applied in clinical settings for capturing the infant’s bio-data, where it has effectively
reduced the overall time needed for data diagnosis [71].
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3.2 Theory of Compressive Sensing
The present popular compression technology Compression Sensing(CS) was first
introduced inthe journal [26]titled “Compressed sensing” breaks the traditional twice–
bandwidth boundary set by the Nyquist-Shannon sampling theorem.Compressed
sensing has the special capacity to reform the data in compressed form and to
reconstruct the actual data from fewer measurements than usually required by the
Nyquist–Shannon sampling theorem.
There are several differences between the latest compression technique CS with the
Nyquist-Shannon Sampling Theorem. Such as, CS mainly emphases on data sensing
and recovering discrete signals with finite length, whereas the Nyquist-Shannon
Sampling Theorem typically considers continuous signals with infinite length. In
addition, CS measures data in the form of inner products between the original signal and
a sensing matrix, while in the Nyquist-Shannon Sampling Theorem the continuous
signal is sampled at equally spaced time points. In CS, the randomness of the sensing
matrix plays a vital role, which will be analyzed in this thesis. Finally, the two
frameworks differ in how the original signal is reconstructed. In CS, signal
reconstruction is achieved by solving nonlinear convex optimization problems, which
may be hardware-demanding and time-consuming. In the Nyquist-Shannon framework,
signal recovery is achieved through sinc interpolation, which is a linear process that
requires little computation and has a simple interpretation. Although the reconstruction
process of the original signal by CS appears to be a problem, it is not a major concern in
CS, since the aim of CS focuses on acquiring less data.
Articles published by Terence Tao, Emmanuel Candes and Justin Romberg on MRI
examined the random sub-sampling of the k-spaces[24-26, 35]. For a compressive
sensing technique to thrive, there have to be at least three significantrequirements:
•Sparse transformation: The intended signal should possess a partitioned
expression in a recognized domain of transformation (that is to say that the signal
should have the ability to undergo compression by transformation encoding)[30].
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•Inconsistency of sub-sampling artifacts: Linear reconstruction of data as a result of sub-sampling the Fourier coefficients should be inconsistent in terms of diffusion transformation field [30].
•Non-linear image reconstruction method: The signal must be reconstructed
using a non-linear approach which applies to a lower density of image expression
plus the coherence of the recreation of the acquired signals. The initial
circumstance is vividly fulfilled using MRI data just as clarified above. The
awareness of the importance of inconsistency, the awareness that the data
obtained from MRI can be designed in order to carry out inconsistent sub-
sampling and the fact which implies the existence of sophisticated and realistic
algorithms for data reconstruction may not be made available in this thesis.
Hence, we are turning to a very simple example. Compressed sensing does not
need a preceding idea of the signal or even any suppositions to take into account
to initiate the reconstruction process. A simple set of measures, from a technical
point of view, is all that is needed to begin the reconstruction process.
The signal must contain more than pure noise. Even if the process is more efficient
when parsimony rates make it possible to define elements like zero, it is still possible
that this theory and its applications remain applicable when the principal coefficients are
close to zero.
The process often begins when a weighted linear combination of samples, called
compression measurement, is taken on a different basis than the one on which the signal
is weak. The comparison process allows you to discover small measures, which usually
contain critical information to reconstruct the signal. As the domain is scanned, it
becomes possible to convert the image back to the intended domain[72].This is a
possibility even if the compression measurements will be less than the number of pixels
or other types of information found in the signal.
For the theory of compressed detection to work effectively, it is sometimes necessary to
apply a parsimony constraint. This process is initiated when a system of
underdetermined or undetermined linear equations must be solved. The application
51
makes it possible to minimize the number of non-zero components and to find the
solution.
The premise of CS is a mathematical projection that attempts to explain how sparsity
signals are recovered when signals are under-sampled in accordance with the theory of
Nyquist.Assuming that the complex space Cn contains a signal denoted as x and also
assuming that the signal is in the Ψ domain such that
𝑠𝑠 = Ψ𝑥𝑥 (3.1)
Ψ is the 𝑚𝑚 × 𝑚𝑚 matrix indicating the sparse transformation. The signal is a Fourier
coefficient and they are well above zero in value. A system of measurement quantifies
the signal z in p dimensional space only by taking m projections of signal x as
𝑧𝑧 = Φ𝑥𝑥 (3.2)
𝑧𝑧 ∈ 𝐶𝐶𝑛𝑛 ,𝐾𝐾 < 𝑝𝑝 < 𝑚𝑚 and Φis the 𝑝𝑝 × 𝑚𝑚 matrix. Φis often sub-sampled in the magnetic
resonance imaging technique. The above expression can be further transformed to
𝑧𝑧 = ΦΨ*s, x =Ψ*s (3.3)
The symbol * means the transpose operation and the signal x is not very condensed in
the Ψregion. Image obtained using the MRI technique may be poorly characterized in
the wavelength domain with the aid of the wavelength transformation matrix.
The Theory of compressive sensing offers an exceptional solution to the problem
through the formula below.
min𝑥𝑥
||Ψ x̂||l1 (3.4)
||Ψx̂||l1∶= ∑ |𝑥𝑥𝛾𝛾|𝑖𝑖 is the l1 norm of x with x1 the ith element of the image reconstruction of
the signal x is achievable if certain mathematical conditions are met.
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3.3 Restricted Isometry in Compression Detection
An essential satisfactory condition for the precise reconstruction of the signal x is the
property called restricted isometry[24, 28, 73]. For the matrix Φ, the Restrictive
Isometry can be expressed as
(1 − δ𝑘𝑘)||𝑥𝑥||𝑑𝑑22 ≤ ‖Φ 𝑥𝑥𝑑𝑑22 ‖ ≤ (1 + δ𝑘𝑘)||𝑥𝑥||𝑑𝑑22 (3.5)
δ𝑘𝑘∈ (0, 1) is a constant known as the Restricted Isometry Property and the term l2 is
described as ||𝑥𝑥||𝑑𝑑2 = (∑ |𝑥𝑥𝑖𝑖 i|2)1/2. The Restrictive Isometry Property is similar to the
equation below[24, 74, 75],
(1 − δ𝑘𝑘) ≤ σ𝑚𝑚𝑖𝑖𝑛𝑛2 [Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾)] ≤ σ𝑚𝑚𝑚𝑚𝑥𝑥
2 [Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘)] ≤ (1− δ𝑘𝑘) (3.6)
The factors σ𝑚𝑚𝑖𝑖𝑛𝑛[Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾)] and σ𝑚𝑚𝑚𝑚𝑥𝑥[Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘)] stands for the lowest and the highest
quantitative values that the factor Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾) can assume.Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾) in itself is the sub-
matrix obtained from the K regions ofΦ. The factorδ2k which is the Restricted Isometry
Property constant is the lowest factor that modify the equation above for each sub-
matrix of Φ and is crucially bound on the space between 1 and all the discrete values
that the factor Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘) can assume. When δ2k<1, it denotes that a signal x is sparse in
the K region and can be precisely reconstructed from the possible values ofΦ.
The factor δ k∈ (0, 1), offers the precise reconstruction of the signal x and the
measurements of δ k defines the stability of the reconstruction of the signal. The
interference of noise which can assume the value 𝜉𝜉, 𝑧𝑧 = Φ𝑥𝑥 + ξ and the reconstructed
signal 𝑥𝑥� fulfils the equation below[76].
�|𝑥𝑥 − 𝑥𝑥�|�𝑑𝑑22≤ 4ξ2
1−δ2𝑘𝑘 (3.7)
Hence, the lower the value of δ k, the lower the error encountered upon the
reconstruction of the signal and the higher the value of δ k, the higher the error
encountered upon the signal reconstruction. But then, the calculation of δk for a possible
53
value of Φ is NP-hard and therefore inflexible. Because δkcritically abound on the space
between unity and the possible values of Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾)s, the scope of the constant δkcan be
measured by the space between unity and σ𝑚𝑚𝑖𝑖𝑛𝑛[Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾)]s and σ𝑚𝑚𝑚𝑚𝑥𝑥[Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾)]s. The
lower space, the lower the value of the constant δk and then the higher the outputΦ.
Because of this equation( (1− δ𝑘𝑘) ≤ σ𝑚𝑚𝑖𝑖𝑛𝑛2 [Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾)] ≤ σ𝑚𝑚𝑚𝑚𝑥𝑥
2 [Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘)] ≤ (1− δ𝑘𝑘) )
must be in place, the values of σ𝑚𝑚𝑖𝑖𝑛𝑛[Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾)] and σ𝑚𝑚𝑚𝑚𝑥𝑥[Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾)] over arbitrarily
sampled Φ𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘)’s are used in [25, 74, 75] to evaluate the performance of δkin a given
measurement matrix Φ. This technique is also implemented in this thesis.
3.4 Incoherence in Compressive Sensing
Anotheressential condition for the precise reconstruction of the signal x is the crucial
factor known as incoherence[30, 77]. For a couple of values of the matrix Φ and
sparsifying transform matrix Ψ, fulfilling Φ*Φ = nl and Ψ*Ψ = l, their incoherence is
outlined as
µ(Φ,Ψ) = 𝑚𝑚𝑎𝑎𝑥𝑥𝑘𝑘.𝑗𝑗| <Φk, ψj>| (3.8)
where ψj and Φk are the jth and kth columns of Φ and Ψ respectively and µ(Φ,Ψ)∈[1,
�𝑚𝑚]. The factor µ(Φ,Ψ) = 1 is known as maximal incoherence. When m ≥ 𝐶𝐶.µ2(Φ,Ψ)
. K .log(𝑚𝑚). andC is a constant then a sparsity signal x in the k-space can be precisely
reconstructed. Hence, µ(Φ,Ψ) decides the least number of measurement essential for
the exact reconstruction of the signal x. The lower the µ(Φ,Ψ), the lower the value of
m required for the precise reconstruction of the signal x. It is paramount to know that
both Incoherence and Restrictive Isometry Property are adequate conditions on the
measurement matrix. Hence, they are similar and both or either of them can be used to
investigate, design and evaluate the measurement matrix for the precise reconstruction
of the signal x.
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3.5 Compressed Sensing in Magnetic Resonance
Imaging
Accelerating the speed of data acquisition in Magnetic Resonance Imaging and
reconstruction by Compressed Sensing is anactive area of research that is increasingly
capturing the attention of researchers and data scientists recently with the sole
objectives being to increase the speed and precision of the image reconstruction
effectively. Compressed sensing is a method that allows for the reconstruction of the
intended signal using the data obtained below the Nyquist-Shannon sampling rate.
Magnetic Resonance Imaging is a perfect scheme for the application of Compressed
Sensing because it acquires images that are already in the Fourier domain instead of the
pixel domain [32, 35]. The CS in MRI is first implemented in [32], where VD random
sub-sampling of phase encoding is proposed as a data sampling technique.The
combination of Parallel MRI Technique with Compressive Sensing has had an
accelerating possibility in the encoding carried out by MRI[27, 78]. Also, the CSMRI
can be used for dynamic imaging while using the k-t space sparsity.
The theory of Compressed Sensing offers the way around certain problems by using
previous information of the compressibility of the signal. This concept promises ideal
reformation of the signal from sub-sampled signals when some conditions are in place.
Images obtained from the biomedical analysis are inherently sparsity in certain
transform region (Fourier, Wavelet, etc.) while angiograms are sparsity in the pixel
region. As reported in some events, the signal may be insufficiently sparsified with
slight loss of information. The mode of acquiring image data using the method of MRI
inherently gains encoded signals instead of the signal values. Therefore sub-sampled
sparse reconstruction techniques can be used in Magnetic Resonance Imaging for the
reconstruction of images from sparse value. It can be demonstrated that the signal in
MRI follows the Fourier formula
s(t) =∫ 𝑚𝑚(𝑟𝑟)𝑅𝑅 ei2πk(t)r dr where k(t) ∞∫ 𝐺𝐺(𝑠𝑠)𝑑𝑑0 𝑑𝑑𝑠𝑠 (3.9)
55
m(r) represents the object while r stands for the co-ordinate in the spatial domain, k(t)
represents the data in the k region while s(t) represents the signal.
3.6 Application of Compressive Sensing
3.6.1 Sparse Error Correction
This method is proficient in identifying and amending errors in data that seems
inaccurate in the channel of transmission. This method of correcting data depends on the
pattern of recurrence of the data, redundancy checks, or closest-neighbour code search.
Let’s take into account a certain scenario where a signal xm with M entries areencoded
by taking length-N encryptions linearly independent (φ1, ... ..,φM) such that N> M and
adding them up utilizing the entries of the signal x as coefficients. The received signal is
encoded of length N code
𝑦𝑦 = ∑ φ𝑖𝑖𝑀𝑀𝑚𝑚=1 𝑥𝑥𝑖𝑖 = Φf,
whereΦ is a matrix with differentcodewords for columns. Let’s say that the channel of
transmission get to corrupt the values of y in anadditive way, in such a way that the
collected data is
𝑦𝑦 = Φ𝑥𝑥 + 𝑒𝑒,
where 𝑒𝑒 represents an error vector. The method implement for the recovery of the sparse
signal in thecontext of compressed sensing offers a number of scheme that is effectively
utilized in the estimation of error vector 𝑒𝑒 , hence, making it possible to rectify it and
acquire the signal x, where the error 𝑒𝑒is adequately sparse. In order to estimate the error,
we construct a matrix Θ that we use as the foundation for the orthogonal subspace of the
span of the matrix Φ, i.e., an (N − M) × N matrix Θ which places Θ Φ = 0. We can work
on the values upon the acquisition of such matrix when we multiply them with the
matrix to get the expression
56
𝑦𝑦� = Θ Φx + Θ𝑒𝑒 = Θ𝑒𝑒.
Assuming that the matrix Θ is perfect for compressive sensing (i.e. is fulfils the
requirement of the RIP) and 𝑒𝑒 is largely sparse, then 𝑒𝑒 can precisely be approximated
with the method of compressive sensing. As soon as 𝑒𝑒 is calculated, the error-free
measurements can be deduced as
𝑦𝑦�= 𝑦𝑦 − �̂�𝑒,
and the recovery of the signal is represented as
𝑥𝑥� = Φϯy− Φϯ�̂�𝑒
For illustration purpose, when the encoded data φm possess identical and random
distribution in the sub-Gaussian entries. A k-sparse error can be rectified when M< 𝑁𝑁 −
𝐶𝐶𝐾𝐾 log𝑁𝑁/𝐾𝐾 for a constant C that satisfies the Restrictive Isometry Property.
3.6.2 Linear Regression and Model Selection
A substantial amount of the algorithms for sparse recovery of signals initially were
designed to solve the problem of model selection and sparse linear regression of
statistical problems. Contextually, data involving a set of input and output variables
were provided to us. We will assume that there exist a sum of N variables, and we
notice that a sum of M input and response pairs.
