research article a ct reconstruction algorithm based on non...

Research ArticleA CT Reconstruction Algorithm Based on Non-AliasingContourlet Transform and Compressive Sensing

Lu-zhen Deng Peng Feng Mian-yi Chen Peng He Quang-sang Vo and Biao Wei

The Key Lab of Optoelectronic Technology and Systems of the Education Ministry of China Chongqing UniversityChongqing 400044 China

Correspondence should be addressed to Peng Feng coe-fpcqueducn and Peng He hepeng vvv163com

Received 11 April 2014 Revised 6 June 2014 Accepted 9 June 2014 Published 30 June 2014

Academic Editor Fenglin Liu

Copyright copy 2014 Lu-zhen Deng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Compressive sensing (CS) theory has great potential for reconstructing CT images from sparse-views projection data Currentlytotal variation (TV-) based CT reconstruction method is a hot research point in medical CT field which uses the gradient operatoras the sparse representation approach during the iteration process However the images reconstructed by this method often sufferthe smoothing problem to improve the quality of reconstructed images this paper proposed a hybrid reconstruction methodcombining TV and non-aliasing Contourlet transform (NACT) and using the Split-Bregman method to solve the optimizationproblem Finally the simulation results show that the proposed algorithm can reconstruct high-quality CT images from few-viewsprojection using less iteration numbers which is more effective in suppressing noise and artefacts than algebraic reconstructiontechnique (ART) and TV-based reconstruction method

1 Introduction

Since computed tomography (CT) [1] technique was bornin 1973 CT has been widely applied in medical diagnoseindustrial nondestructive detection and so forth In medicalCT field how to reconstruct high-quality CT images fromfew-views or sparse-views data is a significant research prob-lem Recently compressive sensing (CS) [2] theory has beenapplied in CT images reconstruction which makes it possibleto reconstruct high-quality images from few-views data InCS theory CT images can be sparsely represented in anappropriate domain such as gradient transform andWavelettransform and the quality of CT reconstructed images will beimproved by some appropriate sparse representations in CTimages reconstruction

Contourlet transform [3] is proposed by Do and Vetterliin 2002 which is a sparse representation for 2D imageswith some properties such asmultiresolutionmultiscale andmultidirection Contourlet transform can also get importantsmooth contour features of the image with few data butthere is frequency aliasing in Contourlet transform Sharpfrequency localization Contourlet transform [4] is firstlyproposed by Lu and Do in 2006 and Feng et al introduced

a detailed explanation and construction in 2009 which isnamed as non-aliasing Contourlet transform (NACT) [5]NACT which can eliminate the frequency aliasing in Con-tourlet transform is more efficient in capturing geometricalstructure and can represent image sparser than traditionalContourlet transform

To solve the optimization problem in CT images recon-struction based on CS Goldstein and Osher proposed Split-Bregman [6]methodwhich is derived fromBregman [7] iter-ation and can accelerate iteration convergence and producebetter reconstruction results Split-Bregman method uses anintermediate variable to split 119871

1regularization and 119871

2regu-

larization into two equations where 1198712and 119871

1regularization

equation can be solved by steepest descent method andthresholding algorithm respectively Based on Split-Bregmanmethod Vandeghinste et al proposed Split-Bregman-basedsparse-view CT reconstruction approach [8] Furthermorean iterative CT reconstruction is proposed using shearlet-based regularization [9] Chu et al proposed multienergy CTreconstruction based on low rank and sparsity with the Split-Bregmanmethod (MLRSS) [10] Chang et al proposed a few-view reweighted sparsity hunting (FRESH) method for CTimages reconstruction [11]

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014 Article ID 753615 9 pageshttpdxdoiorg1011552014753615

2 Computational and Mathematical Methods in Medicine

In this paper we propose a CT reconstruction algorithmbased on NACT and compressive sensing which tries toexplore the sparse capability of NACT in order to recon-struct high-quality CT images In the following section theproposed algorithm will be introduced In the third sectionwe will analyze the experimental results and discuss relevantissues In the last section Section 4 we will conclude thepaper

2 Theory and Method

21 CT Reconstruction Theory Based on Compressive SensingTheoretically the mathematical CT model can be expressedas

119860119891 = 119901 (1)

where 119860 is the system matrix 119891 is the original imageand 119901 is the projection data Traditional CT reconstructionalgorithms such as filtered backprojection (FBP) [12] andalgebraic reconstruction technique (ART) [13] cannot recon-struct high quality CT images with the sparse sampling orlimited projection data

In 2006 Candes and Donoho put forward the CS theorywhich makes it possible to get high quality CT images withsparse projection data The main idea of CS is that a signalcan be reconstructed with far less sampled frequency thanrequired by conventional Nyquist-Shannon sampling fre-quency if the image has a sparsecompressible representationin a transform domain

Compressive sensing theory can be expressed by thefollowing equation

min 100381710038171003817100381711991010038171003817100381710038170

st 119901 = 119860119891 = 119860Φ119867119910 (2)

where Φ is a orthogonal transform Φ119867 is the correspondinginverse transform and 119891 is the CT image to be reconstructedand has a special relationship withΦ119867 that is 119891 = Φ

119867119910 119901 is

the projection data of 119891 through matrix 119860Inspired by CS theory Sidky et al proposed a total

variation (TV-) based CT reconstruction algorithm usinggradient operator as the sparse representation [14] in whichTV is defined as follows

10038171003817100381710038171198911003817100381710038171003817TV = sum

1003816100381610038161003816nabla1198911003816100381610038161003816 = sum

119904119905

radic(nabla119909119891)2

+ (nabla119910119891)2

= sum

119904119905

radic(119891119904119905minus 119891119904minus1119905

)2

+ (119891119904119905minus 119891119904119905minus1

)2

(3)

where nabla119891 represents gradient operator of an image 119891

22 Non-Aliasing Contourlet Transform The traditionalsparse representation such as gradient operator and Wavelettransform [15] cannot get ideal sparse representation of CTimages In 2002 Ni et al proposed Contourlet transform [16]which can utilize intrinsic structure information of imageto represent images more efficiently compared with Wavelettransform However suffering from frequency aliasing Con-tourlet transform does not show good performance in imagedenoising fusion and enhancement In order to solve this

problem a new multiscale analysis method named non-aliasing Contourlet transform (NACT) was proposed NACTconsists of non-aliasing pyramidal filter banks (NPFB) anddirectional filter banks (DFB) NPFB contains two differentfilter banks 119871

0(120596) 119863

0(120596) and 119871

1(120596) 119863

1(120596) 119871

0(120596) and 119871

1(120596)

mean low-pass filters 1198630(120596) and 119863

1(120596) mean high-pass

filters The relationships of two different filter banks are asfollows

1198632

1(120596) +

1198712

1(120596)

4= 1

1198632

0(120596) + 119871

2

0(120596) = 1

(4)

