Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 250689 8 pageshttpdxdoiorg1011552013250689

Research ArticleAn Object-Oriented Simulator for 3D Digital BreastTomosynthesis Imaging System

Saeed Seyyedi1 Kubra Cengiz2 Mustafa Kamasak2 and Isa Yildirim1

1 Department of Electrical and Electronics Engineering Istanbul Technical University 34469 Istanbul Turkey2Department of Computer and Informatics Engineering Istanbul Technical University 34469 Istanbul Turkey

Correspondence should be addressed to Isa Yildirim iyildirimituedutr

Received 13 June 2013 Revised 11 September 2013 Accepted 18 September 2013

Academic Editor Emil Alexov

Copyright copy 2013 Saeed Seyyedi 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

Digital breast tomosynthesis (DBT) is an innovative imaging modality that provides 3D reconstructed images of breast to detectthe breast cancer Projections obtained with an X-ray source moving in a limited angle interval are used to reconstruct 3D image ofbreast Several reconstruction algorithms are available for DBT imaging Filtered back projection algorithm has traditionally beenused to reconstruct images from projections Iterative reconstruction algorithms such as algebraic reconstruction technique (ART)were later developed Recently compressed sensing based methods have been proposed in tomosynthesis imaging problem Wehave developed an object-oriented simulator for 3D digital breast tomosynthesis (DBT) imaging system using C++ programminglanguage The simulator is capable of implementing different iterative and compressed sensing based reconstruction methods on3D digital tomosynthesis data sets and phantom models A user friendly graphical user interface (GUI) helps users to select andrun the desired methods on the designed phantom models or real data sets The simulator has been tested on a phantom studythat simulates breast tomosynthesis imaging problem Results obtained with various methods including algebraic reconstructiontechnique (ART) and total variation regularized reconstruction techniques (ART+TV) are presented Reconstruction results of themethods are compared both visually and quantitatively by evaluating performances of themethods usingmean structural similarity(MSSIM) values

1 Introduction

Breast cancer is one of the three most commonly diagnosedtypes of cancer among thewomen in theUSA in 2012 It is alsoknown as one of themost common causes of cancer deaths inthe USA [1] Diagnosis of this type of cancer in its early stagesmakes the treatment simpler and more likely to be effective

Digital breast tomosynthesis (DBT) is an innovativeimaging modality that provides 3D reconstructed images ofa patientrsquos breast to diagnose the breast cancer [2] Conven-tionally several imaging modalities such as mammographyand ultrasound have been used in diagnosing breast cancersAmong those modalities X-ray mammography has beenregarded as the gold standard for diagnosis Since X-raymammography image is two-dimensional it is limited byoverlapping tissue structure [3] The first study of geometrictomography by Plantes introduced the concept of conven-tional tomosynthesis [4] Garrison et al [5] Richards [6]Miller et al [7] and Grant [8] were the first scientists who

studied three-dimensional tomography Moreover Grant inhis study introduced the term ldquotomosynthesisrdquo system [8]DBT overcomes the overlapping limitation ofmammographyby providing slice images of the breast DBT uses projectionsobtained with an X-ray source moving in a limited angleinterval to reconstruct 3D image of breast

A number of algorithms have been addressed to recon-struct the images Algebraic reconstruction technique (ART)was developed by the Polish mathematician Kaczmarz in1937 [9] Recently it has been proven that a sparse imagecan be reconstructed from an undersampled data set viatotal variation (TV) method [10 11] Figure 1 schematicallyillustrates a digital tomosynthesis system and its three mainparts X-ray source object and detector As shown in thisfigure the X-ray source rotates around the breast in the step-and-shoot (SAS) mode and makes exposure after a completestop at each position The breast is fixed using a set of pedalsto avoid the movement during the scan time and the detectoris capable of high frame rate and has exceptional detective

2 Computational and Mathematical Methods in Medicine

X-ray tube

Breast

Detector

Center

Figure 1 A simple schematic of digital tomosynthesis system

quantum efficiency (DQE) making it well suited to rapidacquisition of a large number of low-dose projection images[2 3]

DBT has the potential to improve the sensitivity in thedetection of breast cancer due to reduced overlap of breasttissues which enables earlier detection It also significantlyimproves the specificity with the 3D data available a 3Danalysis of the distribution of microcalcifications or a 3Danalysis concerning shape determining margins and size oflesions might be easier [3 12]

Several simulators have been implemented to simulate thereconstruction algorithms In 1970 SNARK was developedby Richard Gordon to evaluate different reconstruction algo-rithms Later different versions of SNARK were developedto simulate CT and PET systems [13] In 2010 Hansen et aldevelopedAIRTools package for 2D algebraic reconstructiontechniques on MATLAB [14] Both packages were imple-mented only for 2D models