If we choose to denote the set of input variable observation to be a 𝑀𝑀 × 𝑁𝑁 matrix Φ and
the set of response variable observations as a 𝑀𝑀 × 1 vector y, then according to analysis
using linear regression, y may be an estimate of a linear function of variables.
Nevertheless, if the input variables are big in contrast to the number of observations,
i.e., 𝑁𝑁 ≫ 𝑀𝑀 then it becomes overwhelmingly difficult, and this is because we intend to
approximate N parameters using a relatively few N observations. Broadly speaking, this
may be hard to surmount, however practically speaking, it is frequent that only small
input variables are significant to forecast the response variable. This means that in this
scenario, the signal x which we intend to approximate is sparse and we can be able to
57
use every method we have learned thus far for sparse recovery to approximate the value
of the signal x. Statistically, the sparsity of the signal does not only assist us in our
objective of getting a linear regression, but it also executes model selection by
recognizing the essential variables in forecasting the response.
3.6.3 Group Testing and Data Stream Algorithms
One more case where the method of CS and sparse recovery algorithms can be
potentially useful is in its application to data stream algorithm and group testing and
then the associated cases of data stream computation.
3.6.3.1 Group Test
Some of the most antiquated ideas that address how to recover sparse algorithms were
implemented with the knowledge of combinatorial group testing[79-81]. Assuming that
there exists an N total list of items and K irregular elements in which we attempt to find.
Foe example, we are attempting to identify the problematic products in an industry or
the small group of the infected tissue sample in the body of a patient. In two of the
above scenarios the vector x shows the elements which are defective, i.e. xi ≠ 0 for the
defective K elements and xi = 0. In the easiest real-world scenario, these tests are
denoted with the binary matrix Φ whose entries φij are equivalent to unity on the
condition that the jth list is used in the ith test. When the output of the test is straight in
comparison to the inputs, then it is safe to say that the vector x is very much identical to
the standard recovery problem in CS.
3.6.3.2 Computation on Data Streams
One other area where the knowledge of CS has been found to be very important is in the
data stream computation[82, 83]. As in case of a characteristic data streaming challenge,
assume that xi stands for the number of packets moving through a network router with
destinationi. Storing the vector x is not possible because the sum of the probable
destinations (denoted by 32-bit Internet Protocol Address) is about 𝑁𝑁 = 232. Hence, as
58
a substitute for intending to store x directly, the researcher can rather store𝑦𝑦 = Φ𝑥𝑥. Φ is
an 𝑀𝑀 ×𝑁𝑁 matrix with 𝑁𝑁 ≫ 𝑀𝑀. Vector y is frequently known as a sketch. Explicitly in
the case of the network traffic, we do not have to detect x directly, instead, we measure
increments to xi. Hence we build y iteratively by appending the ith column to y every
time we detect an increment to xi,which is a possibility for us because 𝑦𝑦 = Φ𝑥𝑥 is linear.
The takeover of network traffic by traffic to a small number of destinations, the vector x
is compactable and hence the case of recovering the signal x from the sketch Φ𝑥𝑥 is
again fundamentally the equivalent as the sparse recovery problem in CS.
3.6.4 Coronary Heart Imaging
CS can speed up the rate of acquiring data, permitting the whole heart to be visualized
in just one hold of breadth. Per cardiac prompt, one k-space spiral obtained for every
slice. The restrictions imposed by time always allows CS sub-sampling reconstruction
in the two-level double k-space that eliminates the interference induced by under
sampling without degrading the quality of the image.
3.6.5 Brain Imaging
The concept of Compressive Sensing in performing brain imaging promises to decrease
the time needed for the analysis while increasing the quality of the image so that
appropriate brain activities and structures can be determined without any error.
3.6.6 Rapid 3-D Angiography
CS is particularly suitable in angiography, where it provides advanced temporal and
spatial imaging quality at the cost of under-sampled artifacts[35]. This is one of
thesuperior advantages of the application of compressive sensing in MRI.
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3.7 Some Limitations of CSMRI The following problems have not been fully solved in the present work: In conventional CS-MRI, Fourier matrix as a sensing matrix and Wavelet matrix as a
sparsifying transform matrix respectively are not mutually incoherent. Moreover,
Fourier encodingconcentrates signal energy on the centre of the k-space known as low
frequencies region. This restricts the subsampling to fully sample the low frequency
region and insufficiently sample the high frequency region at high acceleration
factors,resulting in degraded image quality. This limitation can cause a huge loss in
image resolution.
Reduction factor is also crucial for high quality reconstruction in compressed sensing
even with PMRI. When it is increased to a certain label, the artefacts of the image
increaseand the SNR decreases simultaneously.
To overcome the limitations of present works found through literature survey,this thesis
introduces three new procedures.
1. In order to improve the quality of image reconstruction of compressive sensing
for high reduction factor, this thesis has proposed a novel compressive sensing
bunched phase encoding (CSBPE) technique.
2. Due to magnetic field inhomogeneity and imprecise field gradients, the phase of
the zigzag trajectories is different from theoretically calculated ones, which
lowers image quality. To overcome this problem, this thesis has introduced a
new technique called cross-correlation to remove the phase deviation from the
zigzag data position. In addition, to smooth and remove the aliasing from the
image, a modulation map estimation method has been incorporated in compress
sensing based image reconstruction.
3. In order to further improve the image quality of CSBPE, this thesis has
introduced a non-Fourier encoding scheme called chirp modulated Fourier
60
encoding. The simulation result shows that the chirp modulated CSBPE
outperform the Fourier method.
3.8 Conclusion
Compression sampling is a mathematical model that reconstructs signals without
fulfilling Nyquist-Shannon theorem. This technique will undoubtedly revolutionize in
the field of signal processing, which is one of the areas that benefited most from data
compression. Currently, the CS is applied in areas such as biology, geology,
telecommunications and imaging. Moreover, for the proper functioning of the CS, it is
necessary to choose an appropriate technique to transform the data into rare data; this
work has shown that DCT is the best technique for physiological data, as it creates an
ideal sparse space for the needs of CS.
The idea and advances outlined in this thesis could possibly drive the novel uses of
CSMRI, which are previously considered difficult. The CS-MRI technique is still in it is
early stages. A lot of important questions remain unresolved [28]. These include
creating novel inconsistent sparse responses for the sampling operator, optimizing
sampling trajectories, studying the quality of the reconstruction with regards to its
clinical importance and increasing the fastness of the reconstruction algorithms. The
signal processing community has a big advantage here. There exist captivating
hypothetical and practical issues, assuring tangible gains in increasing medical care.
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Chapter 4 Compressive Sensing MRI using Fourier Based Bunched Phase Encoding Compressive Sensing MRI using Fourier Based Bunched Phase Encoding
4.1 Introduction
As discussed in the previous chapter, the data acquisition speed of imaging is critical to
the performance of practical MRI applications. The process of collecting data in MRI is
primarily limited by physical constraints (sweep rate, amplitude and gradient) and
physiological (nerve stimulation). High-speed MRI scanning can save a significant
amount of money and time for the patient, and can reduce patient discomfort and
imaging artifact caused by the patient’s motion. As a result, lots of researchers are
looking for new procedures that can reduce the amount of data requirement without
compromising the quality of the image for any reason.
As we know k-space sampling is performed by phase and frequency encodings. The key
to accelerating MRI data acquisition is to reduce the number of phase encodings, which
leads to k-space under sampling. But, when the k-space is under sampled the
reconstructed image shows aliasing artifacts.Various kind of methods have been
developed to reduce scanning time of MRI. But every methods has some limitation to
obtain certain level of acceleration.
Therefore, in this thesis,we will represent a new acceleration technique for the structural
MRI. This new method is developed by combining compressive sensing (CS) with
Fourier based bunched phased encoding (BPE) technique. Hence we call it CSBPE
method.
62
We intend to exploit the inherent parsimony in magnetic resonance imaging and to
implement a method for integrating different approaches to improving the image
quality.
BPE is a non-linear sampling technique which was proposed by H. Moriguchi in 2006
[63]. This technique asserts that a number of phase encoding line can be reduced, if k-
space data are sampled along a zigzag trajectory during each readout and samples are
acquired at a sampling frequency higher than that of the conventional rectilinear
acquisition. That means we can reduce the number of TR cycles in rectilinear MRI data
acquisition by setting the PE steps size larger than that of the conventional acquisition
technique, and also an increased number of readout samples are acquired along a zigzag
k-space trajectory at a higher sampling rate than in conventional acquisition. This
technique opens opportunities to reduce the total number of TR cycles and hence the
MR scan time.
Among other popular methods, CS-MRI is a new framework for data sampling and
signal recovery. MRI is one of the suitable applications for CS because it meets two
main conditions such as sparsity and incoherency of CS. Hence, the MR images can be
reconstructed using CS technique from data sampled at a rate well below the Nyquist
rate. Because BPE and CS-MRI reduce sampling based on different ancillary
information (non-linear zigzag sampling for BPE and image sparseness for CS), it is
desirable to combine BPE and CS for further reduction of data reduction.
In conventional MRI, compressive sensing is much more efficient due to the facts that
most MR images are sparse in the transforms domain and data in k-space allows a
certain level of incoherent sampling [26, 32, 35, 84]. In CSMRI, the incoherence
between measurement or sensing matrix and sparsifying transform matrix are not
optimal. Moreover, the energy of Fourier encoding usually concentrates in the centre of
the k-space known as the low-frequency region.
This creates a restriction on the subsampling pattern to acquire adequate sample data
from the low-frequency region and insufficiently sample data in the high-frequency
region at high acceleration factors, which results in degraded image quality. Such
limitation can cause a massive loss in image resolution at a higher reduction factor.
63
Besides, some approaches can achieve acceleration without phase array coils.
Bunchedphase encoding is one of them. It works on a wave-like sampling pattern to
reduce sampling data, and reconstruct aliased free image using Papoulis’ generalised
sampling theory [63]. Bunched Phase encoding takes advantage of the theoretical
concept of [85] to reduce the number of phase encodings and Repeating Time (TR)
cycles in Cartesian MRI data acquisitions. In this method, the phase encoding (PE) step
size can be set larger than the conventional rectangular acquisition, hence reducing the
number of phase encodings needed. To acquire sufficient information from the reduced
phase encodings, an additional number of readout samples are acquired, along the
zigzag or oscillating trajectories in the k-space.
This technique can decrease the total number of encodings and TR cycles and hence the
total MRI scan time. However, SNR of BPE approach becomes considerably reduced as
the reduction factor is increased generally if reduction factor is higher than two, the
noise of the reconstructed image becomes very pronounced even through aliasing
artefact are not significant [63]. To take advantages of the coil sensitivity map, Bunch
phase encoding has also been combined with sensitivity encoding (SENSE) in the
readout direction to further accelerate data acquisition in MRI [64]. However, BPE-
SENSE methods also have some crucial problem when the reduction factor is increased.
Even the signal to noise ratio of the BPE approach becomes considerably reduced as the
reduction factor is increased. We observe that if R > 2, the noise of the reconstructed
image is gradually increased even through aliasing artifact are not exist.
In this chapter, we proposed a novel method named CSBPE, which has been partially
presented[86], to combine BPE and CS-MRI for the zigzag trajectory with guaranteed
incoherence. CSBPE sequentially carries out CS reconstruction using sparse MRI for
the aliased image and BPE for the final unfolded image. Simulation and experimental
results show that CSBPE can achieve better resolution a reduction factor higher than
those achieved by CSMRI and BPE individually.
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4.2 BPE and CS Techniques
The sequential step diagram in Figure 4.1 shows the CSBPE method, where image
sampling and reconstruction methods of compressive sensing and bunched phased
encoding technique are used together to form a novel approach to achieve accelerated
MR imaging. In the following, the compressive sensing and bunched phase encoding
technique will be briefly described separately and then the overall scheme of CSBPE
will be represented.
Figure 4.1: Sequential steps of CSBPE image reconstruction process
In bunched phase encoding technique, a subsampled two dimensional image signal Iis
generated by zigzag changes of phase directional gradient and oversampling of
frequency domain data in readout direction to acquire N/R1 rows, where R1 is the
reduction factor and its values always greater than 1. Figure 4.2(a) and (b) illustrate the
rectangular grid line data and the zigzag data, where some datadefined as baseline or
reference data and rest are shifted data, respectively.
BPE Pulse Sequence Design
BP Encoded MRI data acquisition
subsampled K-space data
Further VD subsampling
Aliasing coefficient correction
CS Reconstruction
Image reconstruction
from BP Encoded MRI data
SOS of all channel Output image
65
Figure 4.2:Schematic diagram of k-space trajectories of bunched phase encoding.
(a) Rectilinear sampling data. (b) zigzag sampling datain the BPE scheme.
Bunched phase approach isbased on the Papoulis’ generalized sampling theory[87],
which states that under the condition that the original function f(t) is zero outside a finite
interval (i.e., Time-limited), the signal f(t) can be reconstructed without aliasing
artifactseven though the Nyquist criterion is violated in the Fourier domain. For
example, if n number of bunched samples are collected at 1/n-th the Nyquist rate in the
Fourier domain, then the originalsignal f (t)can reconstruct without aliasing artifacts
[63].
Bunched Phase encoding takes advantage of this theoretical concept to reduce the
number of Repeating Time (TR) cycles in Cartesian MRI data acquisitions. In this
method, the phase encoding (PE) step size can be set larger than in the conventional
acquisition, i.e., 1/FOVy = ∆ky, here FOVy is the field-of-view along the phase direction.
At the same time, an additional number of readout samples are acquired at a higher
sampling rate than in a conventional acquisition, and along with a zigzag or oscillating
trajectories in the k-space domain. This technique enables one to decrease the total
number of TR cycles and hence the total MR scan time. Thek-space trajectories of a
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standard rectilinear and the bunched phase encoding acquisition methods are
schematically shown in Figure 4.2.
In the BPE scheme, k-space data are acquired along a zigzag trajectory, and the distance
between two neighbouring PE lines is qΔky, where q is usually greater than 1. As shown
in Figure 4.2, more data are sampled at higher sampling frequencies using the BPE
technique than a standard sampling method during the same readout period. Because of
the gradient alternation, this is corresponding to acquisition of multiple phase encoding
lines in a single readout to the BPE method. In Figure 4.2, rΔky is defined as the width
of a zigzag k-space trajectory band. In the formerly proposed hybrid fast-scan methods
[88, 89], q - r was less than or equal to 1.Moreover, as the total number of TR cycles
was reduced as q was increased as well as r was usually required to increase. However,
in bunched phase encoding, the width of the phase encoding band, r, can be set to any
value provided that r is greater than 0. But, it is noted that the value of r is independent
of the reduction factor R. The typical sampling rate must satisfy the Nyquist criterion to
reconstruct the signals without aliasing artifacts. Therefore, if the reduction factor of the
bunched phase encoding method is R, the total number of samples acquired for each
TR, m, must be at least N*R.