We assume that 1205961199010

and 1205961199040

represent pass-band fre-quency and stop-band frequency of 119871

0(120596) respectively

Accordingly 1205961199011

and 1205961199041

represent pass-band frequencyand stop-band frequency of 119871

1(120596) respectively In order to

eliminate frequency aliasing the filter parameters shouldmeet (1) 120596

1199041lt 1205872 (2) (120596

1199010+ 1205961199040)2 = 1205872 and (120596

1199011+

1205961199041)2 = 1205874 (3) 120596

1199040le 120587 minus 119886 and 120596

1199041le (120587 minus 119886)2 where 119886

is the maximum width of mixing ingredients in DFB [3]As a sparse representation approach NACT integrate

NPFB and DFB which can decompose image into multidi-rection and multiresolution NPFB decomposes image intoan approximation subband and several detail subbands withdifferent resolutions DFB decomposes the detail subbandsinto directional subbandsThe process of decompositionwith3 levels is shown in Figure 1Wewill use ldquo9-7rdquo filter and ldquopkvardquodirectional filter bank [17] in the study

23 Split Bregman Method In CS theory 1198710norm is the

most ideal regularization norm but it is difficult to solveequations and easily interfered by noise in CT reconstructionso 1198710norm is commonly replaced by 119871

1norm Then the

reconstruction problem depicted by (2) can be converted into

min 100381710038171003817100381711991010038171003817100381710038170

st 119901 = 119860119891 = 119860Φ119867119910 (5a)

min 1003817100381710038171003817Φ11989110038171003817100381710038171

st 119901 = 119860119891 = 119860Φ119867119910 (5b)

where 119910 = Φ119891 Φ is the sparse transform which is normallyused as Wavelet transform Curvelet transform gradientoperator and so forth

Furthermore (5b) can be converted into

119891 = arg min119891

1003817100381710038171003817Φ11989110038171003817100381710038171

+ 1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2 (6)

where 120582 is penalty function weightIn order to solve (6) Goldstein andOsher proposed Split-

Bregman method [6] using an intermediate variable to split1198711regularization and 119871

2regularization into two equations

1198712regularization equation can be solved by gradient descent

method and 1198711regularization equation can be solved by

thresholding algorithm Split-Bregman method contains thefollowing three iteration steps

Step 1

119891119896+1

= arg min119891

1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2+ 120583

10038171003817100381710038171003817119889119896minus Φ119891 minus 119887

11989610038171003817100381710038171003817

2

2 (7)

Computational and Mathematical Methods in Medicine 3

x(n)

L0(120596)

D0(120596)

L1(120596)

L1(120596)

D1(120596)

D1(120596)

darr D2

darr D2

DFB

DFB

DFB

y3(n)

y00(n)simy

07(n)

y10(n)simy

13(n)

y20(n)simy

23(n)

Figure 1 Flowchart of 3 levels of decomposition of NACT

Step 2

119889119896+1

= min119889

1198891 + 12058310038171003817100381710038171003817119889119896minus Φ119891119896+1

minus 11988711989610038171003817100381710038171003817

2

2 (8)

Step 3

119887119896+1

= 119887119896+ (Φ119891

119896+1minus 119889119896+1

) (9)

where 119896 is the Split-Bregman iteration index120583 is convergenceparameter and 119889 and 119887 are intermediate variables with whicheach subproblem can be solved easily

24 Proposed Algorithm According to aforementionedmethods we propose a CT reconstruction algorithm basedon NACT and compressive sensing method which can bedefined as a constrained form (10) or an unconstrained form(11) as follows

min 10038171003817100381710038171198911003817100381710038171003817TV +

1003817100381710038171003817Φ11989110038171003817100381710038171

st 1003817100381710038171003817119860119891 minus 1199011003817100381710038171003817

2

2lt 1205902

(10)

min 10038171003817100381710038171198911003817100381710038171003817TV +

1003817100381710038171003817Φ11989110038171003817100381710038171

+ 1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2 (11)

Applying the Split-Bregman method to (11) we have thefollowing three iteration steps

Step 1

119891119896+1

= arg min119891

1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2+ 120574

10038171003817100381710038171003817119889119896minus nabla119891 minus 119887

11989610038171003817100381710038171003817

2

2

+ 12058310038171003817100381710038171003817119889119896

120593minus Φ119891 minus 119887

119896

120593

10038171003817100381710038171003817

2

2

(12)

Step 2

119889119896+1

= min119889

1198891 + 12057410038171003817100381710038171003817119889119896minus nabla119891119896+1

minus 11988711989610038171003817100381710038171003817

2

2

119889119896+1

120593= min119889120593

10038171003817100381710038171003817119889120593

100381710038171003817100381710038171+ 120583

10038171003817100381710038171003817119889119896

120593minus Φ119891119896+1

minus 119887119896

120593

10038171003817100381710038171003817

2

2

(13)

Step 3

119887119896+1

= 119887119896+ (nabla119891

119896+1minus 119889119896+1

)

119887119896+1

120593= 119887119896

120593+ (Φ119891

119896+1minus 119889119896+1

120593)

(14)

where 120574 is convergence parameter and 119889120593and 119887120593are interme-

diate variables

The steepest descent method is applied to solve (12) Thederivative of (12) is calculated as follows

119892 [119899119898 + 1]

= 2120582119860119879

119898(119860119898119891119898minus 119901119898) minus 2120574nabla

119879(119889119896minus nabla119891119898minus 119887119896)

minus 2120583Φ119879(119889119896

120593minus Φ119891119898minus 119887119896

120593)

119891119898+1

= 119891119898+ 120572119892 [119899119898 + 1]

(15)

where 119899 denotes the iteration index of the steepest descentmethod 119898 = 2 119873data denotes the projection angles 119860

119898

is mth row vector and system matrix 119860 includes 119873data rowvector 119860

119898 Accordingly 119873data row vectors 119901

119898compose the

projection-data vector 119901 120572 is an appropriate step size TheARTmethod is used to get initial image of iteration Equation(13) can be explicitly computed as (16) using the shrinkageoperator as follows

119889119896+1

= shrink(nabla119891119896+1 + 1198871198961

120582)

119889119896+1

120593= shrink(Φ119891119896+1 + 119887

119896

1205931

120583)

(16)

We now describe the iterative steps of the proposed algo-rithm The iteration process contains two loops the outsideloop operate ART and the inside loop solve the optimizationproblem which is constrained by TV and NACTThe outsideloop is labeled by 119899 and the inside loop is labeled by 119896The steps comprising each loop are the DATA-step whichenforces consistency with the projection data the POS-stepwhich ensures a nonnegative imageWe use119891(ART-DATA)[119899119898]

to denote the 119898th DATA-step subiteration with the 119899thiteration and 119891

(ART-POS)[119899] to denote the POS-step with the

119899th iteration in the outside loopWe use119891(NACTTV-DATA)[119896119898]


(NACTTV-POS)[119896] to denote the POS-step with

the 119896th iteration in the inside loopThe steps of the algorithmare as follows