In this study we introduce a simulator for 3D breasttomosynthesis imaging system using C++ programming lan-guageThere are other limited view angle imaging simulatorsavailable such as AIR tools However the former simulationsoftware was typically developed using MATLAB scripts for2D data Our simulator is specially designed to simulate aDBT system that takes projections of an arbitrary phantomand reconstructs it using the acquired images from projec-tions by applying one of the implemented reconstructionmethods in the simulator that is chosen by the user It isalso able to run a set of newly proposed reconstructionmethods with total variation (TV) regularization algorithmsand produce the results such as the image of the layer ofinterest contrast to noise (CNR) root mean square error(RMSE) and structural similarity (SSIM) diagrams

In this paper we briefly describe the reconstructiontechniques used in the simulator and then the characteristicsof the simulator will be provided The results obtained fromthe simulator on a designed sample model are given

2 Methods

21 Arithmetic Reconstruction Technique (ART) The alge-braic reconstruction technique (ART) is an iterative imagereconstruction with a long history and rich literature Firstof all it was designed by Kaczmarz in 1937 [9] and it wasindependently used by Gordon et al in image reconstruction[13] ART is a reconstruction algorithm that uses a set ofprojections to reconstruct the desired object [15 16]

The term ray sum takes the place of the line integral intransform-based methods The ray sum 119910

119894 measured with

the 119894th ray is expressed as

119873

sum

119895=1

119886119894119895119909119895= 119910119894

119894 = 1 2 119872

119895 = 1 2 119873(1)

where 119886119894119895is the weighting parameter which stands for the

influence of 119895th cell on the 119894th ray line integral 119909119895is the

constant intensity value of the 119895th cell119873 is the total numberof cells and 119872 is the total number of rays Conventionmatrix inversion methods mentioned above would be usefulto solve (1) if 119872 and 119873 are small and the problem iswell posed Iterative methods are introduced for ill-posedinversion problems with large values of119873 and119872 Expandedform of (1) can be written as

119886111199091+ 119886121199092+ 119886131199093+ sdot sdot sdot + 119886

1119873119909119873= 1199101

119886211199091+ 119886221199092+ 119886231199093+ sdot sdot sdot + 119886

2119873119909119873= 1199102

11988611987211199091+ 11988611987221199092+ 11988611987231199093+ sdot sdot sdot + 119886119909

119873= 119910119872

(2)

If there is a unique solution to (2) then the intersectionsof the planes to be defined by these equations are a singlepoint in 119873 dimensional space Finding the solution viasubsequent projections is known as the Kaczmarz methodwhich forms the basis of ARTThe implementation procedurestarts with an initial guess (0) at the solution and (0)isprojected on the first plane in (2) giving (1) Then (1) isprojected on the second plane giving (2) thus the initialguess is updated so on This procedure can be formulated asprojection of (119894minus1) on ith plane yields (119894)

119909(119894+1)

119895= 119909(119894)

119895+(119910119894minus sum119873

119896=1119886119894119896sdot 119909(119894)

119896)

sum119873

119896=11198862119894119896

119886119894119895

119895 = 1 2 119873

119894 = 1 2 119872

(3)

Equation (3) states that the previous intensity values ofthe estimated image (119894minus1)s are updated by adding an errorparameter Δ119909(119894)

119895which is the difference between measured

ray sum 119910119894 and the computed ray sum sum119873

119896=1119886119894119896sdot 119909(119894minus1)

119896

normalized by sum119873119896=11198862

119894119896 This process is repeated until all the

projections are considered and all the pixel values convergeto a solution [13 16]

22 Simultaneous Arithmetic Reconstruction Technique(SART) ART method was the first iterative algorithm used

Computational and Mathematical Methods in Medicine 3

Graphical user interface

(GUI)

Configuration Projection ReconstructionParameters Projection results

Load Save

XML file

New objects(rectangular parallelepipedellipsoid and

sphere)

Initial objects(rectangular parallelepipedellipsoid and

sphere)

LOI CNR SSIM and

RMSE diagrams

Layer of interestnumber of iterations

Figure 2 Flow diagram for DBT simulator

Figure 3 Simulatorrsquos configuration part user inserts simulatorparameters manually or by loading an xml file

in CT [13] In 1984 the simultaneous algebraic reconstructiontechnique (SART) was proposed with major alterations inthe ART [17 18] SART as described by Andersen and Kak(1984) is given by

119909(119896+1)

119895= 119909119896

119895+

120596

sum119894119886119894119895

sum

119894

119886119894119895(119910119894minus sum119898119886119894119898119909(119896)

119898)

sum119896119886119894119896

(4)

Figure 4 Simulatorrsquos projection part user can insert more objectsinto the phantom and run the projection method

where 0 lt 120596 lt 2 represents relaxation parameter foriterations 119896 = 0 1 119896 we set 120596 to 1 for our simulationAlthough larger values may speed up convergence if thevalue is too large too much weight is given to the lastprojection which prevents convergence Smaller values causethe algorithm to converge slowly which is not acceptable forreal-time applications and systems with a huge number ofpixels [19]

