medical image analysis image reconstruction figures come from the textbook: medical image analysis,...

Medical Image AnalysisMedical Image AnalysisImage Reconstruction

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Mathematical Preliminaries Mathematical Preliminaries and Basic Reconstruction and Basic Reconstruction MethodsMethods


An original image



Apply the Radon transform



After the inverse Radon transform



An test image



Apply the Radon transform



After the inverse Radon transform



x

y

q

p

p

f(x,y)

P(p,)

Figure 2.8. Line integral projection P(p,q) of the two-dimensional Radon transform.


The Radon transform of an object


),( yxf

dqqpqpf

pJyxfR

)cossin,sincos(

)(),(

cossin

sincos

yxq

yxp

cossin

sincos

qpy

qpx

Central Slice TheoremCentral Slice TheoremThe central slice theorem

◦Called the projection theorem◦A relationship between the Fourier

transform of the object function and the Fourier transform of its Radon transform or projection


Central Slice TheoremCentral Slice Theorem

Figure comes from the Wikipedia, www.wikipedia.org.



dpedqqpqpf

pJFyxfRF

pj 2 )cossin,sincos(

)(),(

dpepJpJFS pj 2 )()()(


u

v

F(u,v)

Sk() S2()

S1()

Figure 5.1. The frequency domain of the Fourier transform F(u,v) with the Fourier transforms, Sq(w) of individual projections Jq(p).



),(

)()(

sincos 2

F

dxdyef(x,y)

pJFS

θ)yθ(xj

sin

cos

u


◦Represents the Fourier transform of the projection that is taken at an angle in the space domain with a rotated coordinate system


)(S

)( pJ

),( qp

Inverse Radon TransformInverse Radon Transform

Where

dudveF(u,v)vuFFyxf vy)(xuj 21 ),(),(ˆ

dpJ

ddeS

dde),F(rf

)y(xj

)y(xj

)'(

)(

),(ˆ

0

0

sincos 2

0

sincos 2

deSpJ )y(xj sincos 2 )()'(

Backprojection MethodBackprojection MethodModified projections

◦Convolution-backprojection◦Filtered-backprojection

)'( pJ

)(

)(

)(

)()'(

1

1

2

sincos 2

pJF

SF

deS

deSpJ

pj

)y(xj

Backprojection MethodBackprojection MethodFrom

')'()'(

)()'( 1

dppJpph

pJFpJ

0

')'()'(),(ˆ ddppJpphyxf

Backprojection MethodBackprojection MethodRamakrishnan and

Lakshiminarayanan

In general

otherwise0

if LRH

otherwise0

if1)(

B

)()( BH


H()

1/2-1/2

1/2

Figure 5.2. A bandlimited filter function .)(H

Backprojection MethodBackprojection MethodThe filter kernel function

◦If the projections are sampled with a time interval of , the projections can be represented as , where is an integer

deHph pi 2)()(

)( kJ k

2

1

Backprojection MethodBackprojection MethodFor the bandlimited projections

with a sampling interval of

Then

2

22 2/

)2/ sin(

4

1

/

)/ sin(

2

1)(

p

p

p

pph

L

i

pJL

yxfi

1

)'(),(ˆ

')'()'()()'( 1 dppJpphpJFpJ

Backprojection MethodBackprojection MethodThe quality of the reconstructed

image◦The number of projections◦The spatial interval of the acquired

projection◦Limited by the detector size and the

scanning procedure◦Suffer from poor signal-to-noise ratio if

there is an insufficient number of photons collected by the detector due to its smaller size

Backprojection MethodBackprojection MethodRamakrishnan and

Lakshiminarayanan filter◦Has sharp cutoffs in the frequency

domain at and ◦Cause modulated ringing artifacts in

the reconstructed imageHamming window function

2/1 2/1

)()2cos()1()(Hamming BH

10


H()

1/2-1/2

Figure 5.3. A Hamming window based filter kernel function in the frequency domain.


h(

hR-L(p)

HHamming(p)

Figure 5.4. A comparison of the and convolution functions in the spatial domain.

)(min ph gHam )( ph LR

Iterative Algebraic Iterative Algebraic Reconstruction MethodsReconstruction MethodsAlgebraic Reconstruction

Techniques (ART)◦The raw projection data from the

scanner are distributed over a prespecified image reconstruction grid such that the error between the computed projections from the reconstructed image and the actual acquired projections is minimized



Raywith ray sum pi

f1 f2 f3

fN

Overlapping area for defining wi,j

Figure 5.5. Reconstruction grid with a ray defining the ray sum for ART.

Iterative Algebraic Iterative Algebraic Reconstruction MethodsReconstruction Methods : the projection data : the pixels of the image : weights

◦Determined by geometrical consideration as the ratio of the area overlapping with the scanning ray to the total area of the pixel


ip

jf

jiw ,

N

jjjii fwp

1, Mi ,...1

Iterative Algebraic Iterative Algebraic Reconstruction MethodsReconstruction Methods : the computed ray sum in

the iteration


kiq thk

jiN

lli

kiik

jkj w

w

qpff ,

1

2,

1

Mi ,...1

N

lli

kl

ki wfq

1,

1

Iterative Algebraic Iterative Algebraic Reconstruction MethodsReconstruction MethodsThe iterative ART

◦Deal with the noise and random fluctuations in the projection data caused by detector inefficiency and scattering


Estimation MethodsEstimation MethodsStatistical estimation

◦Assume a certain distribution of the measured photons

◦Find the parameters for attenuation function (in the case of transmission scans such as X-ray CT) or emitter density (in the case of emission scans such as PET)


Estimation MethodsEstimation Methods : measurement

vector : the random variable

representing the number of photons collected by the detector for the ray

: the blank scan factor : the attenuation coefficients


thi

],...,,[ 21 NJJJJ

iJ

im

Ldlzyx

ii emJE),,(

][ Ni ,...2,1

Estimation MethodsEstimation MethodsA line integral or ray sum for

ray

The Poisson distribution model for the photon counts

thi

N

i

N

ij

ji

j

ii i

ii jejJPjJP

1 1

)(

!