(A) initialization

119899 = 1 119891(ART-DATA)

[119899 1] = 0 (17)

(B) data projection iteration for119898 = 2 119873data

119891(ART-DATA)

[119899119898] = 119891(ART-DATA)

[119899119898 minus 1]

+ 119860119898

119901119898minus 119860119898sdot 119891(ART-DATA)

[119899119898 minus 1]

119860119898sdot 119860119898

(18)


Figure 2 Head phantom

(C) positivity constraint

(119891119894119895)(ART-POS)

[119899]

= (119891119894119895)(ART-DATA)

[119899119873data] (119891119894119895)(ART-DATA)

[119899119873data] ge 0

0 (119891119894119895)(ART-DATA)

[119899119873data] lt 0

(19)

(D) initialization of Split-Bregman

119896 = 1

119889 (119899) =10038171003817100381710038171003817119891(ART-DATA)

[119899 1] minus 119891(ART-POS)

[119899]100381710038171003817100381710038172

119891(NACTTV-DATA)

[119896 1] = 119891(ART-POS)

[119899]

119889119896

119909= nabla119909119891(ART-POS)

[119899]

119889119896

119910= nabla119910119891(ART-POS)

[119899]

119889119896

120593= Φ119891(ART-POS)

[119899]

119887119896

119909= 119887119896

119910= 119887119896

120593= 0

(20)

(E) iteration for119898 = 2 119873data

119889119901= 119860119898119891(NACTTV-DATA)

[119896119898 minus 1] minus 119901119898minus1

119892 [119896119898 minus 1]

= 2120582119860119898119889119901minus 2120574nabla

119879

119909(119889119896


119909)

minus 2120574nabla119879

119910(119889119896


119910)

minus 2120583Φ119879(119889119896

120593minus Φ119891 minus 119887

119896

120593)10038161003816100381610038161003816119891=119891(NACTTV-DATA)

[119896119898minus1]

119892 [119896119898 minus 1] =119892 [119896119898 minus 1]

1003816100381610038161003816119892 [119896119898 minus 1]1003816100381610038161003816

119891(NACTTV-DATA)

[119896119898]

= 119891(NACTTV-DATA)

[119896119898 minus 1] minus 119886119889 (119899) 119892 [119896119898 minus 1]

(21)

(F) positivity constraint

(119891119894119895)(NACTTV-POS)

[119896 + 1]

= (119891119894119895)(NACTTV-DATA)

[119896119873data] (119891119894119895)(NACTTV-DATA)

[119896119873data] ge 0

0 (119891119894119895)(NACTTV-DATA)

[119896119873data] lt 0

(22)

(G) update 119889119909 119889119910 119889120593 119887119909 119887119910 119887120593 increase 119896 and return to

step (E) until 119896 = 119870NACTTV as follows

119889119896+1

119909= shrink(nabla

119909119891(NACTTV-POS)

[119896 + 1 1] + 119887119896

1199091

120582)

119887119896+1


119910119891(NACTTV-POS)

[119896 + 1 1] + 119887119896

1199101

120582)

119889119896+1

120593= shrink(Φ119891(NACTTV-POS) [119896 + 1 1] + 119887

119896

1205931

120583)

119887119896+1

119909= 119887119896

119909+ (nabla119909119891(NACTTV-POS)

[119896 + 1 1] minus 119889119896+1

119909)

119887119896+1

119910= 119887119896

119910+ (nabla119910119891(NACTTV-POS)

[119896 + 1 1] minus 119889119896+1

119910)

119887119896+1

120593= 119887119896

120593+ (Φ119891

(NACTTV-POS)[119896 + 1 1] minus 119889

119896+1

120593)

(23)

(H) initialize next loop

119891(ART-DATA)

[119899 + 1 1] = 119891(NACTTV-POS)

[119870NACTTV 1] (24)

increase 119899 and return to step (B) The iteration is stoppedwhen 119860119891 minus 119901

2

2lt 1205902 In our study we selected 120582 = 1000

120574 = 30 120583 = 30 119886 = 02 and 119870NACTTV = 10 which canstrike a good balance in the steepest descent and generategood reconstruction results in the experiments

3 Experimental Results

31 The Image Quality Evaluation This paper uses the rootmean square errors (RMSE) and universal quality index(UQI) [18] to evaluate the quality of the reconstructed images

RMSE is the most widely applied way to evaluate imagequality and RMSE is defined as

RMSE = radic

1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891119877

119894119895)2

(25)

where 119891119894119895

is the pixel value of original image and 119891119877

119894119895is the

pixel value of reconstructed imageWang and Bovic proposed UQI mode which evaluates

images distortion problem including correlation distortionbrightness distortion and contrast distortion The value of


(a) (b) (c)

(d) (e) (f)

Figure 3The reconstructed images using three different reconstruction algorithms from the noise-free and noisy data Top row is for noise-free data and bottom row is for noisy data (a) and (d) are reconstructed by ART (b) and (e) are reconstructed by ART-TV and (c) and (f)are reconstructed by SpBr-NACT method

UQI is between minus1 and 1 When the reconstructed image isthe same as the original image the value of UQI is 1 UQI isdefined as

UQI =4120590119891119891119877119891 times 119891

119877

(1205902

119891+ 1205902

119891119877) [(119891)

2

+ (119891119877

)

2

]

(26)

where

119891 =1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119894119895

119891119877

=1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119877

119894119895

1205902

119891=

1

119872 times119873 minus 1sum

0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891)2

1205902

119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119877

119894119895minus 119891119877

)

2

120590119891119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891) (119891

119877

119894119895minus 119891119877

)

(27)

32 Numerical Simulation In this section a head phantomas shown in Figure 2 is used to reconstruct and compareby 3 different methods ART ART-TV and our proposedalgorithm (SpBr-NACT)The size of phantom image is 200 times200 We assume that the CT system was viewed as in a typicalpencil-beam geometry and the scanning range was from 1∘to 360∘ with a 120579 angular increment projection angles can beindicated as

120579119894= 1 + 360 times

(119894 minus 1)

119873view 119894 = 1 2

119873view2

120579119894= 182 + 360 times

(119894 minus 119873view2)

119873view

119894 =119873view2

119873view2

+ 1 119873view

(28)

In the simulation we reconstruct the head phantomfrom noise-free and noisy projection data To obtain noisyprojection data we add 10 dB Gaussian noise into noise-free projection data Projection number 119873view is 60 anditeration numbers for all reconstruction algorithms are 50The reconstructed images are shown in Figure 3 and theprofile of line 140 in different reconstructed images is plottedin Figure 4


0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART

Noisy-free

(a)

OriginalART

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(b)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART-TV

Noisy-free

(c)

OriginalART-TV

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(d)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalSpBr-NACT

Noisy-free

(e)

SpBr-NACT

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

Original

(f)