However, SNR of BPE approach becomes considerably reduced as the reduction factor
is increased.Often when R>2 the noise of the reconstructed image is significantly
increased even through aliasing artefact does not exist[63]. To overcome this difficulty
we will try to further decrease the scan time while attaining noiseless aliasing free
image by combine BPE with other new approaches such as compressed sensing, and
SENSE, and non-Fourier encoding method such as noiselet, and chirp modulation
encoding.
67
Figure 4.3: Generalized 1D bunched phase encoding technique in k-space
Figure 4.3 show the generalized 1D BPE technique in k-space where the data
subsampledat a factor 2 and shifted from its original position 𝛼𝛼Δ𝑘𝑘.The mathematical
derivation of the generalized BPE technique will be explained step by step in the
following. Nyquist sampling and Fourier transform of the generalised 1D signal can be
represented as
𝑀𝑀�𝑘𝑘𝑗𝑗� = �𝐼𝐼(𝑥𝑥𝑖𝑖)𝑛𝑛
𝑖𝑖=1
𝑒𝑒−𝑖𝑖2𝜋𝜋𝑥𝑥𝑖𝑖𝑘𝑘𝑗𝑗 (4.1)
𝐼𝐼(𝑥𝑥𝑖𝑖) = �𝑀𝑀�𝑘𝑘𝑗𝑗�𝑛𝑛
𝑖𝑖=1
𝑒𝑒𝑖𝑖2𝜋𝜋𝑥𝑥𝑖𝑖𝑘𝑘𝑗𝑗(4.2)
Where 𝐼𝐼(𝑥𝑥𝑖𝑖) and 𝑀𝑀�𝑘𝑘𝑗𝑗� are image and k-space data respectively, i = 1,2,3….n, j =
1,2,3….nand𝑘𝑘𝑗𝑗 = 𝑗𝑗∆𝑘𝑘. The equation of the aliasing signal due to downsampling is given
as
𝐼𝐼𝑑𝑑(𝑥𝑥𝑖𝑖) = �𝑀𝑀𝑑𝑑(𝑘𝑘𝑑𝑑)
𝑁𝑁𝑝𝑝
𝑑𝑑=1
𝑒𝑒𝑖𝑖2𝜋𝜋𝑥𝑥𝑖𝑖𝑘𝑘𝑑𝑑
68
= � � 𝐼𝐼(𝑥𝑥𝑚𝑚)𝑒𝑒−𝑖𝑖2𝜋𝜋𝑥𝑥𝑚𝑚𝑘𝑘𝑑𝑑𝑒𝑒𝑖𝑖2𝜋𝜋𝑥𝑥𝑖𝑖𝑘𝑘𝑑𝑑𝑁𝑁
𝑚𝑚=1
𝑁𝑁𝑝𝑝
𝑑𝑑=1
(4.3)
where 𝑘𝑘𝑑𝑑 = 𝑑𝑑.𝑝𝑝∆𝑘𝑘. If we consider the bunched Nyquist sampling, the equation will be
𝐼𝐼𝑠𝑠1(𝑥𝑥𝑖𝑖) = �𝑀𝑀𝑠𝑠1�𝑘𝑘𝑗𝑗 + 𝛼𝛼∆𝑘𝑘�𝑁𝑁
𝑗𝑗=1
𝑒𝑒𝑖𝑖2𝜋𝜋𝑥𝑥𝑖𝑖�𝑘𝑘𝑗𝑗+𝛼𝛼∆𝑘𝑘�
= � � 𝐼𝐼(𝑥𝑥𝑚𝑚)𝑒𝑒−𝑖𝑖2𝜋𝜋𝑥𝑥𝑚𝑚�𝑘𝑘𝑗𝑗+𝛼𝛼∆𝑘𝑘�𝑒𝑒𝑖𝑖2𝜋𝜋𝑥𝑥𝑖𝑖�𝑘𝑘𝑗𝑗+𝛼𝛼∆𝑘𝑘�𝑁𝑁
𝑚𝑚=1
𝑁𝑁
𝑗𝑗=1
(4.4)
Now, the equation can be represented for the bunched phased down sampling
trajectories as
𝐼𝐼𝑠𝑠𝑑𝑑1(𝑥𝑥𝑖𝑖) = �𝑀𝑀𝑠𝑠𝑑𝑑1(𝑘𝑘𝑑𝑑 + 𝛼𝛼∆𝑘𝑘)
𝑁𝑁𝑝𝑝
𝑑𝑑=1
𝑒𝑒𝑖𝑖2𝜋𝜋𝑥𝑥𝑖𝑖(𝑘𝑘𝑑𝑑+𝛼𝛼∆𝑘𝑘)
= � � 𝐼𝐼(𝑥𝑥𝑚𝑚)𝑒𝑒−𝑖𝑖2𝜋𝜋𝑥𝑥𝑚𝑚(𝑘𝑘𝑑𝑑+𝛼𝛼∆𝑘𝑘)𝑒𝑒𝑖𝑖2𝜋𝜋𝑥𝑥𝑖𝑖(𝑘𝑘𝑑𝑑+𝛼𝛼∆𝑘𝑘)𝑁𝑁
𝑚𝑚=1
𝑁𝑁𝑝𝑝
𝑑𝑑=1
(4.5)
Equation (4.5) can be represented in matrix form
⎝
⎜⎜⎛ 𝑀𝑀𝛼𝛼0𝑀𝑀𝛼𝛼1⋮
𝑀𝑀𝛼𝛼𝑏𝑏−1⎠
⎟⎟⎞
=1𝑝𝑝
⎝
⎜⎜⎛
1 𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼0
𝑝𝑝⋯ 𝑒𝑒
𝑖𝑖2𝜋𝜋𝛼𝛼0(𝑝𝑝−1)𝑝𝑝
1 𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼1𝑝𝑝
… 𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼1(𝑝𝑝−1)
𝑝𝑝
⋮1
⋮
𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼𝑏𝑏−1
𝑝𝑝…
⋮
𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼𝑏𝑏−1(𝑝𝑝−1)
𝑝𝑝 ⎠
⎟⎟⎞
×
⎝
⎜⎜⎜⎜⎛
𝐼𝐼(𝑥𝑥)
𝐼𝐼 �𝑥𝑥 − 𝐹𝐹𝑝𝑝�
⋮𝐼𝐼(𝑥𝑥 − 𝐹𝐹(𝑝𝑝−1)
𝑝𝑝 ⎠
⎟⎟⎟⎟⎞
(4.6)
69
In Equation (4.6), I(x), I(x-F/p), . . . , I(x-F(p-1)/p) (all for 0 ≤ x ≤ F/p) can be found by
computing an inverse matrix (when p =m) or a pseudo inverse matrix (when p< m)
from Equation (4.6). Hence I(x) for the entire range –F/2 ≤ x≤ F/2 can be reconstructed
without aliasing artifacts as
⎝
⎜⎜⎜⎜⎛
𝐼𝐼(𝑥𝑥)
𝐼𝐼 �𝑥𝑥 − 𝐹𝐹𝑝𝑝�
⋮𝐼𝐼(𝑥𝑥 − 𝐹𝐹(𝑝𝑝−1)
𝑝𝑝 ⎠
⎟⎟⎟⎟⎞
= 𝑝𝑝
⎝
⎜⎜⎛
1 𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼0
𝑝𝑝⋯ 𝑒𝑒
𝑖𝑖2𝜋𝜋𝛼𝛼0(𝑝𝑝−1)𝑝𝑝
1 𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼1𝑝𝑝
… 𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼1(𝑝𝑝−1)
𝑝𝑝
⋮1
⋮
𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼𝑏𝑏−1
𝑝𝑝…
⋮
𝑒𝑒𝑖𝑖2𝜋𝜋𝛼𝛼𝑏𝑏−1(𝑝𝑝−1)
𝑝𝑝 ⎠
⎟⎟⎞
−1
×
⎝
⎜⎜⎛ 𝑀𝑀𝛼𝛼0𝑀𝑀𝛼𝛼1⋮
𝑀𝑀𝛼𝛼𝑏𝑏−1⎠
⎟⎟⎞
or in simpler form,
𝐼𝐼1 = 𝐶𝐶𝑇𝑇𝐷𝐷1 (4.7)
where𝐼𝐼1 is the reconstructed 1D image data matrix, 𝐷𝐷1 is the BP encoded 1D data matrix
and C is the aliasing coefficient matrix constructed based on the zigzag trajectories and
the reduction factor R1for the 1D case.
Similarly for the 2D case, if the image signal is I(x,y) then the Fourier transformed k-
space signal can be represente as
𝑀𝑀�𝑘𝑘𝑖𝑖 , 𝑙𝑙𝑗𝑗� = ��𝐼𝐼�𝑥𝑥𝑖𝑖, 𝑦𝑦𝑗𝑗�𝑒𝑒−𝑖𝑖2𝜋𝜋𝑁𝑁 〖(𝑥𝑥〗𝑖𝑖,𝑘𝑘𝑖𝑖+𝑥𝑥𝑗𝑗𝑑𝑑𝑗𝑗�
𝑁𝑁
𝑗𝑗=1
(4.8)𝑁𝑁
𝑖𝑖=1
where 𝑘𝑘𝑖𝑖 = 𝛾𝛾∆𝑘𝑘 and 𝑙𝑙𝑗𝑗 = 𝑗𝑗∆𝑘𝑘.
If we apply inverse Fourier transform onfully encoded 2D MRI data, the equation will
be
𝐼𝐼�𝑥𝑥𝑖𝑖,𝑦𝑦𝑗𝑗� =1𝑚𝑚2��𝑀𝑀�𝑘𝑘𝑖𝑖, 𝑙𝑙𝑗𝑗�𝑒𝑒
𝑖𝑖2𝜋𝜋𝑁𝑁 (𝑥𝑥𝑖𝑖,𝑘𝑘𝑖𝑖+𝑥𝑥𝑗𝑗𝑑𝑑𝑗𝑗) (4.9)
𝑁𝑁
𝑗𝑗=1
𝑁𝑁
𝑖𝑖=1
70
Similarly, if we apply IFFT on subsampled 2D MRI data,the reconstructed image will
be aliasing and the equation will be
𝐼𝐼𝑑𝑑�𝑥𝑥𝑖𝑖,𝑦𝑦𝑗𝑗� =1𝑚𝑚2��𝑀𝑀𝑑𝑑�𝑘𝑘𝑖𝑖, 𝑙𝑙𝑗𝑗� 𝑒𝑒
𝑖𝑖2𝜋𝜋(𝑥𝑥𝑖𝑖,𝑘𝑘𝑖𝑖𝑁𝑁 𝑝𝑝�
+𝑥𝑥𝑗𝑗𝑙𝑙𝑗𝑗𝑁𝑁 )
(4.10)𝑁𝑁
𝑗𝑗=1
𝑁𝑁𝑝𝑝
𝑖𝑖=1
where p is the reduction factor.
Now,if we apply IFFT on 2D Bunched Nyquist sampled MRI data, the equation will be,
𝐼𝐼𝑠𝑠�𝑥𝑥𝑖𝑖,𝑦𝑦𝑗𝑗� =1𝑚𝑚2��𝑀𝑀𝑠𝑠�𝑘𝑘𝑖𝑖 + 𝛼𝛼∆𝑘𝑘, 𝑙𝑙𝑗𝑗 + 𝛽𝛽∆𝑘𝑘� 𝑒𝑒𝑖𝑖2𝜋𝜋(
𝑥𝑥𝑖𝑖(𝑘𝑘𝑖𝑖+𝛼𝛼∆𝑘𝑘)𝑁𝑁 +
𝑥𝑥𝑗𝑗(𝑙𝑙𝑗𝑗+𝛽𝛽∆𝑘𝑘)𝑁𝑁 ) (4.11)
𝑁𝑁
𝑗𝑗=1
𝑁𝑁
𝑖𝑖=1
Also, if we apply the down-sampled 2D Bunched phase encoded MRI data, the equation
will be,
𝐼𝐼𝑠𝑠𝑑𝑑�𝑥𝑥𝑖𝑖,𝑦𝑦𝑗𝑗� =1𝑚𝑚2��𝑀𝑀𝑠𝑠𝑑𝑑�𝑘𝑘𝑖𝑖 + 𝛼𝛼∆𝑘𝑘, 𝑙𝑙𝑗𝑗 + 𝛽𝛽∆𝑘𝑘� 𝑒𝑒
𝑖𝑖2𝜋𝜋(𝑥𝑥𝑖𝑖�𝑘𝑘𝑖𝑖+𝛼𝛼∆𝑘𝑘�
𝑁𝑁 𝑝𝑝�+𝑥𝑥𝑗𝑗�𝑙𝑙𝑗𝑗+𝛽𝛽∆𝑘𝑘�
𝑁𝑁 )𝑁𝑁
𝑗𝑗=1
𝑁𝑁𝑝𝑝
𝑖𝑖=1
(412)
Now, IFFT of the acquired data gives the aliased image data D with N/R1 rows. With
linear equation based reconstruction[63], the image I can be recovered by solvingthe
following equation for the image I,
𝐼𝐼 = 𝐶𝐶𝑇𝑇𝐷𝐷, (4.13)
where C is the aliasing coefficient matrix constructed based on the zigzag trajectories
and the reduction factor R1.Linear equation based reconstruction technique can produce
aliasing free image from the single receive or multi-receivecoil data without using coil
sensitivity.
71
Now for reminiscing the compressive sensing method, the methamatical model will be
explained from the begaining. Suppose an n-dimensionalcomplex signalIin the space
ℂ𝑛𝑛 can be sparsely represented in sparsifying transformation𝛹𝛹 as
𝑥𝑥 = 𝛹𝛹𝐼𝐼 (4.14)
where the matrix 𝛹𝛹is the orthogonal wavelet basis or the frame of column vectors.