)]([];[];[

i

N

kkikL Aadlzyx

p

i][),,(

1

iAii emJ ][)(

Estimation MethodsEstimation MethodsThe Maximum Likelihood (ML)

estimate◦The log likelihood function

!log)(log)(

!

)]([log

];[log];[log)(

1

1

)(

1

i

i

ii

ji

N

iii

N

ij

ji

J

ii

N

i

JjJ

Je

jJPjJPL

iAii emJ ][)(

Estimation MethodsEstimation MethodsThe Maximum Likelihood (ML)

estimate◦The log likelihood function

◦Find

!log)][ (log)(1

][ ii jii

N

ii

Ai AmjemL

)(maxargˆ0

L

pN

kkiki aA

1

][

Estimation MethodsEstimation MethodsPenalty functions

◦Additional constraints such as smoothness

◦Find

)( )(maxargˆ0

RL

K

kkk CwR

1

2][2

1)(

Estimation MethodsEstimation MethodsOptimization methods

◦Expectation Maximization (EM)◦Complex conjugate gradient◦Gradient descent optimization◦Grouped coordinated ascent◦Fast gradient based Bayesian

reconstruction◦Ordered-subsets algorithms

Fourier Reconstruction Fourier Reconstruction MethodsMethodsDirect Fourier reconstruction

◦Use the central slice theorem◦Resampling the frequency domain

information from a polar to a Cartesian grid

◦Developing sinc-based interpolation method for the bandlimited functions in the radial direction

Image Reconstruction in Image Reconstruction in Medical Imaging ModalitiesMedical Imaging ModalitiesChoice

◦Filtered backprojection (X-ray CT)◦Statistical estimation



C

B

A

S

F

R

T

D

E

x

y

O

G

Figure 5.6. A 2-D divergent beam geometry.

X-ray Computed X-ray Computed TomographyTomography : angular step : radial distance between the

source and the origin : the angle that the source

makes with its central reference axis

: a fan projection from the divergent beam


D

)(K

X-ray Computed X-ray Computed TomographyTomography Objective: convert fan

projections into the parallel-beam projection

Sorted cos)()(' DKKii

)(K

)( pJ

sinDpOG

X-ray Computed X-ray Computed TomographyTomography Backprojected

: the total number of source positions

: the angle of the divergent beam ray passing through the point

: the distance between the source and the point for the source position

)()()( ' hKQii

)'();,(

1),(

12

iQ

yxLyxf

N

i i

N

'

L

),( yx

),( yx

i

Nuclear Emission Computed Nuclear Emission Computed Tomography: SPECT and PETTomography: SPECT and PETX-ray CT

◦Estimate the attenuation coefficient map

SPECT or PET◦Reconstruct the source emission map

within the object from the statistical distribution of photons that have gone through attenuation within the object but detected outside the object

◦Attenuation correctionFigures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Nuclear Emission Computed Nuclear Emission Computed Tomography: SPECT and PETTomography: SPECT and PETThe transmission scans in SPECT

◦Computing attenuation coefficient parameter

◦The iterative ML estimation-based algorithms have provided better results


Multi-Grid EM AlgorithmMulti-Grid EM AlgorithmImage reconstruction in PET


InitializeGrid level, k = 0Iterations, i = 0

i = i +1

NO

YES

NO

YES

i = 0

WaveletDecomposition

Final ReconstructedImage

NO

YESIs grid

optimizationmeasure

satisfied ?

Is intra-levelperformance

measuresatisfied ?

k = k + 1

Final ReconstructedImage

Is currentgrid resolution

>detector

resolution?

Initial Solution0

0

= EM( ,n)ki+1 i

k

= INTERP( )Wavelet Interpolation

i

kk+10

Figure 5.7. A flowchart of the MGEM algorithm for PET image reconstruction.


image

convolution with thelow-pass wavelet

decompositionfilter

convolution with thehigh-pass wavelet

decompositionfilter

bicubicinterpolation

bicubicinterpolation

convolution with thehigh-pass wavelet

reconstructionfilter

convolution with thelow-pass wavelet

reconstructionfilter

interpolatedimage

Figure 5.8. Wavelet based interpolation method.

Figure 5.9. Shepp and Logan phantom (top left) and reconstructed phantom images using WMREM algorithm (top right), ML-EM algorithm (bottom left) and filtered backprojection method (bottom right).

Figure 5.10. Four reconstructed brain images of a patient with a tumor from a PET scan. Images in the top row are reconstructed using filtered backprojection method, images in the middle row are reconstructed using WMREM algorithm. Images in the bottom row are reconstructed using a generalized ML-EM algorithm.

Image Reconstruction MRIImage Reconstruction MRI


dxdydzezyxMS zyxizyx

zyx )(0 ),,(),,(

zyxzyxi

zyx dddeSMzyx zyx )(0 ),,(),,(

Image Reconstruction Image Reconstruction Ultrasound ImagingUltrasound ImagingPoint measurements

◦Line scan◦Reduction of speckle noise

Image averaging Image filtering: weighted median,

Wiener filters


medical image analysis image reconstruction figures come from the textbook: medical image analysis,...

Documents

ieee press

central slice theoremfigures

twodimensional radon

fourier transforms

bandlimited filter function

line integral projection

space domain

object function