Figure 4 The profile of line 140 in different reconstructed images Left column is for noise-free date and right column is for noisy data (a)and (b) are for ART method (c) and (d) are for ART-TV method (e) and (f) are for SpBr-NACT method


0 50 100

002

004

006

008

01

012

Iteration numbers

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

0 50 100Iteration numbers

Noisy-free

ARTART-TVSpBr-NACT

09

092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100002

004

006

008

01

012

Iteration numbers

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT

Iteration numbers0 50 100

09

092

094

096

098

1U

QI

Noisy

(d)

Figure 5 The relationship of RMSE and UQI with respect to iteration number (a) and (b) are the RMSE and UQI of reconstructed imagesfrom noise-free data with ART ART-TV and SpBr-NACT method respectively (c) and (d) are the RMSE and UQI of reconstructed imagesfrom noisy data with ART ART-TV and SpBr-NACT method respectively

From Figures 3 and 4 we can see that the reconstructedimages using ART and ART-TV methods contain a lot ofnoise and artifacts while the reconstructed images usingSpBr-NACTmethod contain less noise and artifacts and haveclearer edges

Table 1 lists all the RMSE andUQI calculated from recon-structed images It is obvious that the RMSE of reconstructedimages using SpBr-NACT method is much smaller than thatof reconstructed images using ART and ART-TV methodsthe UQI is much bigger Thus SpBr-NACT method canreconstruct higher quality images

Figure 5 plots the change of RMSE and UQI with respectto iteration number Figure 6 plots the change of RMSE and

UQIwith respect to projection number119873view In both figuresART ART-TV and the proposed SpBr-NACT approach areused to reconstruct images from noise-free and noisy dataThe blue-solid line is for ART the green-dashed line is forART-TV and the red dashed line is SpBr-NACT For Figure 5the projection number is fixed and 119873view is 60 For figure6 the iteration number is fixed and equals 50 From bothFigures it is easy to find that with the increase of projectionnumber or iteration number SpBr-NACT approach canalways get the minimum RMSE and maximum UQI whichmeans that the quality of reconstructed images with SpBr-NACT is better than those with ART and ART-TV And alsowe see from Figure 5 when the iteration number is relatively


0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

Projection angles

Noisy-free

ARTART-TVSpBr-NACT

0 50 100 150 200092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT

0 50 100 150 200Projection angles

092

094

096

098

1U

QI

Noisy

(d)

Figure 6 The relationship of RMSE and UQI with respect to projection number119873view (a) and (b) are the RMSE and UQI of reconstructedimages from noise-free data with ART ART-TV and SpBr-NACT method respectively (c) and (d) are the RMSE and UQI of reconstructedimages from noisy data with ART ART-TV and SpBr-NACT method respectively

Table 1 RMSE and UQI of reconstructed images using three different algorithms

RMSE UQIMethods ART ART-TV SpBr-NACT ART ART-TV SpBr-NACTNoisy-free 00502 00321 00196 09869 09947 09980Noisy 00554 00406 00318 09839 09914 09948

small 3 methods that almost have the same RMSE and UQIwhich implies that our proposed method has no advantage ifthe iteration step does not converge

4 Conclusion

In this study we proposed a CT reconstruction algorithmbased on NACT and compressive sensing The experimental

results demonstrate that the proposed method can recon-struct high-quality images from few-views data and has apotential for reducing the radiation dose in clinical appli-cation In the further research we will try to explore moredirectional information from NACT so as to improve theperformance of SpBr-NACT algorithm especially when theprojection number is far more below what we setup in thecurrent experiment


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (no 61201346) and the Fun-damental Research Funds for the Central Universities (no106112013CDJZR120020 and no CDJZR14125501)

References

[1] G Wang H Yu and B de Man ldquoAn outlook on X-ray CTresearch and developmentrdquo Medical Physics vol 35 no 3 pp1051ndash1064 2008

[2] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[3] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEETransactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[4] Y Lu and M N Do ldquoA new contourlet transform with sharpfrequency localizationrdquo in Proceedings of the IEEE InternationalConference on Image Processing (ICIP 06) vol 2 pp 1629ndash1632Atlanta Ga USA October 2006

[5] P Feng B Wei Y J Pan and D L Mi ldquoConstruction of non-aliasing pyramidal transformrdquo Acta Electronica Sinica vol 37no 11 pp 2510ndash2514 2009

[6] T Goldstein and S Osher ldquoThe split Bregman method for 1198711-regularized problemsrdquo SIAM Journal on Imaging Sciences vol2 no 2 pp 323ndash343 2009

[7] L Bregman ldquoThe relaxation method of finding the commonpoints of convex sets and its application to the solution ofproblems in convex optimizationrdquo USSR Computational Math-ematics and Mathematical Physics vol 7 pp 200ndash217 1967

[8] B Vandeghinste B Goossens J de Beenhouwer et al ldquoSplit-Bregman-based sparse-view CT reconstructionrdquo in Proceedingsof the 11th International Meeting on Fully Three-DimensionalImage Reconstruction in Radiology and Nuclear Medicine (Fully3D rsquo11) pp 431ndash434 2011

[9] B Vandeghinste B Goossens R van Holen et al ldquoIterative CTreconstruction using shearlet-based regularizationrdquo inMedicalImaging 2012 Physics of Medical Imaging vol 8313 of Proceed-ings of SPIE p 83133I San Diego Calif USA February 2012

[10] J Chu L Li Z Chen G Wang and H Gao ldquoMulti-energyCT reconstruction based on low rank and sparsity with thesplit-bregman method (MLRSS)rdquo in Proceedings of the IEEENuclear Science Symposium and Medical Imaging ConferenceRecord (NSSMIC rsquo12) pp 2411ndash2414 Anaheim Calif USANovember 2012

[11] M Chang L Li Z Chen Y Xiao L Zhang and G Wang ldquoAfew-view reweighted sparsity hunting (FRESH) method for CTimage reconstructionrdquo Journal of X-Ray Science and Technologyvol 21 no 2 pp 161ndash176 2013

[12] S L Zhang W B Li and G F Tang ldquoStudy on imagereconstruction algorithm of filtered backprojectionrdquo Journal ofXianyang Normal University vol 23 no 4 pp 47ndash49 2008

[13] R Gordon R Bender and G T Herman ldquoAlgebraic recon-struction techniques (ART) for three-dimensional electron

microscopy and X-ray photographyrdquo Journal of TheoreticalBiology vol 29 no 3 pp 471ndash481 1970

[14] E Y Sidky C Kao and X Pan ldquoAccurate image reconstructionfrom few-views and limited-angle data in divergent-beam CTrdquoJournal of X-Ray Science and Technology vol 14 no 2 pp 119ndash139 2006

[15] M Abramowitz and C A Stegun A Wavelet Tour of SignalProcessing Academic Press San Diego Calif USA 3rd edition2008