That means a number of coefficients k in signal x are sparse. A mathematical
measurement scheme can measure mdimensional signal yby selectingonly mprojections
of the signal Ias
𝑦𝑦 = Φ𝐼𝐼 (4.15)
where Φ ∈ ℂ𝑚𝑚×𝑛𝑛 is the sensing matrix. In a specific type of signal such as MRI, 𝐼𝐼 ∈ ℂ𝑛𝑛
is the vectorised complex image data with the dimension 𝑚𝑚 = 𝑟𝑟𝑟𝑟for a𝑟𝑟 × 𝑟𝑟dimensional
image, and Φ ∈ ℂ𝑚𝑚×𝑛𝑛 is usually a restricted randomly subsampled DFT matrix obtained
from the encoding process, where𝑚𝑚 = 𝑝𝑝𝑟𝑟 and 𝑝𝑝 < 𝑟𝑟 is the number of phase encoding
linesused to acquire y.Therefore, for the image dimension𝑟𝑟 × 𝑟𝑟, the subsampling ratio
(𝑛𝑛𝑚𝑚
= 𝑟𝑟𝑝𝑝
) is determined by the number of phase encoding lines, p, which ultimately
determines the acquisition time [32]. Now, Eq (1) can be further represented as
𝑦𝑦 = ΦΨ∗𝑥𝑥, where 𝐼𝐼 = Ψ∗𝑥𝑥 (4.16)
where * denotes the conjugate transpose operator and the signal I is sparse in the 𝛹𝛹
domain. Image signal can be sparsely represented in the wavelet domain using the
wavelet transformmatrix. The signal equation becomes under-determined when the data
are under-sampled. Therefore, it is near impossible to recover the exact signal using
linear reconstruction approaches. The compressive sensing technique provides full
recovery of a sparse signal from subsampled measurements by providing a unique
solution to the ill-posed nonlinear recovery problem:
min𝑉𝑉�Ψ𝐼𝐼�𝑑𝑑1 𝑠𝑠. 𝑡𝑡. �𝑦𝑦 −Φ𝐼𝐼�
2𝑑𝑑2≤ 𝜀𝜀2 (4.17)
72
Where �𝐼𝐼�𝑑𝑑1 ∶= ∑ |𝐼𝐼𝑖𝑖|𝑖𝑖 is the l1norm of I,Iiis the ith element of I, Ψ is the sparsifying
transform matrix, and𝜀𝜀 is the allowed data discrepancy. The constrained optimisation
problem of Eq. (4.17) is converted to an unconstrained problem with regularisation
penalties for computation purposes:
arg𝑚𝑚𝛾𝛾𝑚𝑚 ��𝑦𝑦 − Φ𝐼𝐼�2𝑑𝑑2
+ 𝜆𝜆1�Ψ𝐼𝐼�𝑑𝑑1 + 𝜆𝜆2𝑇𝑇𝑉𝑉(𝐼𝐼)� (4.18)
where TV is the total variation of the signal and 𝜆𝜆1, 𝜆𝜆2 are the sparsity and TV
regularisation penalties respectively [32]. However, accurate reconstruction of the
signalIis only achievable if the conditions of restricted isometry property and
incoherency are satisfied in compressive sensing.
Besides R1 reduction in BPE, we can also use an additional variable density (VD)
undersampling to randomly omit the BPE rows by a factor R2> 1, i.e. the rows of Dcan
be randomly subsampled by a factor R2 to acquire
𝐷𝐷𝑠𝑠 = 𝐸𝐸𝐷𝐷, (4.19)
using the undersampling matrix E. The Ds thus acquired can be used in the iterative CS
reconstruction given below to recover the aliased image𝐷𝐷� .
arg𝑚𝑚𝛾𝛾𝑚𝑚𝐷𝐷� ��𝐷𝐷𝑠𝑠 − 𝐷𝐷�𝐸𝐸�𝑑𝑑22
+ 𝜆𝜆1�ΨD��𝑑𝑑1 + 𝜆𝜆2�𝐷𝐷��𝑇𝑇𝑉𝑉� (4.20)
Where 𝜆𝜆1 and 𝜆𝜆1 are the weighting parameters that can be tuned to control the sparsity
and smoothness of the solution 𝐷𝐷� . The reconstructed aliased image data matrix D� is in
turn used in
𝐼𝐼 = 𝐶𝐶𝑇𝑇𝐷𝐷� (4.21)
to reconstruct the image I.
73
N×N image I, ∆Ky =1/FOVy, ∆Kx =1/FOVx, R1 > 1, r > 0
BPE reconstruction D:
Solve 𝐼𝐼 = 𝐶𝐶𝑇𝑇𝐷𝐷
Reduction Factor R2
(Variable down-sampling)
CS+ BPE reconstruction
using Ds:
arg𝑚𝑚𝛾𝛾𝑚𝑚𝐷𝐷� ��𝐷𝐷𝑠𝑠 − 𝐷𝐷�𝐸𝐸�𝑑𝑑22
+ 𝜆𝜆1�ΨD��𝑑𝑑1 + 𝜆𝜆2�𝐷𝐷��𝑇𝑇𝑉𝑉�
𝑆𝑆𝑆𝑆𝑙𝑙𝑆𝑆𝑒𝑒 𝐼𝐼 = 𝐶𝐶𝑇𝑇𝐷𝐷�
Figure 4.4: Schematic diagram of compressive sensing BPE
74
The physically acquired data in this approach isDs,with overall reduction 𝑅𝑅 = 𝑅𝑅1𝑅𝑅2.
The image I can be precisely reconstructed by the above described two step CS plus
BPE reconstructions. Therefore, we call this approach CSBPE. Shown schematicallyin
Figure 4.4.
4.3 Multichannel CSBPFE
For a multichannelCSBPFE system, multiple measurements of the signalare acquired
simultaneously from all independent channels. Each channel in the multichannelCSMRI
has a sensitivity map associated with the location of this channel with respect to the
object scanned. The acquired signal from N-receiver MRI system can be modelled as
follows,
𝐷𝐷𝑠𝑠𝑖𝑖 = E𝑆𝑆𝑖𝑖𝐷𝐷, i ∈ [1, L] (4.22)
where, 𝐷𝐷 is the BPE encoded k-space data set, S𝑖𝑖 = diag[γ𝑖𝑖𝑗𝑗]𝑗𝑗=1,2,…,𝑛𝑛 is the complex-
valued of sensitivity map matrixof theith receiver channel with γ𝑖𝑖𝑗𝑗 being the sensitivity
of the ith channel at the jth pixel of the vectorised image, 𝐷𝐷𝑠𝑠𝑖𝑖 is the subsampled data
acquired from the coil of the ith channel, and L denotes the number of the receiver
channels, Therefore, Dsi can be represented in a matrix form,
𝔇𝔇𝑠𝑠 ≔
⎣⎢⎢⎢⎢⎢⎢⎢⎡𝐷𝐷𝐴𝐴1
𝐷𝐷𝑠𝑠2
.
.
.𝐷𝐷𝑠𝑠𝐿𝐿⎦
⎥⎥⎥⎥⎥⎥⎥⎤
=
⎣⎢⎢⎢⎢⎢⎢⎡E𝑆𝑆1
E𝑆𝑆2
:..
ES𝐿𝐿⎦⎥⎥⎥⎥⎥⎥⎤
𝐷𝐷 =:𝐹𝐹𝐷𝐷(4.23)
As obtained from the above equation, the multichannel measurement matrix, denoted as
F for D, comes from the measurement matrix ES𝐿𝐿 of L column with dimension Lm×n.
75
The sub-matrices, ES𝐿𝐿 s, share a common measurement matrix E ∈ ℂ𝑚𝑚×𝑛𝑛 resulting
from the encoding process, so they measure the same I simultaneously with the same
under-sampling pattern and under-sampling ratio n/m. The under-sampling ratio is
defined by the number of PE lines used in data acquisition and it does not depend on the
number of channel L.
Sensitivity map SLis complex-valued and 𝑆𝑆𝑖𝑖 ≠ 𝑆𝑆𝑗𝑗for 𝛾𝛾 ≠ 𝑗𝑗 in general. Therefore, E𝑆𝑆𝑖𝑖 ≠
E𝑆𝑆𝑗𝑗for 𝛾𝛾 ≠ 𝑗𝑗 and they are independent base on the specific values of 𝑆𝑆𝑖𝑖 and 𝑆𝑆𝑗𝑗. Hence,
measurement matrix F can provide more individual measurement than single-channel
matrix 𝐸𝐸, and multichannel measurement can reduce the number of measurements, m,
which is essential for each channel for exact reconstruction of 𝐷𝐷� . Hence, the
multichannel CSBPE is considered for reconstructing the image I from the multichannel
measurement of MRI. The equation of the multichannel CSBPE MRI is given by
Arg𝑚𝑚𝛾𝛾𝑚𝑚 ��𝔇𝔇𝑠𝑠 − 𝐹𝐹𝐷𝐷��2𝑑𝑑2
+ 𝜆𝜆1�Ψ𝐷𝐷��𝑑𝑑1 + 𝜆𝜆2�𝑇𝑇𝑉𝑉�𝐷𝐷���𝑑𝑑1� (4.24)
where TV is the total variation of the signal and 𝜆𝜆1, 𝜆𝜆2 are the sparsity and TV
regularisation penalties respectively [32]. However, accurate reconstruction of the
signal 𝐷𝐷� is only achievable if the conditions of restricted isometry property and
incoherency are satisfied in compressive sensing.The compressive sensing reconstructed
aliased image data matrix 𝐷𝐷� are used in following equation to reconstruct the final
image I.
𝐼𝐼 = 𝐶𝐶𝑇𝑇𝐷𝐷� (4.25)
Daubechies-4 (db-4) wavelet are used all the simulation and reconstruction of this work,
because the performance of db-4 is superior in sparsifying the MR results. But here,
there is some issues to obtain high resolution precise image. In the next step, we will
discuss these issues and their effective solutions.
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4.4 Aliasing Coefficient Correction
In BPE imaging, the k-space trajectory for each RO line (x-direction) is modified to
make a variation to form the zigzag trajectories from the rectlinear trajectories. A
precise knowledge of the zigzag coordinate is needed in order to reconstruct artifact free
images from BPE k-space data. Butthe physically realizedzigzag trajectories are
generally not the same as the theoritically calculated ones due to gradient imprecision
and field inhomogeneity, which results in poor image quality. A solution to this problem
is to measure the actualcoordinates of the trajectories and use them to construct the
coefficient matrix C. However such measurements generally requiresprescan calibration
that complicates the process and increases operation cost.
To overcome this difficulty, we canuse a technique known as cross-correlation to
estimate theactual coordinates of BPE trajectories from standard scan data.Cross
correlation is a technique used in signal processing to measure the degree of similarity
and the Euclidean distance between two signals [90].
The basic concept of obtaining a quantitative measure of similarity between two signals
is fundamentally the same as statistical correlation of two arbitrary variables. However,
it is not quantified explicitly by statistical measures like convariance or standard
deviations. Generally, the quantative comparison between any two image signals or
waveforms may be based upon the amount of the component of one signal contained in
the other signal. Consider two signals x(m) and y(m)of finite energy. It is possible to
make a correlation sequence that would indicate the similarity of the two signals at
differents time intants. If the two signals are diffetrent from each other, the correlation
sequence is known as cross correletion and is represented as
𝜑𝜑𝑥𝑥𝑥𝑥(𝑙𝑙) = � 𝑥𝑥(𝑚𝑚− 𝑙𝑙) 𝑦𝑦 (𝑚𝑚)∞
𝑚𝑚=−∞
𝑙𝑙 = 0, ±1, ±2, … … (4.25)
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where l is the shift or lag parameter.If the length of the signal is not infinite, the length
of the correlation is not infinite either. In that case, the finite cross-correlation equation
is given as
𝜑𝜑𝑥𝑥𝑥𝑥(𝑙𝑙) = � 𝑥𝑥(𝑚𝑚 + 𝑙𝑙) 𝑦𝑦 (𝑚𝑚)𝑀𝑀
𝑚𝑚=1
𝑙𝑙 ∈ [0,𝑀𝑀 − 1] (4.26)
𝜑𝜑𝑥𝑥𝑥𝑥(𝑙𝑙) can be used to measure the shift between two signals. If 𝑦𝑦 (𝑘𝑘) = 𝑥𝑥(𝑘𝑘 + 𝜏𝜏)
where 𝑘𝑘𝑘𝑘[1,𝑀𝑀]and 𝜏𝜏 are constant, the shift 𝜏𝜏 between x(k) and y(k) can be found by
finding 𝑙𝑙∗ = 𝜏𝜏 such that
𝜑𝜑𝑥𝑥𝑥𝑥(𝑙𝑙∗) = max𝑑𝑑∈[1,𝑀𝑀]
𝜑𝜑𝑥𝑥𝑥𝑥�𝑙𝑙 � (4.27)
In the frequency domain, we can represent cross-correlationof 1D signalsas
Φ𝑥𝑥𝑥𝑥 = 𝐹𝐹𝑇𝑇�𝜑𝜑𝑥𝑥𝑥𝑥(𝑙𝑙)� = ℱ(𝑥𝑥).ℱ(𝑦𝑦) (4.28)
where ℱ denotes the Fourier transform. When M is large, computation of 𝜑𝜑𝑥𝑥𝑥𝑥(𝑙𝑙)cab be
very heavily. Computing 𝜑𝜑𝑥𝑥𝑥𝑥(𝑙𝑙) using ℱ -transform can speed up computation,
especially in MRI which requires complex valued computations. Cross correlation can
also be defined for 2D signals. For 2D signals x(m,n) and y(m,n), their cross correlation
is defined as
𝐶𝐶(𝑘𝑘, 𝑙𝑙) = � �𝑥𝑥(𝑚𝑚,𝑚𝑚) 𝑦𝑦 (𝑚𝑚− 𝑘𝑘,𝑚𝑚 − 𝑙𝑙)𝑁𝑁
𝑛𝑛=1
𝑀𝑀
𝑚𝑚=1
(4.29)
The cross-correlation of 2D signals can be computed efficiently in the spatial domain by
𝐶𝐶(𝑘𝑘, 𝑙𝑙) = ℱ−1 { ℱ (𝑥𝑥)ℱ∗(𝑦𝑦)} (4.30)
whereℱ is the Fourier transform. The complex conjugate of ℱ accomplished reversal of
the feature via the Fourier transform property ℱ𝑦𝑦∗(−𝑚𝑚) = 𝐹𝐹∗𝑦𝑦(𝜔𝜔).
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k-space
…… Image-
space
Figure 4.5: Baseline data and shifted data in the Zigzag trajectory of Bunched
phase encoding. The image from all PE baseline data on left and rest of all
images from shifted PE line data and RO line data for all four sets of image. The
whole 256 × 1028 BPE data image is showing on below.
79
Similarly in 1D case, the shift between x(m,n) and y(m,n) 𝑦𝑦(𝑚𝑚, 𝑚𝑚) = 𝑥𝑥(𝑚𝑚 − 𝜏𝜏𝑚𝑚 , 𝑚𝑚 −
𝜏𝜏𝑛𝑛) can be found by finding 𝑘𝑘∗ = 𝜏𝜏𝑚𝑚 and 𝑙𝑙∗ = 𝜏𝜏𝑛𝑛 such that
𝐶𝐶(𝑘𝑘∗, 𝑙𝑙∗) = max𝑘𝑘∈[1,𝑀𝑀]𝑑𝑑∈[1,𝑁𝑁]
𝐶𝐶 �𝑘𝑘, 𝑙𝑙 � (4.31)
We will use the cross correlation technique described above to determine the actual
zigzag trajectories of the acquired BPE k-space data.
In order to determine the individual data positional deviation of acquired BPE k-space
data from the reference calculated data position is decomposed into eight individualk-
space volumes of matrix size N x Ns by sorting the PE k-space lines into individual
matrices.Each k-space matrix represents the same aliased image with the entire k-space
shifted by the step sizedk(n). Figure 4.5 shows the acquired zigzag data orientation and
subsampled image.