[16] X Ni H LWang L Chen and JMWang ldquoImage compressedsensing based on sparse representation using Contourlet direc-tional subbandsrdquoApplication Research of Computers vol 30 no6 pp 1889ndash1898 2013

[17] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004

[18] Y R Liu Research on Objective Full-Reference Image QualityEvaluation Method Computer Science amp Technology NanjingChina 2010

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


In this paper we propose a CT reconstruction algorithmbased on NACT and compressive sensing which tries toexplore the sparse capability of NACT in order to recon-struct high-quality CT images In the following section theproposed algorithm will be introduced In the third sectionwe will analyze the experimental results and discuss relevantissues In the last section Section 4 we will conclude thepaper

2 Theory and Method

21 CT Reconstruction Theory Based on Compressive SensingTheoretically the mathematical CT model can be expressedas

119860119891 = 119901 (1)

where 119860 is the system matrix 119891 is the original imageand 119901 is the projection data Traditional CT reconstructionalgorithms such as filtered backprojection (FBP) [12] andalgebraic reconstruction technique (ART) [13] cannot recon-struct high quality CT images with the sparse sampling orlimited projection data

In 2006 Candes and Donoho put forward the CS theorywhich makes it possible to get high quality CT images withsparse projection data The main idea of CS is that a signalcan be reconstructed with far less sampled frequency thanrequired by conventional Nyquist-Shannon sampling fre-quency if the image has a sparsecompressible representationin a transform domain

Compressive sensing theory can be expressed by thefollowing equation

min 100381710038171003817100381711991010038171003817100381710038170

st 119901 = 119860119891 = 119860Φ119867119910 (2)

where Φ is a orthogonal transform Φ119867 is the correspondinginverse transform and 119891 is the CT image to be reconstructedand has a special relationship withΦ119867 that is 119891 = Φ

119867119910 119901 is

the projection data of 119891 through matrix 119860Inspired by CS theory Sidky et al proposed a total

variation (TV-) based CT reconstruction algorithm usinggradient operator as the sparse representation [14] in whichTV is defined as follows

10038171003817100381710038171198911003817100381710038171003817TV = sum

1003816100381610038161003816nabla1198911003816100381610038161003816 = sum

119904119905

radic(nabla119909119891)2

+ (nabla119910119891)2

= sum

119904119905

radic(119891119904119905minus 119891119904minus1119905

)2

+ (119891119904119905minus 119891119904119905minus1

)2

(3)

where nabla119891 represents gradient operator of an image 119891

22 Non-Aliasing Contourlet Transform The traditionalsparse representation such as gradient operator and Wavelettransform [15] cannot get ideal sparse representation of CTimages In 2002 Ni et al proposed Contourlet transform [16]which can utilize intrinsic structure information of imageto represent images more efficiently compared with Wavelettransform However suffering from frequency aliasing Con-tourlet transform does not show good performance in imagedenoising fusion and enhancement In order to solve this

problem a new multiscale analysis method named non-aliasing Contourlet transform (NACT) was proposed NACTconsists of non-aliasing pyramidal filter banks (NPFB) anddirectional filter banks (DFB) NPFB contains two differentfilter banks 119871

0(120596) 119863

0(120596) and 119871

1(120596) 119863

1(120596) 119871

0(120596) and 119871

1(120596)

mean low-pass filters 1198630(120596) and 119863

1(120596) mean high-pass

filters The relationships of two different filter banks are asfollows

1198632

1(120596) +

1198712

1(120596)

4= 1

1198632

0(120596) + 119871

2

0(120596) = 1

(4)

We assume that 1205961199010

and 1205961199040

represent pass-band fre-quency and stop-band frequency of 119871

0(120596) respectively

Accordingly 1205961199011

and 1205961199041

represent pass-band frequencyand stop-band frequency of 119871

1(120596) respectively In order to

eliminate frequency aliasing the filter parameters shouldmeet (1) 120596

1199041lt 1205872 (2) (120596

1199010+ 1205961199040)2 = 1205872 and (120596

1199011+

1205961199041)2 = 1205874 (3) 120596

1199040le 120587 minus 119886 and 120596

1199041le (120587 minus 119886)2 where 119886

is the maximum width of mixing ingredients in DFB [3]As a sparse representation approach NACT integrate

NPFB and DFB which can decompose image into multidi-rection and multiresolution NPFB decomposes image intoan approximation subband and several detail subbands withdifferent resolutions DFB decomposes the detail subbandsinto directional subbandsThe process of decompositionwith3 levels is shown in Figure 1Wewill use ldquo9-7rdquo filter and ldquopkvardquodirectional filter bank [17] in the study

23 Split Bregman Method In CS theory 1198710norm is the

most ideal regularization norm but it is difficult to solveequations and easily interfered by noise in CT reconstructionso 1198710norm is commonly replaced by 119871

1norm Then the

reconstruction problem depicted by (2) can be converted into

min 100381710038171003817100381711991010038171003817100381710038170

st 119901 = 119860119891 = 119860Φ119867119910 (5a)

min 1003817100381710038171003817Φ11989110038171003817100381710038171

st 119901 = 119860119891 = 119860Φ119867119910 (5b)

where 119910 = Φ119891 Φ is the sparse transform which is normallyused as Wavelet transform Curvelet transform gradientoperator and so forth

Furthermore (5b) can be converted into

119891 = arg min119891

1003817100381710038171003817Φ11989110038171003817100381710038171

+ 1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2 (6)

where 120582 is penalty function weightIn order to solve (6) Goldstein andOsher proposed Split-

Bregman method [6] using an intermediate variable to split1198711regularization and 119871

2regularization into two equations

1198712regularization equation can be solved by gradient descent

method and 1198711regularization equation can be solved by

thresholding algorithm Split-Bregman method contains thefollowing three iteration steps

Step 1

119891119896+1

= arg min119891

1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2+ 120583

10038171003817100381710038171003817119889119896minus Φ119891 minus 119887

11989610038171003817100381710038171003817

2

2 (7)


x(n)

L0(120596)

D0(120596)

L1(120596)

L1(120596)

D1(120596)

D1(120596)

darr D2

darr D2

DFB

DFB

DFB

y3(n)

y00(n)simy

07(n)

y10(n)simy

13(n)

y20(n)simy

23(n)


Step 2

119889119896+1

= min119889

1198891 + 12058310038171003817100381710038171003817119889119896minus Φ119891119896+1

minus 11988711989610038171003817100381710038171003817

2

2 (8)

Step 3

119887119896+1

= 119887119896+ (Φ119891

119896+1minus 119889119896+1

) (9)



min 10038171003817100381710038171198911003817100381710038171003817TV +

1003817100381710038171003817Φ11989110038171003817100381710038171

st 1003817100381710038171003817119860119891 minus 1199011003817100381710038171003817

2

2lt 1205902

(10)

min 10038171003817100381710038171198911003817100381710038171003817TV +

1003817100381710038171003817Φ11989110038171003817100381710038171

+ 1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2 (11)


Step 1

119891119896+1

= arg min119891

1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2+ 120574


11989610038171003817100381710038171003817

2

2

+ 12058310038171003817100381710038171003817119889119896

120593minus Φ119891 minus 119887

119896

120593

10038171003817100381710038171003817

2

2

(12)

Step 2

119889119896+1

= min119889

1198891 + 12057410038171003817100381710038171003817119889119896minus nabla119891119896+1

minus 11988711989610038171003817100381710038171003817

2

2

119889119896+1

120593= min119889120593

10038171003817100381710038171003817119889120593

100381710038171003817100381710038171+ 120583

10038171003817100381710038171003817119889119896

120593minus Φ119891119896+1

minus 119887119896

120593

10038171003817100381710038171003817

2

2

(13)

Step 3

119887119896+1

= 119887119896+ (nabla119891

119896+1minus 119889119896+1

)

119887119896+1

120593= 119887119896

120593+ (Φ119891

119896+1minus 119889119896+1

120593)

(14)


diate variables


119892 [119899119898 + 1]

= 2120582119860119879



minus 2120583Φ119879(119889119896

120593minus Φ119891119898minus 119887119896

120593)

119891119898+1

= 119891119898+ 120572119892 [119899119898 + 1]

(15)


119898



119898compose the


119889119896+1

= shrink(nabla119891119896+1 + 1198871198961

120582)

119889119896+1

120593= shrink(Φ119891119896+1 + 119887

119896

1205931

120583)

(16)








(A) initialization

119899 = 1 119891(ART-DATA)

[119899 1] = 0 (17)


119891(ART-DATA)

[119899119898] = 119891(ART-DATA)

[119899119898 minus 1]

+ 119860119898


[119899119898 minus 1]

119860119898sdot 119860119898

(18)




(119891119894119895)(ART-POS)

[119899]

= (119891119894119895)(ART-DATA)

[119899119873data] (119891119894119895)(ART-DATA)

[119899119873data] ge 0

0 (119891119894119895)(ART-DATA)

[119899119873data] lt 0

(19)


119896 = 1

119889 (119899) =10038171003817100381710038171003817119891(ART-DATA)

[119899 1] minus 119891(ART-POS)

[119899]100381710038171003817100381710038172

119891(NACTTV-DATA)

[119896 1] = 119891(ART-POS)

[119899]

119889119896

119909= nabla119909119891(ART-POS)

[119899]

119889119896

119910= nabla119910119891(ART-POS)

[119899]

119889119896

120593= Φ119891(ART-POS)

[119899]

119887119896

119909= 119887119896

119910= 119887119896

120593= 0

(20)


119889119901= 119860119898119891(NACTTV-DATA)


119892 [119896119898 minus 1]

= 2120582119860119898119889119901minus 2120574nabla

119879

119909(119889119896


119909)


119910(119889119896


119910)

minus 2120583Φ119879(119889119896

120593minus Φ119891 minus 119887

119896

120593)10038161003816100381610038161003816119891=119891(NACTTV-DATA)

[119896119898minus1]

119892 [119896119898 minus 1] =119892 [119896119898 minus 1]

1003816100381610038161003816119892 [119896119898 minus 1]1003816100381610038161003816

119891(NACTTV-DATA)

[119896119898]



(21)


(119891119894119895)(NACTTV-POS)

[119896 + 1]

= (119891119894119895)(NACTTV-DATA)

[119896119873data] (119891119894119895)(NACTTV-DATA)

[119896119873data] ge 0

0 (119891119894119895)(NACTTV-DATA)

[119896119873data] lt 0

(22)



119889119896+1


119909119891(NACTTV-POS)

[119896 + 1 1] + 119887119896

1199091

120582)

119887119896+1


119910119891(NACTTV-POS)

[119896 + 1 1] + 119887119896

1199101

120582)

119889119896+1


119896

1205931

120583)

119887119896+1

119909= 119887119896

119909+ (nabla119909119891(NACTTV-POS)

[119896 + 1 1] minus 119889119896+1

119909)

119887119896+1

119910= 119887119896

119910+ (nabla119910119891(NACTTV-POS)

[119896 + 1 1] minus 119889119896+1

119910)

119887119896+1

120593= 119887119896

120593+ (Φ119891

(NACTTV-POS)[119896 + 1 1] minus 119889

119896+1

120593)

(23)


119891(ART-DATA)

[119899 + 1 1] = 119891(NACTTV-POS)

[119870NACTTV 1] (24)


2






RMSE = radic

1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891119877

119894119895)2

(25)

where 119891119894119895


119894119895is the




(a) (b) (c)

(d) (e) (f)



UQI =4120590119891119891119877119891 times 119891

119877

(1205902

119891+ 1205902

119891119877) [(119891)

2

+ (119891119877

)

2

]

(26)

where

119891 =1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119894119895

119891119877

=1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119877

119894119895

1205902

119891=

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891)2

1205902

119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119877

119894119895minus 119891119877

)

2

120590119891119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891) (119891

119877

119894119895minus 119891119877

)

(27)


120579119894= 1 + 360 times

(119894 minus 1)

119873view 119894 = 1 2

119873view2

120579119894= 182 + 360 times

(119894 minus 119873view2)

119873view

119894 =119873view2

119873view2

+ 1 119873view

(28)



0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART

Noisy-free

(a)

OriginalART

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(b)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART-TV

Noisy-free

(c)

OriginalART-TV

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(d)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalSpBr-NACT

Noisy-free

(e)

SpBr-NACT

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

Original

(f)



0 50 100

002

004

006

008

01

012

Iteration numbers

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)


Noisy-free

ARTART-TVSpBr-NACT

09

092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100002

004

006

008

01

012

Iteration numbers

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


09

092

094

096

098

1U

QI

Noisy

(d)







0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

Projection angles

Noisy-free

ARTART-TVSpBr-NACT

0 50 100 150 200092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


092

094

096

098

1U

QI

Noisy

(d)





4 Conclusion






Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















x(n)

L0(120596)

D0(120596)

L1(120596)

L1(120596)

D1(120596)

D1(120596)

darr D2

darr D2

DFB

DFB

DFB

y3(n)

y00(n)simy

07(n)

y10(n)simy

13(n)

y20(n)simy

23(n)


Step 2

119889119896+1

= min119889

1198891 + 12058310038171003817100381710038171003817119889119896minus Φ119891119896+1

minus 11988711989610038171003817100381710038171003817

2

2 (8)

Step 3

119887119896+1

= 119887119896+ (Φ119891

119896+1minus 119889119896+1

) (9)



min 10038171003817100381710038171198911003817100381710038171003817TV +

1003817100381710038171003817Φ11989110038171003817100381710038171

st 1003817100381710038171003817119860119891 minus 1199011003817100381710038171003817

2

2lt 1205902

(10)

min 10038171003817100381710038171198911003817100381710038171003817TV +

1003817100381710038171003817Φ11989110038171003817100381710038171

+ 1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2 (11)