Figure 4.6 shows the deviation along the PE direction as line plots along or near the
center ofk-space for each individual k-space matrix. The deviation in PE direction from
the cartesian RO k-spaceline can be computed from two matrices by cross-correlation.
The peak position of the cross correlationmatrix then corresponds to the pixel shift
between the two k-space matrices.
The deviation of the k-space data from the calculated zigzag trajectories can be
computed usingthe cross-correlation shown in Eq. (4.29).The peak position of the cross-
correlation matrix then corresponds to the pixel shift between the two k-space matrices.
That means we will find a maximum column value and a maximum row value. Then a
sub-pixel analysis of the cross-correlation matrices using a weighted centre of mass
approach can be applied to determine with a high sub-pixel resolution ofthe BPE k-
space shifts in every step. The deviaion of the matrices N =1: ncan be determined
relative to the baseline with N = 0. In the method section, we will elaborate the cross-
correlation technique to recover the precise zigzag coordinate of k-space.
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Figure 4.6: K-space lines on or near the k-space centre for eight
individual k-space volume of the matrix represents the same image
with the entire k-space shifted by the step size dk(n) and n=[0:7],
top: n=0 and bottom: n=7.
4.5 Smoothness Enhancing
There is another issue that we have observed: the smoothness of the reconstructed
image in compress sensing relies on the accuracy of the inverse Fourier transform of
shifted k-space data is known as modulation maps. Usually, data shift in k-space will
create a modulation in the image domain. Therefore, the proper estimation of
modulation maps is necessary for unaliasing and smoothing of the image. We can
mathematically represent modulation maps as,
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M(x𝑖𝑖, y𝑖𝑖) = ��𝐼𝐼(𝑘𝑘𝑚𝑚 ,𝑘𝑘𝑑𝑑)𝑒𝑒−𝑖𝑖𝜔𝜔𝑥𝑥𝑖𝑖(𝑘𝑘𝑚𝑚+𝛼𝛼∆𝑘𝑘𝑚𝑚)𝑒𝑒𝑖𝑖𝜔𝜔𝑥𝑥𝑖𝑖(𝑘𝑘𝑙𝑙+𝛼𝛼∆𝑘𝑘𝑙𝑙)𝑁𝑁
𝑑𝑑=1
𝑁𝑁
𝑚𝑚=1
（4.32)
where M(x𝑖𝑖 , y𝑖𝑖) is the image domain and 𝐼𝐼(𝑘𝑘𝑚𝑚 ,𝑘𝑘𝑑𝑑) is the frequency domain data set.
The incoherency in multichannel CS cannot be guaranteed in bunch phase encoding
because the encoding matrix is channel dependent and can vary from scan to scan. Thus,
an accurate estimation of phase modulation maps can solve this problem. We can
estimate thel-th coil modulation map, Ml,via
𝑀𝑀�𝑑𝑑 ≜ 𝑎𝑎𝑟𝑟𝑎𝑎𝑚𝑚 𝑚𝑚𝛾𝛾𝑚𝑚12‖𝑆𝑆𝑑𝑑 − 𝑑𝑑𝛾𝛾𝑎𝑎𝑎𝑎{𝑆𝑆𝑧𝑧𝑒𝑒𝑟𝑟𝑧𝑧}𝑀𝑀‖2 + 𝛽𝛽𝑅𝑅(𝑚𝑚)（4.33）
where𝛾𝛾 ∈ [1,2,3 … . . 𝐿𝐿l] and𝑅𝑅(𝑚𝑚) is a spatial roughness penalty function with weighting
factor β. The reference image,𝑆𝑆𝑧𝑧𝑒𝑒𝑟𝑟𝑧𝑧 can be obtained by taking the sum of squares or
geometric mean of individual coil images 𝑆𝑆𝑖𝑖’s. The modulation maps were estimated
from subsampled BPE image using the method of [78]. The phase of one coil’s image
can further be incorporated into this term to prevent inclusion of the underlying object’s
phase in the modulation map. The smoothness of the modulation map can be easily
improvedby varying β. Further, we have used a variable density randomly subsampled
technique in PE direction to reduce the data sample and reconstruct the image using CS
reconstruction technique, where we use individual channel reconstruction and add them
together to obtain the final image.In the Multichannel CS part, estimated modulation
maps are applied to improve the robustness of the image.
4.6 Methods
Both simulated BPE data and real BPE scan data were used in our studies. For
simulated BPE data, the in vivo data was used to simulate the BPE DAT. The in vivo
data was obtained from a 2D spin-echo brain scan of a healthy volunteer on 3T Skyra
(Siemens Healthcare, Erlangen, Germany) with the 32-channel head coil (FOV: 240
mm, Flip angle: 10o, image matrix: 256 x 256). The 256 x 256 k-space data were
interpolated to a 1024 x 1024 matrix, with the added data points to simulate the
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underlying continuous k-space within the original sampling grid. The interpolated data
matrix were zigzag-sampled, with R1 = 2, r = 1 (confer figure 4.2), and then variable
downsampled(VD) along rows by R2 = 2 to obtain the 64 x 1024 data matrix DS with
total reduction R = R1×R2 = 4. The DSthus obtained is used in the iterative CS
reconstruction in Eq.(4.25) to compute the image I. We also used variable density
random sampling CS with R = 2, 3,4 and FFT2 to reconstruct the images from the
original gradient trajectories k-space data.
Pulse sequence was designed to obtain real BPE data for the study of CSBPE. The pulse
sequence design of CSBPE method is implemented by minor modification of the
gradient waveforms of the standard sequence. In BPE, a zigzag gradient is incorporated
in the phase encoding direction during readout and the sampling rate is increased more
than double. Figure 4.7 shows a timing diagram of a sample BPE sequence.
In this study, it is considered that the reconstructed image size is N in Phase encoding
(PE) direction and N in Readout (RO) direction. That means a standard rectilinear
sampling method acquired N samples in each PE. If NBconsiders as a total number of
TR cycles in the BPE method, the reduction factor will be R = N/NB.
Figure 4.7: A sample pulse sequence design of bunched phase encoding; (Gse) Gradient in slice direction; (Gpe) Gradient in phase encoding direction; (Gro) Gradient in readout direction. A zigzag gradient is incorporated in the phase encoding direction during readout. The amplitude and period of the oscillatory
83
gradient in this figure are not to scale.
Figure 4.8:(a)Inverse Fourier transform of the acquired all 32 channel data gives the aliased image; (b) Inverse Fourier transform of the acquired channel 4 data
gives the aliased image; (c) k-space of channel 4.
Figure 4.9: Phase encoded data sampling technique; (a) Random sampling; (b) Variable density Random sampling.
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In this data acquisition technique, k-space data are acquired along a zigzag trajectory in
the PE direction. Also, four time’s oversampled data are taken in a readout direction.
The distance between two adjacent PE lines is R1Δky, where R1 is typically higher than
1. As shown inFigure 4-4, more data are sampled at a higher sampling rate than the
standard acquisition method during the same readout period. Inverse Fourier transform
of the acquired data gives the aliased image shown in Figure 4.8, which can be further
processed by the linear equation based (LEB) reconstruction technique to produce the
unaliased image.
Also, an aliased free image can be produced from the multi-coil data without using a
coil sensitivity map. A variable density random sampling pattern hasbeen used in the
phase encoding direction according to a Gaussian distribution function in this research
which is shown in Figure 4.9 because the most important information of the image is
preserved in the centre of k-space. The nonlinear iterative method (4.24)is solved to
reconstruct the aliased image for acceleration factors of 2, 4, 6, and 8. In this case, the
encoding matrix E does not have any sensitivity mapinformation (i.e. E = Ф).
It has been obseved in the experiment, that some trajectory deviationhasoccuredin the
physical trajectories compared to the calculated ones due to field inhomogeneity and
imprecise gradient.Such deviationhas developedan artifact in the reconstructed image as
shown inFigure 4.13 and Figure 4.14.
To solve the trajectory deviation of real data, a new recovery technique has been applied
to reconstructan artifact-free image. Suppose, a schematic diagram of BPE k-space data
isas shown inFigure 4.10, where the red dashed lines are the calculated (simulated)
trajectory, ×’s are the measured data positions. The solid black horizontal line in the
figure is actually an underlying rectangular baseline grid, which contains the baseline
data points shown by the black x’s.
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Figure 4.10: BPE k-space data acquisition. Ccalculated (simulated) trajectorywhich
shown in red marks (----);Original (measured) trajectory where data positions are
marked as (×’s, ×’s, ×’s, ×’s), baseline (zero shift) data points are marked as ×’s,
measured shifted data points are marked as ×’s, ×’s, ×’s.
As explained earlier in Sect 4.4, if the image signal of M is band limited, the other data
points, shown by the blue x’s, green x’s and red x’s can be regarded as the baseline data,
i.e. the black x’s shifted in the x and y directions. After applying BPE in MRI data
acquisition, all the data are shifted towards PE ditrection to form a zigzag pattern except
the black x’s which is consider as the baseline data of zigzag trajectory.The baseline
data points can be used as a reference to calculate the x and y directional shift.
For example, there are four sets of k-space data: black, blue, green and red as shown in
Figure4.10 and the black x’s areconsideredas a reference or baseline data.Using the
BPE acquisition technique some data are shifted to the different position so the data
alignment will be zigzag and mathematically the data position should be on red dash(---
) line. But, due to some factors, the data are shifted to different positions from the
mathematically calculated position as shown in Figure 4.10.
The 2D cross-correlation technique described in section 4.4 has been used to estimate
the correct data position of themeasured shifted data with respect to the baseline (black
x’s) data. In order to determine the individual k-space shifts, the whole k-space is
decomposed into eight individual k-space sub matrix size 256 x 128 by sorting the PE
k-space lines into individual matrices such that for N=[0 :7].The matrix kn includes
k=128 PE k-space lines kn=[n+1,n+2,…,n+k].Each 256×128 submatrix of the 256×1028
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data matrix represents the same aliased image but shifted in k-space by the step size
dk(n).
Figure 4.10shows that the data shift along the PE direction as line plots along or near
the centre of k-space for each individual k-space matrix. The data shift in PE direction
from the cartesian RO k-space line can be computed from two matrices by cross-
correlation. Hence, the data deviation from its original calculated position can be
determined by applying the cross-correlation technique.
The peak position of the cross-correlationmatrix then corresponds to the pixel shift
between the two k-space matrices. A sub-pixel analysis of thecross-correlation matrices
using a weighted centre-of-mass approach can be used to determine with highsub-pixel
resolution the BPE k-space deviation in every step. The shifts of the matrices N=1:7
weredetermined relative to the first with n=0. The sequential steps of the cross-
correlation technique is described in Figure 4.11.
Figure 4.12 shows a deviation between the Synthetic and the actual measured k-space
data position. The figure shows that calculated data follow the estimated sin(x) graph
but the measured data points are not symmetrical with the estimated sin(x) line.
Themathematically calculated k-space shifts were Kc = [0.0, 0.13021, 0.5, 0.86979, 1.0,
0.86979, 0.5, 0.13021], almost representing asinusoidal function.
The calculation of the cross-correlations was carried out for each matrix, N = 1...7,
withrespect to the first matrix, n=0 , and for every channel separately. The final values
were then averagedover all channels and resulted in the following k-space shifts Km
=[0.0, 0.1191, 0.5634, 0.8235, 0.8514, 0.7382, 0.1978, 0.0341]. These also
approximately followed a sinusoidal function.These parameters are used to reducethe
artifacts in the image reconstruction using the linear equation, but the result for
suppression of the artifactswas not optimal.
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Figure 4.11: The sequential steps of Aliasing Coefficient Correction by cross correlation technique
Step 1• Load the BPE k-space data ( Data array size: 256×1024×Number of
Channel).
Step 2• Split k-space data in 8 zigzag sub k-spaces for each channel.• Puts those data into the individual k-space volumes.
Step 3
• Calulate the offset in k-space between w.r.t the first zero shift k-space loop over number of zigzag steps upto 8.
• Select the central part of k-space with the peak.
Step 4
• Calculate the cross correlation between zero shift or baseline k-space with rest of 7 shifted k-space images. The resulting matrix can be analysed for a shift between the k-space images by finding the location/ index of its peak.
Step 5 • Observed near the center or k-space to gain sub-pixel resolution precision.
Step 6• Calculate weighted center of mass• Calculate deviation from reference zero shift.
Step 7• Calculate average shift for all 32 channel.• Results
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Figure 4.12:Calculated (red) and measured (blue) BPE k-space positions.
The data were shifted to be symmetrical around zero PE shift.
Table 1: Calculated trajectory vs corrected trajectory of all 32 channel.
Calculated
Trajectory
Corrected
trajectory
(Ch 1)
Corrected
trajectory
(Ch 2)
………..
Corrected
trajectory
(Ch 32)
Corrected
trajectory
(Mean)
0.00000 0.00000 0.00000 ………. 0.00000 0.00000
0.13021 0.12665 0.11766 ……….. 0.11721 0.1191
0.50000 0.32806 0.72221 …………. 0.60141 0.5634
0.86979 0.83531 0.87044 …………. 0.7866 0.8235
1.00000 0.86801 0.90716 …………. 0.81606 0.8514
0.86979 0.75521 0.78237 …………. 0.71495 0.7382
0.50000 0.23095 0.17618 ………… 0.24334 0.1978
0.13021 0.053406 0.030958 …………. 0.037712 0.0341
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It was found empirically that an additional multiplicative factor had to be applied in
order to optimize the image qualities. This factor had to be chosen such that the peak-to-
peak amplitude of the measured parameters would result in a value of 1√2
as opposed to a
value of 1 as set for the synthetic data. In Figure 4.12, both data sets were positioned
symmetrically around the shift value S=0. Estimated fits of sine functions to the two
data sets are shown for a better comparison of the data.
Figure 4.13 and Figure 4.14show reconstructed phase maps and images respectively
using the two parameter sets calculated BPE shifts, Kc , and measured and adjusted BPE
shifts, Km. The images reconstructed with the measured and adjusted BPE shifts show
sharply reduced artifacts compared to the images reconstructed with the calculated BPE
shifts and appear to be close to artifact free in PE direction as can be seen in the single
channel phase images Figure 4.13, right. Remaining artifacts in RO direction are weak
and might be specific to other parameters of the BPE method.
Figure 4.13: BPE image reconstruction (channel 4), phase images, using the
synthetic BPE k-space PE shifts, Ks (left) and using the measured and corrected
BPE k-space PE shifts, Kc (right).
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Figure 4.14: BPE image reconstruction, all channels combined, R = 2, matrix 128 x 128, using the synthetic BPE k-space PE shifts, Ks (left), and using the measured and
corrected BPE k-space PE shifts, Kc (right).