Step 1

119891119896+1

= arg min119891

1205821003817100381710038171003817119860119891 minus 119901

1003817100381710038171003817

2

2+ 120574


11989610038171003817100381710038171003817

2

2

+ 12058310038171003817100381710038171003817119889119896

120593minus Φ119891 minus 119887

119896

120593

10038171003817100381710038171003817

2

2

(12)

Step 2

119889119896+1

= min119889

1198891 + 12057410038171003817100381710038171003817119889119896minus nabla119891119896+1

minus 11988711989610038171003817100381710038171003817

2

2

119889119896+1

120593= min119889120593

10038171003817100381710038171003817119889120593

100381710038171003817100381710038171+ 120583

10038171003817100381710038171003817119889119896

120593minus Φ119891119896+1

minus 119887119896

120593

10038171003817100381710038171003817

2

2

(13)

Step 3

119887119896+1

= 119887119896+ (nabla119891

119896+1minus 119889119896+1

)

119887119896+1

120593= 119887119896

120593+ (Φ119891

119896+1minus 119889119896+1

120593)

(14)


diate variables


119892 [119899119898 + 1]

= 2120582119860119879



minus 2120583Φ119879(119889119896

120593minus Φ119891119898minus 119887119896

120593)

119891119898+1

= 119891119898+ 120572119892 [119899119898 + 1]

(15)


119898



119898compose the


119889119896+1

= shrink(nabla119891119896+1 + 1198871198961

120582)

119889119896+1

120593= shrink(Φ119891119896+1 + 119887

119896

1205931

120583)

(16)








(A) initialization

119899 = 1 119891(ART-DATA)

[119899 1] = 0 (17)


119891(ART-DATA)

[119899119898] = 119891(ART-DATA)

[119899119898 minus 1]

+ 119860119898


[119899119898 minus 1]

119860119898sdot 119860119898

(18)




(119891119894119895)(ART-POS)

[119899]

= (119891119894119895)(ART-DATA)

[119899119873data] (119891119894119895)(ART-DATA)

[119899119873data] ge 0

0 (119891119894119895)(ART-DATA)

[119899119873data] lt 0

(19)


119896 = 1

119889 (119899) =10038171003817100381710038171003817119891(ART-DATA)

[119899 1] minus 119891(ART-POS)

[119899]100381710038171003817100381710038172

119891(NACTTV-DATA)

[119896 1] = 119891(ART-POS)

[119899]

119889119896

119909= nabla119909119891(ART-POS)

[119899]

119889119896

119910= nabla119910119891(ART-POS)

[119899]

119889119896

120593= Φ119891(ART-POS)

[119899]

119887119896

119909= 119887119896

119910= 119887119896

120593= 0

(20)


119889119901= 119860119898119891(NACTTV-DATA)


119892 [119896119898 minus 1]

= 2120582119860119898119889119901minus 2120574nabla

119879

119909(119889119896


119909)


119910(119889119896


119910)

minus 2120583Φ119879(119889119896

120593minus Φ119891 minus 119887

119896

120593)10038161003816100381610038161003816119891=119891(NACTTV-DATA)

[119896119898minus1]

119892 [119896119898 minus 1] =119892 [119896119898 minus 1]

1003816100381610038161003816119892 [119896119898 minus 1]1003816100381610038161003816

119891(NACTTV-DATA)

[119896119898]



(21)


(119891119894119895)(NACTTV-POS)

[119896 + 1]

= (119891119894119895)(NACTTV-DATA)

[119896119873data] (119891119894119895)(NACTTV-DATA)

[119896119873data] ge 0

0 (119891119894119895)(NACTTV-DATA)

[119896119873data] lt 0

(22)



119889119896+1


119909119891(NACTTV-POS)

[119896 + 1 1] + 119887119896

1199091

120582)

119887119896+1


119910119891(NACTTV-POS)

[119896 + 1 1] + 119887119896

1199101

120582)

119889119896+1


119896

1205931

120583)

119887119896+1

119909= 119887119896

119909+ (nabla119909119891(NACTTV-POS)

[119896 + 1 1] minus 119889119896+1

119909)

119887119896+1

119910= 119887119896

119910+ (nabla119910119891(NACTTV-POS)

[119896 + 1 1] minus 119889119896+1

119910)

119887119896+1

120593= 119887119896

120593+ (Φ119891

(NACTTV-POS)[119896 + 1 1] minus 119889

119896+1

120593)

(23)


119891(ART-DATA)

[119899 + 1 1] = 119891(NACTTV-POS)

[119870NACTTV 1] (24)


2






RMSE = radic

1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891119877

119894119895)2

(25)

where 119891119894119895


119894119895is the




(a) (b) (c)

(d) (e) (f)



UQI =4120590119891119891119877119891 times 119891

119877

(1205902

119891+ 1205902

119891119877) [(119891)

2

+ (119891119877

)

2

]

(26)

where

119891 =1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119894119895

119891119877

=1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119877

119894119895

1205902

119891=

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891)2

1205902

119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119877

119894119895minus 119891119877

)

2

120590119891119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891) (119891

119877

119894119895minus 119891119877

)

(27)


120579119894= 1 + 360 times

(119894 minus 1)

119873view 119894 = 1 2

119873view2

120579119894= 182 + 360 times

(119894 minus 119873view2)

119873view

119894 =119873view2

119873view2

+ 1 119873view

(28)



0 50 100 150 2000

05

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15

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y va

lue

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Noisy-free

(a)

OriginalART

0 50 100 150 2000

05

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y va

lue

Noisy

(b)

0 50 100 150 2000

05

1

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Pixel number

Gre

y va

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OriginalART-TV

Noisy-free

(c)

OriginalART-TV

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(d)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalSpBr-NACT

Noisy-free

(e)

SpBr-NACT

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

Original

(f)



0 50 100

002

004

006

008

01

012

Iteration numbers

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)


Noisy-free

ARTART-TVSpBr-NACT

09

092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100002

004

006

008

01

012

Iteration numbers

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


09

092

094

096

098

1U

QI

Noisy

(d)







0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

Projection angles

Noisy-free

ARTART-TVSpBr-NACT

0 50 100 150 200092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


092

094

096

098

1U

QI

Noisy

(d)





4 Conclusion






Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















(119891119894119895)(ART-POS)

[119899]

= (119891119894119895)(ART-DATA)

[119899119873data] (119891119894119895)(ART-DATA)

[119899119873data] ge 0

0 (119891119894119895)(ART-DATA)

[119899119873data] lt 0

(19)


119896 = 1

119889 (119899) =10038171003817100381710038171003817119891(ART-DATA)

[119899 1] minus 119891(ART-POS)

[119899]100381710038171003817100381710038172

119891(NACTTV-DATA)

[119896 1] = 119891(ART-POS)

[119899]

119889119896

119909= nabla119909119891(ART-POS)

[119899]

119889119896

119910= nabla119910119891(ART-POS)

[119899]

119889119896

120593= Φ119891(ART-POS)

[119899]

119887119896

119909= 119887119896

119910= 119887119896

120593= 0

(20)


119889119901= 119860119898119891(NACTTV-DATA)


119892 [119896119898 minus 1]

= 2120582119860119898119889119901minus 2120574nabla

119879

119909(119889119896


119909)


119910(119889119896


119910)

minus 2120583Φ119879(119889119896

120593minus Φ119891 minus 119887

119896

120593)10038161003816100381610038161003816119891=119891(NACTTV-DATA)

[119896119898minus1]

119892 [119896119898 minus 1] =119892 [119896119898 minus 1]

1003816100381610038161003816119892 [119896119898 minus 1]1003816100381610038161003816

119891(NACTTV-DATA)

[119896119898]



(21)


(119891119894119895)(NACTTV-POS)

[119896 + 1]

= (119891119894119895)(NACTTV-DATA)

[119896119873data] (119891119894119895)(NACTTV-DATA)

[119896119873data] ge 0

0 (119891119894119895)(NACTTV-DATA)

[119896119873data] lt 0

(22)



119889119896+1


119909119891(NACTTV-POS)

[119896 + 1 1] + 119887119896

1199091

120582)

119887119896+1


119910119891(NACTTV-POS)

[119896 + 1 1] + 119887119896

1199101

120582)

119889119896+1


119896

1205931

120583)

119887119896+1

119909= 119887119896

119909+ (nabla119909119891(NACTTV-POS)

[119896 + 1 1] minus 119889119896+1

119909)

119887119896+1

119910= 119887119896

119910+ (nabla119910119891(NACTTV-POS)

[119896 + 1 1] minus 119889119896+1

119910)

119887119896+1

120593= 119887119896

120593+ (Φ119891

(NACTTV-POS)[119896 + 1 1] minus 119889

119896+1

120593)

(23)


119891(ART-DATA)

[119899 + 1 1] = 119891(NACTTV-POS)

[119870NACTTV 1] (24)


2






RMSE = radic

1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891119877

119894119895)2

(25)

where 119891119894119895


119894119895is the




(a) (b) (c)

(d) (e) (f)



UQI =4120590119891119891119877119891 times 119891

119877

(1205902

119891+ 1205902

119891119877) [(119891)

2

+ (119891119877

)

2

]

(26)

where

119891 =1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119894119895

119891119877

=1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119877

119894119895

1205902

119891=

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891)2

1205902

119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119877

119894119895minus 119891119877

)

2

120590119891119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891) (119891

119877

119894119895minus 119891119877

)

(27)


120579119894= 1 + 360 times

(119894 minus 1)

119873view 119894 = 1 2

119873view2

120579119894= 182 + 360 times

(119894 minus 119873view2)

119873view

119894 =119873view2

119873view2

+ 1 119873view

(28)



0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART

Noisy-free

(a)

OriginalART

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(b)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART-TV

Noisy-free

(c)

OriginalART-TV

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(d)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalSpBr-NACT

Noisy-free

(e)

SpBr-NACT

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

Original

(f)



0 50 100

002

004

006

008

01

012

Iteration numbers

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)


Noisy-free

ARTART-TVSpBr-NACT

09

092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100002

004

006

008

01

012

Iteration numbers

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


09

092

094

096

098

1U

QI

Noisy

(d)







0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

Projection angles

Noisy-free

ARTART-TVSpBr-NACT

0 50 100 150 200092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


092

094

096

098

1U

QI

Noisy

(d)





4 Conclusion






Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b) (c)

(d) (e) (f)



UQI =4120590119891119891119877119891 times 119891

119877

(1205902

119891+ 1205902

119891119877) [(119891)

2

+ (119891119877

)

2

]

(26)

where

119891 =1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119894119895

119891119877

=1

119872 times119873sum

0le119894lt119873

sum

0le119895lt119872

119891119877

119894119895

1205902

119891=

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891)2

1205902

119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119877

119894119895minus 119891119877

)

2

120590119891119891119877 =

1


0le119894lt119873

sum

0le119895lt119872

(119891119894119895minus 119891) (119891

119877

119894119895minus 119891119877

)

(27)


120579119894= 1 + 360 times

(119894 minus 1)

119873view 119894 = 1 2

119873view2

120579119894= 182 + 360 times

(119894 minus 119873view2)

119873view

119894 =119873view2

119873view2

+ 1 119873view

(28)



0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART

Noisy-free

(a)

OriginalART

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(b)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART-TV

Noisy-free

(c)

OriginalART-TV

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(d)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalSpBr-NACT

Noisy-free

(e)

SpBr-NACT

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

Original

(f)



0 50 100

002

004

006

008

01

012

Iteration numbers

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)


Noisy-free

ARTART-TVSpBr-NACT

09

092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100002

004

006

008

01

012

Iteration numbers

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


09

092

094

096

098

1U

QI

Noisy

(d)







0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

Projection angles

Noisy-free

ARTART-TVSpBr-NACT

0 50 100 150 200092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


092

094

096

098

1U

QI

Noisy

(d)





4 Conclusion






Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART

Noisy-free

(a)

OriginalART

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(b)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalART-TV

Noisy-free

(c)

OriginalART-TV

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

(d)

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

OriginalSpBr-NACT

Noisy-free

(e)

SpBr-NACT

0 50 100 150 2000

05

1

15

Pixel number

Gre

y va

lue

Noisy

Original

(f)



0 50 100

002

004

006

008

01

012

Iteration numbers

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)


Noisy-free

ARTART-TVSpBr-NACT

09

092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100002

004

006

008

01

012

Iteration numbers

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


09

092

094

096

098

1U

QI

Noisy

(d)







0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

Projection angles

Noisy-free

ARTART-TVSpBr-NACT

0 50 100 150 200092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


092

094

096

098

1U

QI

Noisy

(d)





4 Conclusion






Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 50 100

002

004

006

008

01

012

Iteration numbers

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)


Noisy-free

ARTART-TVSpBr-NACT

09

092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100002

004

006

008

01

012

Iteration numbers

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


09

092

094

096

098

1U

QI

Noisy

(d)







0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

Projection angles

Noisy-free

ARTART-TVSpBr-NACT

0 50 100 150 200092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


092

094

096

098

1U

QI

Noisy

(d)





4 Conclusion






Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy-free

ARTART-TVSpBr-NACT

(a)

Projection angles

Noisy-free

ARTART-TVSpBr-NACT

0 50 100 150 200092

094

096

098

1

UQ

I

(b)

ARTART-TVSpBr-NACT

0 50 100 150 2000

002

004

006

008

01

012

Projection angles

RMSE

Noisy

(c)

ARTART-TVSpBr-NACT


092

094

096

098

1U

QI

Noisy

(d)





4 Conclusion






Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















research article a ct reconstruction algorithm based on non...

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