Now, using the trajectory corrected BPE data and the coefficient matrix C, we can
reconstruct the image I.The 2D cross-correlation technique is a useful and reliable
method for estimating the relative shifts of 2D signals. It can be performed by
convolution and also by Fourier transform for fast computation.
There was another issue that has been observed in compress sensing part which is
responsible for some aliasing and roughness in the output image. In BPE, the phase-
encode lines are regularly under-sampled by a reduction factor of R1 (= 2). After
obtaining k-space data through BPE, we can estimate modulation maps using equation
(4.33) and that maps can be used in the CS part where the phase-encoded lines are
undersampled by the reduction factor R2 = 2, 3, 4 etc.But in compressed sensing part
under sampling should be variably dense because most of the image information
preserved in of k-space. Thus, the overall reduction factor will be R=R1×R2= 4, 6, 8.
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4.7 Simulations and Experimental Results
We have categorized our study into two stages:(i) A simulation study with a
multichannel data using Fourier based CSBPE reconstructiontechnique and then
compare with CSMRI reconstruction. Simulations were performed on a (256 × 256×8)
multichannelbrain image to study the performance of CSMRI using Fourier based
bunched phase encoding technique. (ii) Anexperimental study with a multichannel
encoded data usingFourier based CSBPE reconstructiontechnique and then compare
with multichannel CSMRI reconstruction image.The experimental study was
implemented on a (256 × 256×32) multichannel brain image. Before present the result
we will observe some results after using correction technique in different stage of
CSBPE MRI technique.
4.7.1 Multichannel CSBPE Simulation
The simulation was performed using a MATLAB R2017b software and a workstation of
Intel ® Xeron ® CPU E5-16600 @ 3.30 GHz, 64 bit Operating System, and 64.0 GB
memory. In addition, the Rice_Wavelet_Toolbox_2.4 is used to simulate the
compressed sensing reconstruction. A brain image (size: 256 × 256× 8) was used to
analyze the performance of Fourier encoding in CSMRI and CSBPE MRI for different
acceleration factors. The simulation were perform by using eight complex sensitivity
maps which are obtained from the head coil of a Siemens Skyra 3T scanner shown in
Figure 4.15. The multichannel data was simulated by calculating the phase shift
position in CSBP encoding and reconstruct the image using equation (4.23) and (4.24)
with a differentdownsampling factor.
Quantitative analysis of the relative error in the CSBPE reconstructed image and
CSMRI reconstructed image is defined as a metric:
𝑅𝑅𝑒𝑒𝑙𝑙𝑎𝑎𝑡𝑡𝛾𝛾𝑆𝑆𝑒𝑒 𝑒𝑒𝑟𝑟𝑟𝑟𝑆𝑆𝑟𝑟 =‖𝑥𝑥0−𝑥𝑥�‖𝑙𝑙2‖𝑥𝑥0‖𝑙𝑙2
(4.25)
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Figure 4.15: The coil Sensitivity maps used in CSBPE simulation. The sensitivity maps are estimated from data acquired on MR scanner
The graph shows in Figure 4.16, the mean relative error of CSBPE is less than the
CSMRI reconstruction method at difference reduction factor. There is another way to
calculate the image reconstruction error, which is calculated by artifact power (AP)
method based on the following equation:
𝐴𝐴𝑃𝑃 = �∑𝑁𝑁𝑧𝑧𝑧𝑧=1 ∑ ∑ �𝑥𝑥𝑥𝑥𝑥𝑥𝑧𝑧 −𝑥𝑥𝑥𝑥𝑥𝑥𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟�
2𝑁𝑁𝑥𝑥𝑥𝑥=1
𝑁𝑁𝑥𝑥𝑥𝑥=1
𝑁𝑁𝑧𝑧 ×𝑁𝑁𝑥𝑥×𝑁𝑁𝑥𝑥 (4.26)
where 𝑥𝑥𝑥𝑥𝑥𝑥𝑧𝑧 is the CSBPE reconstructed image data and 𝑥𝑥𝑥𝑥𝑥𝑥𝑧𝑧𝑟𝑟𝑒𝑒𝑟𝑟 is the CSMRI
reconstructed data. Nx is the total number of frequency encodinglines, Ny is the total
number of phase encoding lines, and Nz is the totalnumber of slices. The artifact power
of CSBPE MRI at different reduction factor is quite better than general CSMRI, which
is present in table 2.
93
Figure 4.16: The mean relative error versus the acceleration factor in CS-
MRI, CSBPE for simulation data.
Table 2: Comparison of the artifact power (AP) between proposed CSBPE-MRI methods and the reference CS-MRI in a simulated brain image at
different acceleration factor (R).
Acceleration factor (R)
Artifact Power (AP)
CSBPE-MRI CS-MRI
R2 0.0052 0.0116
R4 0.0113 0.0137
R6 0.0128 0.0172
R8 0.0226 0.0264
Figure 4.17 and Figure 4.18 show the simulation results of Fourier based compressed
sensing reconstruction using BPE data. Figure 4.18 is magnitude image and Figure 4.18
is phase image respectively. The BPE acquired data images are all aliased images which
has obtained before the CS reconstruction and the middle column show the aliased
images after CS reconstructionat reduction factor R = 4, 6, 8 and 10 respectively. Right
column shows the error images of Fourier based CS reconstruction for acceleration
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 2 4 6 8 1 0 1 2 1 4
MEA
N R
ELAT
IVES
ERR
OR
FOR
1000
TR
IALS
ACCELERATION FACTOR
cs
CSBPE(Fourier)
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factors of 4, 6, 8 and 10 respectively; The result show the CS reconstructed images from
BPEdata can preserve image resolution.
Before CS
reconstruction
(Magnitude)
CS reconstructed images (Magnitude)
Error images (Magnitude)
(a)
R = 4
(b)
R = 6
(c)
R = 8
(d)
R = 10
Figure 4.19: Simulation results of Fourier based compressed sensing reconstruction using BPE data; (a)-(d) (left two column) comparing the BPE acquired data image and CS reconstructed images at reduction factor R = 4, 6, 8 and 10 respectively; (a)-(d) (right column) error images of Fourier based CS reconstruction for acceleration factors of 4, 6, 8 and 10 respectively; The result show the CS reconstructed images from BPEdata can preserve image resolution.
95
Before CS
reconstruction
(Phase)
After CS reconstructed images (Phase)
Error images (Phase)
(a)
R = 4
(b)
R = 6
(c)
R = 8
(d)
R = 10
Figure 4.20: Simulation results (phase image)of Fourier based compressed
sensing reconstruction using BPE data; (a)-(d) (left two column) comparing the
BPE acquired data phase image and CS reconstructed phase images at reduction
factor R = 4, 6, 8 and 10 respectively; (a)-(d) (right column) error images of
Fourier based CS reconstruction for acceleration factors of 4, 6, 8 and 10
respectively; The result show the CS reconstructed images from BPE data can
preserve image resolution.
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CSMRI CSBPE-MRI Reconstructed Error Reconstucted Error
Figure 4.21: Simulation results for comparing the conventional CS and Fourier
based CSBP encoding technique. (a) reference image; (b)-(d) reconstructed images
ofFourier based CS for acceleration factors of 4, 6, and 8 respectively; (e)-(g) error
images of Fourier based CS for acceleration factors of 4, 6, and 8 respectively; (h)-
(j) reconstructed images ofFourier based CSBP encoding for acceleration factors
of 4, 6, and 8 respectively; (k)-(m) error images of Fourier based CSBP encoding
for acceleration factors of 4, 6, and 8 respectively; The result of reconstructed
images of CSBPE outperforms the CS for preserving image resolution.
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Figure 4.19shows the reconstructedimages using Fourier encoded CS and CSBPE for
the acceleration factors 4, 6 and 8 respectively. The simulation results indicate that
Fourier based CSBP encoding technique is more capable of preserving resolution than
Fourier encoded CS at a higher reduction factor. These results suggest that CSBPE is
better in terms of preserving the resolution of the reconstructed images from under
sampled k-space data.
4.7.2 Multichannel CSBPE Experiment The experiments were performed on the phantom and the brain image datasets to
validate the proposed method. The data set was acquired on a Siemens Skyra 3T MRI
scanner using 32 channel head coils with a maximum gradient strength of 40 mT/m and
a maximum slew rate of 200 mT/m/sec. A256 points RF excitation pulse have used.
The flip angle was 10° calculated by the equal integral rule, with the SAR level checked
by Siemens’ RF pulse programming software IDEA to be well below the safety limit
and about 5% higher than that of Fourier encoding RF pulse. We also acquired the
Fourier encoded data using the spin echo (SE) sequence to compare the quality of the
reconstructed image from the data acquired by the bunched phase encoding sequence.
A slice selective sinc RF excitation pulse was used in the spin echo sequence with a flip
angle of 10°. The protocol parameters of the bunched phase encoding of the Fourier
encoding SE sequence were in vivo experiments FOV = 240 mm, TE/TR = 26/750 ms,
image matrix = 256 × 256. In general,bunched phase encoding is sensitive to field
inhomogeneity, but careful design of the sequence and correcting phase shifting which
has discussed in section 4.5 can result in high-quality images. To reconstruct the
bunched phaseencoded data, the inverse Fourier transform was taken along both phase
and frequency encoding axis.
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Figure 4.22: Comparison between with and without aliasing coefficient
correction of reconstructed MRI images from BPE acquired in vivo k-space
data; (a) reference image; (b) Reconstructed MRI image using calculated
(simulated) trajectories; (c) Reconstructed MRI image after aliased coefficient
correction using cross-correlation technique; (d) and (e) are error images. The
maximum error in the calculated trajectory is 0.2095 and in corrected trajectory
is 0.0423.
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Figure 4.23 : Comparison between CSBPE image reconstruction using
estimated modulation maps and non-estimated modulation maps; (a)
Original image (b) CSBPE image reconstruction without an estimated
modulation map at reduction factor R = 4, (c) CSBPE image
reconstruction using the estimated modulation map at reduction factor
R = 4. (d) and (e) are error images.
100
Figure 4.20 shows the comparison between reconstructed image with and without
aliasing coefficient correction after obtaining BP encoded data. The reconstructed image
without correction shows ghost and noise in the image but after correction most of the
noise and ghost has been disappear. The image (d) and (e) shows the error image. The
maximum error in the calculated trajectory is 0.2095 and in corrected trajectory is
0.0423
Similarly,Figure 4.21(b)also shows some ghost and noise in the reconstructed image
when bunched phase encoding technique are applied before randomly subsampling for
CS reconstruction. However, we can remove noise in image and make a smooth image
using estimated modulation maps using equation (4.33). InFigure 4.21(c) shows that the
reconstructed image using estimation modulation has no ghost image and the image is
sharper than the conventional reconstruction technique.
4.7.3 Phantom Scan Data Experiment
Figure 4.22 shows experimental results of Fourier based CSBPEphantom data and the
reconstructed images for the different subsampling of experimental phantom data set
using CSBPE technique. The results show that the CSBPE reconstructed image using
high reduction factor is possible and this technique can produce artifact-free images.The
left column top imagerepresentsthe reference image and rest of the left column shows
reconstruction images using Fourier based CSBPE for down sampling R = 2, 4, 6, 8
respectively and the right column show the error images. The error for the reduction
factor 2, 4, 6 and 8 are 0, 0.021, 0.3637, and 0.3867 respectively. Phantom results are
confirmed that Fourier based CSBPE can preserved resolution of image better that
CSMRI reconstructed images
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Reference Image
Reconstructed Image Error image Reduction Factor
R = 2
R = 4
R = 6
R = 8
Figure 4.24: Experimental results of Fourier based CSBPEphantom data. The left column top imagerepresentsthe reference image and Fig. Rest of the left column shows reconstruction images using Fourier based CSBPE for down sampling R =
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2, 4, 6, 8 respectively and the right column show the error images. The error for the reduction factor 2, 4, 6 and 8 are 0, 0.021, 0.3637, and 0.3867 respectively.
Figure 4.25: Experimental results of Fourier based CSBP encoding. The left
column represents the reference image and Fig. (a) show image reconstruction
usingFourier based CSBPE for down sampling R = 4 and Fig. (b) – (c) show
reconstruction imagesof Fourier based CSBPencoding for acceleration factors of R
=6, 8 respectively and Fig.(d) -(f) (right Column): show the error images. The error
for the reduction factor 4, 6 and 8 are 0.021, 0.478 and 0.523 respectively.
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4.7.4 Brain in vivo Scan Data Experiment
Figure 4.23 shows the reconstructedimages from the subsampledbunched phase
encoded data sets. These results demonstrate that the bunched phaseencoding
reconstructions can produce artifact-free images. After retrospective under-sampling,
the unconstrained optimization program equation (4.23) is solved using the non-linear
conjugate gradient method to reconstruct the desired image for different acceleration
factors.
Figure 4.23 (a) -(c)shows the reconstructed images for the acceleration factors of 4, 6
and 8 on the CSBPE data while Figure 4.23 (d) – (f)shows the corresponding error
images. The relative errors at reduction factor 4, 6 and 8 are 0.021, 0.478 and 0.523,
respectively.
4.8 Computation Complexity of CSBPE as Compared
with CSMRI.
Roughly speaking, the computation of CSMRI is mainly that of the CS iterative
reconstruction algorithms (4.22)-(4.24), while CSBPE needs to use both (4.22)-(4.24)
and (4.25) in image reconstruction. The additional BPE unpacking computation in
(4.25) does increase computation complexity and hence computation time. But this is
the price we have to pay in order to accelerate imaging speed and enhance image
quality. MR image reconstruction is generally performed offline due the physical
constraints in MR signal generation and data acquisition process. The extra computation
time incurred by more complex computation is therefore not a major problem. In the
clinic applications where time critical image reconstruction is needed, the computation
can be accelerated by modern parallel computing hardware, e.g. GPU (Graphics
Processing Unit) array, which is now widely available on state-of-the-art MRI scanners.
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4.9 Conclusion
This chapter haspresented a new methodto accelerate data acquisition for MR
imaging.This method uses a novel compressed sensing approach in Fourier based
zigzag sampling of k-space and a new way to remove the artifactand noise from
physical scanning data at an acceleration factor up to 8. Fourier encoded CSMRI has
previously been used only for magnitude images. In the proposed method, both phase
and magnitude information is utilised in the reconstruction process.
Here, we have used the cross-correlation technique to estimate theexact coordinates of
non-rectangular periodic trajectories from standard scan data and reconstruct a high-
quality image from ZIGZAG in vivo data without prescan. Also, we have used the
estimatedmodulation map to smoothing the image and remove aliasing in compressed
sensing reconstruction stage.
The simulation and experimental results demonstrate that the new acceleration and
artifact removing techniquesof CSBPE-MRI outperform the conventional Fourier
encoded CSMRI. The encoding and reconstruction technique of the proposed CSBPE
scheme preserves the spatial resolution far better than the conventional Fourier encoded
CSMRI scheme. The proposed method is applicable to general non-rectangular periodic
trajectories. This CSBPE method has a significant potential for MRIacceleration.
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Chapter 5 Compressive Sensing MRI using Non Fourier Based Bunched Phase Encoding Compressive Sensing MRI using Non-Fourier based Bunched Phase Encoding
5.1 Introduction
There are three continuous steps used for data acquisition in MRI. RF pulse excitation is the
first step, which is typically a constant excitation of the field of view (FOV), and encoding
typically Fourier encoding is the second stage which is achieved by the imaging gradients. At
the last stage, data are acquired by analogue to digital converter. In fact, some specific case, a
selective RF pulse can also be used to perform excitation and encoding simultaneously [91,
92]. The spatially selective RF pulse was used in different wavelet-based encoding, and SVD
encoding was applied to increase the speed of MRI scans[93, 94].
Non-Fourier encoding technique can be combined with the parallel imaging method as
second steps, which relies on the sensitivity information of the receiver coil [95]. Besides, the
performance of the compressive sensing depends on the incoherence [30] between the
measurement domain and the sparsifying transform matrices some non-Fourier encoding
methods such as Noiselet encoding [96], random encoding [33, 97, 98], and chirp encoding
[99, 100]have better incoherence than Fourier encoding. Among them, the chirp modulation
technique has good strength to preserve incoherency compare to others technique.
Basically, chirp modulation can spread the k-space signal energy throughout the region k-
space, whereas Fourier encoding only concentrates the signal energy in the low-frequency
region. Chirp modulation can modify the sensing matrix and also can be used as a
controllable spread spectrum technique to achieve energy spread. Recently, there are some
non-Fourier encoding schemes such as Gaussian random encoding, Toeplitz random
encoding have been proposed for compressive sensing MRI have been investigated for
106
incoherent sampling. But those methods are challenging to implement and requires long RF
pulse and unrealistic computation power and storage memory for reconstruction [33, 97,
101]. In our work, we have applied the chirp modulation technique due to the better
incoherency performance for CSMRI. In the CSBPE method, the use of chirp modulated RF
pulses to spread the energy along the phase encoding direction by exciting the scanned object
along phase encoding direction with different profile at each excitation.Therefore, application
of non-Fourier encoding to CSBPE is expected to outperform conventional Fourier encoding.
5.2 Chirp Modulationbased CSBP Encoding
In paper [99]the authors suggested a new spread spectrum procedure using Chirp modulation.
Spread spectrum method uses second-order shim coils to reform the radio frequency pulse. In
order to distribute the signal energy in the k-space and allow for incoherent sampling, a
spread spectrum technique is also presented to modify the encoding matrix. Chirp radio
frequency (RF) modulation can be used as a controllable spread spectrum method to achieve
an energy spread [96, 99, 100, 102].To reduce the reconstruction error for unsystematically
sampled k-space. The Fourier encoded MRI signal can be described as,
𝑌𝑌 = F𝑝𝑝𝑒𝑒I 𝐹𝐹𝑟𝑟𝑧𝑧 (5.1)
where Yis the acquired k-space matrix, I is the desired 2D image matrix; Fpe is the encoding
matrix in phase encoding direction and Fro is the encoding matrix in readout direction.
Inconventional MRI dataacquisition,Fpe and Fro are the Fourier transform matrices in the PE
and RO direction, respectively. Here Fpedata matrix is replaced by chirp modulated encoding
matrix Fc, which is defined as:
𝐹𝐹𝑐𝑐 = φ𝐹𝐹𝑝𝑝𝑒𝑒 (5.2)
where𝜑𝜑 is used to modulate the Fourier transform with its diagonal entry given as:
𝜑𝜑(𝑟𝑟, 𝑟𝑟) = 𝑑𝑑𝛾𝛾𝑎𝑎𝑎𝑎 [ 𝑒𝑒−𝑖𝑖�Δ𝑐𝑐𝑟𝑟2+Δ𝑐𝑐�] (5.3)
107
where ∆c is the strength of the Chirp modulation, and r ∈ [0,1, … . N], ris the location of the
acquired point along the direction of encoding [103].The Fourier transform of each row of the
chirp encoding matrix Fc, constitutes a unique RF pulse that is subsequently applied along
the primary PE direction during data acquisition. After that, a non-linear conjugate gradient
iteration has used to reconstruct images from under sampled k-space data for different
acceleration factors.
Chirpmodulation spreads the energy of the k-space along the phase encoding direction. This
thesis proposesa random subsample the k-space along the phase encoding direction with a
uniform probability using chirp modulation to increase the sampling incoherency and hence,
increase the image quality of thereconstructed image of the CSBPE technique.
Figure 5.1: Sequential steps of Chirp modulated CSBPE image reconstruction process
BPE Pulse Sequence Design
BP Encoded MRI data acquisition
Chirp modulation of BPE data
subsampled K-space data
Further VD subsampling
Aliasing coefficient correction
CS Reconstruction
Image reconstruction
from BP Encoded MRI
data
SOS of all channel
Output image
108
In this research, we have used the chirp modulation method to modify the bunched phase
encoded k-space energy. Therefore, the chirp modulation allows spreading the energy of the
bunched phaseencoded k-space along the phase encoding direction.Figure 5.1 shows the
sequential steps of chirp modulated CSBPE image reconstruction process.
According to equation (4.7) in chapter 4, the chirp modulated BPE equation will be
𝐼𝐼 = 𝐶𝐶𝑇𝑇𝐷𝐷𝑐𝑐 (5.4)
where 𝐷𝐷𝑐𝑐 ∈ 𝕔𝕔𝑀𝑀 is bunched phase and chirp modulated Fourier base (CMFB) encoded k-
space dataset𝐷𝐷𝑐𝑐 = 𝐹𝐹𝑐𝑐𝐷𝐷 ,𝐶𝐶𝑇𝑇 ∈ 𝕔𝕔𝑁𝑁×𝑁𝑁 is the appropriate chirp modulated co-efficient matrix
constructed based on the sample co-ordinates. Besides R1 reduction in chirp modulated BPE,
we can also use an additional random undersampling to randomly omit the chirp modulated
Fourier based BPE rows by a factor R2> 1, i.e. the rows of 𝐷𝐷𝑐𝑐 can be randomly subsampled
by a factor R2 to acquire
𝐷𝐷𝑠𝑠 = 𝐸𝐸𝐷𝐷𝑐𝑐 (5.5)
where E is the undersampling matrix. The Ds thus acquired can be used in the iterative CS
reconstruction given below to recover the aliased image 𝐷𝐷�𝑐𝑐 . Atypical chirp modulated
CSBPE-MRI reconstruction attempts to solve
arg𝑚𝑚𝛾𝛾𝑚𝑚𝐷𝐷�𝑐𝑐 ��𝐷𝐷𝑠𝑠 − 𝐸𝐸𝐷𝐷�𝑐𝑐�𝑑𝑑22
+ 𝜆𝜆1�ΨD�𝑐𝑐�𝑑𝑑1 + 𝜆𝜆2�𝐷𝐷�𝑐𝑐�𝑇𝑇𝑉𝑉� (5.6)
where 𝜆𝜆1 and 𝜆𝜆1 are the weighting parameters that can be tuned to control the sparsity and
smoothness of the solution 𝐷𝐷�𝑐𝑐 . The reconstructed aliased image data matrix 𝐷𝐷�𝑐𝑐 is in turn
used in
𝐼𝐼 = 𝐶𝐶𝑇𝑇𝐷𝐷�𝑐𝑐 (5.7)
to reconstruct the image I.
109
(a)
(b)
(c)
Figure 5.2: (a) Fourier modulated k-space(Mesh view), (b)Chirp modulated k-
space(Mesh view), (c) Chirp modulated k-space
110
5.3 Chirp Modulation ofMultichannel CSBPE
For a multi-receiver MRI system, multiple measurements of the signal can be acquired
simultaneously from different independent channels. Each channel in the multi-receiver MRI
has a sensitivity map associated with the location of this channel with respect to the object
scanned. The acquired signal from N-receiver MRI system can be modelled as follows,
𝐷𝐷𝑐𝑐𝑠𝑠𝑖𝑖 = E𝑆𝑆𝑖𝑖𝐷𝐷𝑐𝑐 , i ∈ [1, L] (5.8)
where,𝐷𝐷𝑐𝑐 is the chirp modulated BPE encoded k-space data set,S𝑖𝑖 = diag[γ𝑖𝑖𝑗𝑗]𝑗𝑗=1,2,…,𝑛𝑛 is the
complex-valued of sensitivity map matrixof theith receiver channel with γ𝑖𝑖𝑗𝑗 being the
sensitivity of the ith channel at the jth pixel of the vectorised image, 𝐷𝐷𝑐𝑐𝑠𝑠𝑖𝑖 is the chirp
modulated subsampled data acquired from the coil of the ith channel, and L denoted as the
number of the receiver channels, Therefore, Dsi can be represented as a matrix form,
𝔇𝔇𝑐𝑐𝑠𝑠 ≔
⎣⎢⎢⎢⎢⎢⎢⎢⎡𝐷𝐷𝑐𝑐𝑠𝑠1
𝐷𝐷𝑐𝑐𝑠𝑠2
.
.
.𝐷𝐷𝑐𝑐𝑠𝑠𝐿𝐿⎦
⎥⎥⎥⎥⎥⎥⎥⎤
=
⎣⎢⎢⎢⎢⎢⎢⎡E𝑆𝑆1
E𝑆𝑆2
:..
ES𝐿𝐿⎦⎥⎥⎥⎥⎥⎥⎤
𝐷𝐷𝑐𝑐 =:𝐹𝐹𝐷𝐷𝑐𝑐(5.9)
As obtained from the above equation, the multichannel measurement matrix denoted as F for
𝐷𝐷𝑐𝑐, which is come from the measurement matrix ES𝐿𝐿 of L column and dimension is Lm×n.
The sub-matrices, ES𝐿𝐿s, share a common measurement matrix E ∈ ℂ𝑚𝑚×𝑛𝑛 resulting from the
encoding process, so they measure the same 𝐷𝐷𝑐𝑐 simultaneously with the same under-
sampling pattern and under-sampling pattern ratio n/m. The under-sampling ratio is defined
by the number of PE lines used in data acquisition and it does not depend on the number of
channel L.
111
Sensitivity map SLis complex-valued and 𝑆𝑆𝑖𝑖 ≠ 𝑆𝑆𝑗𝑗for 𝛾𝛾 ≠ 𝑗𝑗 in general. Therefore, E𝑆𝑆𝑖𝑖 ≠ E𝑆𝑆𝑗𝑗for
𝛾𝛾 ≠ 𝑗𝑗 and they are independent base on the specific values of 𝑆𝑆𝑖𝑖and𝑆𝑆𝑗𝑗. Hence, measurement
matrix F can provide more individual measurement than single-channel matrix 𝐸𝐸, and
multichannel measurement can reduce the number of measurements, m, which is essential for
each channel for exact reconstruction of 𝐷𝐷�𝑐𝑐. Hence, the multichannel CSBPE is considered
for reconstructing the aliased image 𝐷𝐷�𝑐𝑐 from the multichannel measurement of MRI. The
equation of the chirp modulated multichannel CSBPE MRI can denote as
arg𝑚𝑚𝛾𝛾𝑚𝑚 ��𝔇𝔇𝑐𝑐𝑠𝑠 − 𝐹𝐹𝐷𝐷�𝑐𝑐�2𝑑𝑑2
+ 𝜆𝜆1�Ψ𝐷𝐷�𝑐𝑐�𝑑𝑑1 + 𝜆𝜆2�𝑇𝑇𝑉𝑉(𝐷𝐷�𝑐𝑐)�𝑑𝑑1� (5.10)
where TV is the total variation of the signal and 𝜆𝜆1, 𝜆𝜆2 are the sparsity and TV regularisation
penalties respectively [32]. However, accurate reconstruction of the signal 𝐷𝐷�𝑐𝑐 is only
achievable if the conditions of restricted isometry property and incoherency are satisfied in
compressive sensing.Hence , the final image I can be reconstructed using the following
equation,
𝐼𝐼 = 𝐶𝐶𝑇𝑇𝐷𝐷� 𝑐𝑐 (5.11)
Daubechies-4 (db-4) wavelet are used all the simulation and reconstruction of this work. The
performance of db-4 is superior in sparsifying the MR results.
5.4 Chirp Modulated CSBPE Sampling Pattern The energy of Fourier encoded signal concentrate in the centre of the k-space and it is not
universal. So Fourier encoding is weakly incoherent with some sparsifying transform such as
Daubechies-4 (db-4) wavelet. In CSBPE scheme the variable density probability distribution
function is used to design a pattern for sampling the k-space that the low spatial frequency
always has to be fully sampled. However, sampling pattern using variable density probability
distribution function unable to adequate sampling of the high spatial frequency at high
acceleration factor. But, the proposed chirp modulated encoding method is designed to
optimally spread the energy of the signal along the desired under-sampling direction and
hence allowing the uniformly random sampling.Figure5.3 shows that the Fourier encoded
variable density sampling pattern and the chirp modulated random sampling pattern which
energy is spread along the phase direction.
112
Figure 5.3: (a) shows that the Fourier encoded variable density sampling
pattern and (b) shows the chirp modulated random sampling pattern which
energy is spread along the phase direction.
5.5 Simulation and Experimental Results Simulations were performed to observethe proposal of chirp encoded CSBPE encoding
scheme in the multi-channel CS-MEI framework. Figure 5.2shows the acquired k-space data
for chirp modulated k-space. It is proved that the energy is spread out in PE direction for the
chirp encoded data, and it increases with chirping factor. The characteristic of energy
spreading of chirp modulation is a favourable condition for CS data acquisition and
reconstruction. Simulations of CSBPE using chirp modulated Fourier base was implemented
on a (256 × 256) brain image to study the performance of CS-MRI using chirp encoded based
BPE. The simulation study was divided into two parts: a simulation study with a multi-
channel chirp modulated BPE data set,and an experimental study with multiple channels
chirp modulated BPE data set, where the sensitivity profiles wereestimated from the
reference image and calculated phase modulation maps for BPE reconstruction.
A chirp modulation of the image is taken in the phase encoding direction to obtain chirp
encoding. Anabsolutely random sampling pattern need to use to sample the chirp encoded
data in the phase encoding direction. A non-linear solution of Eq. (5.11) need to solve to
reconstruct the final image for acceleration factors of 2, 4, 5.3, 6, and 8. Here, the encoding
matrix E does not have any sensitivity information (i.e. E = Ф).
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Simulations were performed on a 256×256×8complex brain image. The brainimage data for
Fourier encoding was simulated by variable density random under-sampling and taking the
2D Fourier transform of the images. While for simulating chirp modulation encoded k-space
data, a chirp transform was taken in phase directionfollowed by a Fourier transform in
readout direction. The pseudo-random under samplingscheme that is known to be optimal for
the Fourier encoding scheme was performed to under sample the Fourier encoded data. While
complete random under sampling was performed for chirp encoded data. Equation (5.11) was
solved using non-linear conjugate gradient iterations to reconstruct images from under
sampled k-space data for different acceleration factors.
The quantitative performance of both the encoding schemes can determine by used the
relative error defined as a metric:
𝑅𝑅𝑒𝑒𝑙𝑙𝑎𝑎𝑡𝑡𝛾𝛾𝑆𝑆𝑒𝑒 𝐸𝐸𝑟𝑟𝑟𝑟𝑆𝑆𝑟𝑟 = ‖𝑥𝑥0−𝑥𝑥�‖𝑙𝑙2‖𝑥𝑥0‖𝑙𝑙2
Figure 5.4The mean relative error versus the acceleration factor in CS-MRI, CSBPE
Fourier encoding and CSBPE chirp encoding for simulation data.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 2 4 6 8 1 0 1 2 1 4
MEA
N R
ELAT
IVES
ERR
OR
FOR
1000
TRI
ALS
ACCELERATION FACTOR
cs
CSBPE(Fourier)
CSBPE(chirp)
114
5.5.1 Simulation Results
The mean relative errorversus the acceleration factor is plotted in Figure 5.4and highlights
that CSBPE using chirp encodingoutperforms CSBPE using Fourier encoding for
acceleration factors up to R = 6 after that the image resolution going down in chirp encoding
technique than Fourier encoding.The mean relative error of output images is showing that the
error will increase as the reduction factor R increases. In addition, chirp modulation encoding
outperforms Fourier encoding.But both scheme is far better than convention CS scheme for
all acceleration factors.
Figure 5.5and Figure 5.6 show the simulation results of chirp modulated Fourier based
compressed sensing reconstruction using BPE data. Figure 5.5is magnitude image and Figure
5.6 is phase image respectively. The BPE acquired data images are all aliased images which
has obtained before the CS reconstruction and the middle column show the aliased images
after CS reconstructionat reduction factor R = 4, 6, 8 and 10 respectively. Right column
shows the error images of chirp modulated Fourier based CS reconstruction for acceleration
factors of 4, 6, 8 and 10 respectively; The result show the CS reconstructed images from
BPEdata can preserve image resolution.
Figure 5.7 shows the reconstructed images using chirp modulated Fourier based CSBP
encoding for the acceleration factors of 2, 4, 5.3, 6 and 8. In addition, the simulation results
illustrate that chirp modulated Fourier based CSBP encoded images is able to preserve
resolution better than Fourier based CSBP encoded imagesas is explicit from the enlarged
region of the reconstructed images. This result suggests that chirp encoding is better in term
of preserving the resolution of the reconstructed images from under sampled k-space data.
These simulation results show that chirp modulation encoding can preserve the sensitivity
information while performing CS. Also, table 3 shows the less error of chirp modulated
CSBPE compare to the Fourier based CSBPE and therefore, chirp modulation is better
encoding scheme for CSBPE-MRI. We have investigated the performance of Fourier
encoding, and the chirp modulated Fourier encoding schemes using single and multi-channel
data.
115
Figure 5.8shows the comparison of image quality between Fourier encodedCSBPE and chirp
modulated CSBPE method at different reduction factor and it shows that the quality of chirp
modulated Fourier based CSBP encoded images are better than Fourier based CSBP encoded
images.Also Figure 5.9 shows the comparison of image quality between chirp encoded
CSBPE and CS-MRI method at different reduction factor.The result of reconstructed images
of chirp modulated CSBPE outperforms the CSMRI for preserving image resolution.
Table 3: Compare the simulation results of Fourier based CSBPE and chirp modulated CSBPE for single channel data.
Reduction factor (R) Error using Fourier based
CSBPE (Mean relative error)
Error using Chirp modulated
CSBPE (Mean relative error)
R =2 0 0
R =4 0.0515 0.0461
R = 6 0.0738 0.0719
R = 8 0.0925 0.0992
R = 10 0.1244 0.1197
5.5.2 Experimental Results
Figure 5.5 shows the comparing real phantom results between the Fourier based CSBPE and
chirp modulation based CSBP encoding schemes.Figure 5.6(b)-(d) showChirp modulated
Fourier based CSBPE reconstructed images at different acceleration factors.Figure 5.7(e)-(g)
show error images with Chirp modulated Fourier based CSBPE for acceleration factors of 4,
6, and 8 respectively.Figure 5.8(h)-(j) show images reconstructed with Fourier encoding
based CSBPE using for acceleration factors of 4, 6, and 8 respectively.Figure 5.9(k)-(m):
show error images with Fourier encoded based CSBPE for acceleration factors of 4, 6, and 8
respectively. But, for the real phantom data case as shown in Figure 5.10, The Chirp
modulated CSBPE results are not significantly high quality compare to Fourier based CSBPE
methods.
116
Chirp based BPE
images before CS
reconstruction
Chirp based BPE
images after CS
reconstructed image Error image
(a)
R = 4
(b)
R = 6
(c)
R = 8
(d)
R = 10
Figure 5.10:Simulation results for performance evaluation of chirp based compressed
sensing reconstruction using BPE data; (a)-(d) (left two column) comparing the chirp
based BPE acquired data image and CS reconstructed images at reduction factor R =
4, 6, 8 and 10 respectively; (a)-(d) (right column) error images of Fourier based CS
reconstruction for acceleration factors of 4, 6, 8 and 10 respectively; The result show
the CS reconstructed images from chirp based BPE data can preserve image
resolution.
117
Chirp based BPE images before CS
reconstruction(Phase)
Chirp based BPE images CS
reconstructed image (Phase)
Error image
(a)
R = 4
(b)
R = 6
(c)
R = 8
(d)
R = 10
Figure 5.11: Simulation results for performance evaluation of chirp based compressed
sensing reconstruction using BPE data; (a)-(d) (left two column) comparing the chirp
based BPE acquired data image and CS reconstructed images at reduction factor R =
4, 6, 8 and 10 respectively; (a)-(d) (right column) error images of Fourier based CS
reconstruction for acceleration factors of 4, 6, 8 and 10 respectively; The result show
the CS reconstructed images from chirp based BPE data can preserve image
resolution.
118
Figure 5.12: Simulation results for Chirp modulated Fourier encoding based CSBPE.
Left column represents the reference image 256 × 256 (up/down: phase encodes,
left/right: frequency encode), Fig. (a) - (d) (middle column): shows reconstructed
images byChirp modulated Fourier basedCSBPE with random down sampling
patterns for acceleration factors of R= 4, 5.7, 6, 8 respectively and Fig.(a)-(d) (right
Column): shows the error images.
(a)
(b)
(c)
(d)
R =
R =
R =
R =
119
Chirp encoded CSBPE Fourier encoded CSBPE Reconstructed image Error Reconstructed image Error
Figure 5.13: Comparing results between the Fourier based CSBPE and chirp
modulation based CSBPE encoding schemes (up/down: phase encodes, left/right:
frequency encode). (b)-(d): show images reconstructed withChirp modulated based
CSBPE for acceleration factors of 4, 6, and 8 respectively; (e)-(g): show error images
withChirp modulated based CSBPE for acceleration factors of 4, 6, and 8
respectively; (h)-(j): show images reconstructed withFourier encoding based CSBPE
using for acceleration factors of 4, 6, and 8 respectively; (k)-(m): show error images
withFourier encoded based CSBPE for acceleration factors of 4, 6, and 8 respectively;
b
c
d
e
f
g
R = 4
R = 6
R = 8
120
Chirp encoded CSBPE CS-MRI Reconstructed image Error Reconstructed image Error
Figure 5.14: Comparing results between the CS-MRI and chirp modulation based
CSBPE encoding schemes (up/down: phase encodes, left/right: frequency encode).(b)-
(d): show images reconstructed with Fourier encoding based CSBPE using for
acceleration factors of 4, 6, and 8 respectively; (e)-(g): show error images with Fourier
encoded based CSBPE for acceleration factors of 4, 6, and 8 respectively; (h)-(j):
show images reconstructed with Chirp modulated based CSBPE for acceleration
factors of 4, 6, and 8 respectively; (k)-(m): show error images with Chirp modulated
based CSBPE for acceleration factors of 4, 6, and 8 respectively;
h
i
J
k
l
m
R = 4
R = 6
R = 8
121
Chirp encoded CSBPE Fourier encoded CSBPE
Reconstructed image Error Reconstructed image Error
Figure 5.15: Compare the results between the Fourier based CSBPE and chirp
modulation based CSBPE encoding schemes (up/down: phase encodes, left/right:
frequency encode). (b)-(d): show images reconstructed with Chirp modulated based
CSBPE for acceleration factors of 4, 6, and 8 respectively; (e)-(g): show error images
with Chirp modulated based CSBPE for acceleration factors of 4, 6, and 8
respectively; (h)-(j): show images reconstructed with Fourier encoding based CSBPE
using for acceleration factors of 4, 6, and 8 respectively; (k)-(m): show error images
with Fourier encoded based CSBPE for acceleration factors of 4, 6, and 8 respectively;
R = 4
R = 6
R = 8 R = 8
R = 6
R = 4
122
5.6 Conclusion
The study presented in this chapter was motivated by the fact that non-Fourier encoding
schemes have demonstrated better performance in CS reconstruction. There are a number of
previous works, including random spectrum, spread spectrum, Toeplitz encoding and
Noiselet encoding, that have validated the advantages of non-Fourier encoding for CS
imaging. Most of the studies presented before have used non-Fourier encoding for parallel
MRI or CS MRI, which have improved performance. Here we have introduced chirp
modulated non-Fourier in CSBPE to spread the signal energy in BPE space, aiming to further
improve the image quality of CSBPE method (Figure 5.3).
A key ingredient of CS MRI is to under-sample the Fourier encoded k-space that has most
signal energy at the central low frequencies. To acquire more energy, the under-sampling is
dense at the central low frequencies and sparse at the peripheral high frequencies. The k-
space data thus acquired results in the loss of high frequency energy and hence the loss of
image details. In contrast, chirp modulated non-Fourier encoding can spread the energy
evenly in the k-space, and hence allows for uniform random under-sampling to acquire the
signal energy evenly over the entire frequency range.
The k-space data thus acquired gives more image details and hence better image quality. In
principle, spreading signal energy in BPE space by chirp modulation should have the same
effect as it does in Fourier encoded k-space. However, the experiment results of this chapter
have shown that its impact on the improvement of image quality is minor. This is because the
signals in BPE space is distributed in non-rectangular grid, e.g. zig-zag grid, with its energy
already spread, to some extent, by the irregular grid.
123
Chapter 6 Conclusion and Future work
Conclusion and Future work In this chapter, we will review the main contributions of this thesis and discuss some possible
future direction based on our research.
6.1 Conclusion MRI is an innovative medical imaging technology widely used in contemporary clinical
diagnosis and biological research. It has the capability to non-invasively produce high-
resolution images of the internal soft organ of human or animal in any direction. However, it
suffers from its lengthy data acquisition time caused by physical constraints. As discussed in
Chapters 1-3, despite the great research effort in recent years to shorten the data acquisition
time, it is still far away from achieving this goal. This thesis has presented a number of new
solutions to address this issue.
In Chapter 4, we have developed a new class of CSBPE methods to further reduce the data
acquisition time of MRI without compromising the image quality. These methods combine
the regular under-sampling scheme of BPE with the random under-sampling scheme of CS to
acquire significantly less k-space data than that of conventional BPE and CS MRI schemes.
The k-space data thus acquired are used in the iterative CS image reconstruction and then
BPE image reconstruction to obtain high quality image. For simulated zig-zag k-space data,
these new methods have produced high quality image at the high reduction (acceleration)
factor up to eight, which outperforms the conventional BPE and CS MRI schemes.
To overcome the difficulty of image ghost caused by the imprecise BPE trajectories in real
MRI scan, we have used cross-correlation technique to develop a new aliasing coefficient
correction method. This new method only requires the k-space data from normal scan to
effectively and precisely calculate the physically realised true trajectories. Hence, it avoids
the pre-scan calibration commonly required by the conventional aliasing coefficient
correction methods, and can significantly reduce the preparation time and operation cost of
MRI. The experiments on the real zig-zag k-space data have shown that this new correction
124
method can effectively remove the image ghost of CSBPE caused by the imprecise BPE
trajectories in real MRI scans, yielding high quality images.
In chapter 5, we have investigated further improvement of CSBPE scheme by using chirp
modulated non-Fourier BPE. This investigation is based on the fact that chirp modulation can
spread the signal energy in the Fourier encoded k-space where the data is acquired/measured,
and hence improve the sampling incoherence and image quality of CS reconstruction. The
experiments on simulated data have shown that such approach has only resulted in some
minor improvement of image quality. This is because BPE itself can spread some signal
energy in the Fourier encoded k-space and hence improve the incoherence of signal sampling.
The effect of chirp modulation in BPE is therefore limited. Nevertheless, the chirp
modulation in CSBPE still yields some improvement in image quality.
6.2 Future Work Listed below are some possible directions for future research.
• The CSBP encoding sequence can be designed in 3D volume imaging. The simulation
results of CSBPE demonstrate that images reconstructed from 3D sub-sampled
Fourier and non-Fourier encoded data are of better quality than those from 2D sub-
sampled CSBPE data. The first step is to design the pulse sequence of Fourier
bunched phase encoding or chirp modulated non-Fourier bunched phase encoding for
volume imaging. After that, phantom and human body scanning should be conducted
in order to test the performance of the pulse sequence. The reconstructed image from
volume imaging data could be used to determine whether the expected image can be
obtained by directly applying the inverse Fourier transform and inverse chirp
transform on the volume imaging data.
• Zig-zag GRAPPA can be introduced into CSBPE to reconstruct the image and
compared with the CSBPE method presented in this thesis. We may randomly under-
sample the k-space encoded by the zig-zag BPE, and use CS reconstruction to
generate the aliased image. We can then use zig-zag GRAPPA to reconstruct the
original image. The image thus reconstructed and be then compared with that of
125
CSBPE. This might lead to a zig-zag GRAPPA based CSBPE method with better
image quality.
• Some other non-Fourier encoding such as Noiselet can also be used to study the
performance of CSBPE method.
126
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List of Publications
1. Jingxin Zhang and Kazi Rafiqul Islam, "Compressed Sensing MRI Using
Bunched Phase Encoding", Presented in ISMRM 25th Annual Meeting and
Exhibition, Honolulu, USA, 30 May-05 June, 2017.
2. Kazi Rafiqul Islam and Jingxin Zhang, "Simple and effective trajectory
estimation for image reconstruction of accelerated k-space acquisition on
non-rectangular periodic trajectories", Presented in ISMRM 27th Annual
Meeting and Exhibition, Vancouver, Canada, 11May-17May, 2019.
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THE END