4 Computational and Mathematical Methods in Medicine

Figure 5 Simulatorrsquos reconstruction part user can choose areconstruction method insert the layer of interest and number ofiterations and change the initial objects characteristics then run thedesired method

23 Compressed Sensing (CS) Compressed sensing (CS)image reconstruction is used to reconstruct a sparse image byminimizing the 1198971 norm of the sparse image There are somesignificant factors in original CS method to be considered(1) the image must be sparse (2) reconstruction of the imagemust be done using a nonlinear method and (3) the standardlinear reconstruction method should generate incoherentview aliasing artifacts by applying the sparsifying transformin (7) [10 11] The image can be sparsified using sparsifyingtransform (Ψ) which is a linear transform operator and isused to transform nonsparse version of image 119883 to thesparsified version Equation (5) shows the constrained min-imization problem which CS image reconstruction theorytries to solve iteratively

min Ψ1198831 (5)

st 119860119883 = 119884 (6)

24 Prior Image Constrained Compressed Sensing (PICCS)PICCS method considers a high quality prior image 119883

119875to

reconstruct the image 119883 from an undersampled data set bysolving the following constrained minimization problem

min [1205721003817100381710038171003817Ψ1 (119883 minus 119883119884)10038171003817100381710038171 + (1 minus 120572)

1003817100381710038171003817Ψ2 (119883)10038171003817100381710038171] (7)

where 119860119883 = 119884 is assumed and Ψ1and Ψ

2can be any

transform like those used in CS and they can be the sameor different transforms and 120572 is the regularization parameterthat can be selected between 0 and 1 for 120572 = 0 the PICCSalgorithm is equivalent to the known CS method [10 11]

The constrainedminimization problemof PICCSmethodis numerically implemented using arithmetic reconstruction

technique (ART) and the total variation (TV) regularizationmethods respectively ART is used to reconstruct the image119883 by considering the consistency condition 119860119883 = 119884 andTV regularization of 119883 is defined as 1198971 norm of the discretegradient of the image Equation (8) shows the 2D TV of pixel119883(119894 119895) in the image

TV2D (119883 (119894 119895)) =

119873

sum

119894119895=1

10038161003816100381610038161003816nabla119894119895(119883 (119894 119895))

100381610038161003816100381610038161 (8)

The discrete gradient of image in pixel (119894 119895) is defined as

10038161003816100381610038161003816nabla119894119895119883(119894 119895)

10038161003816100381610038161003816= radic(119863

119909119883)2+ (119863119910119883)2

(9)

where 119883(119894 119895) is the intensity value at pixel (119894 119895) 119863119909119883 =

119883(119894 119895) minus 119883(119894 + 1 119895) and119863119910119883 = 119883(119894 119895) minus 119883(119894 119895 + 1)

TV regularization can be assumed in 3D objects whereit shows better performance in the 119911-axis neighborhood oraxial direction of the object (10) shows the 3DTVof the voxel119883(119894 119895 119896) in 3D object

TV3D (119883 (119894 119895 119896)) =

119873

sum

119894119895119896=1

10038161003816100381610038161003816nabla119894119895119896(119883 (119894 119895 119896))

100381610038161003816100381610038161 (10)

The discrete gradient of image in voxel (119894 119895 119896) is shown in

10038161003816100381610038161003816nabla119894119895119896119883(119894 119895 119896)

10038161003816100381610038161003816= radic(119863

119909119883)2+ (119863119910119883)2

+ (119863119911119883)2 (11)

where 119883(119894 119895 119896) is the intensity value at voxel (119894 119895 119896) 119863119909119883 =

119883(119894 119895 119896) minus119883(119894 + 1 119895 119896)119863119910119883 = 119883(119894 119895 119896) minus119883(119894 119895 + 1 119896) and

119863119911119883 = 119883(119894 119895 119896) minus 119883(119894 119895 119896 + 1)The TV method is applied after each iteration of ART

method After applying TV the forward projection runsagain TV method can be applied to 2D or 3D data 2D TVis applied for each layer of 3D object but the 3D version ofTV regularization is applied to the whole of the 3D object atthe end of each iterationThe pseudocode of the ARTwith 3DTV or SART with 3D TV implementation is shown below

119883119875larr Prior Image

119884119875larr Forward Projection of119883

119875

119884 larrMeasured Projections

while (119884119875minus 119884 gt 120576)

for each iteration

Calculate Δ119883119896 Using ARTUpdate119883(119896)

119875(119883(119896+1)119875

= 119883(119896)

119875+ Δ119883(119896))

end for each iteration3D Total Variation Regularization119884119875larr Forward Projection of119883

119875

end while

Computational and Mathematical Methods in Medicine 5

20 40 60 80 100 120

20

40

60

80

100

1200

5

10

15

(a)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

5

(b)

Figure 6 Original 3D phantom (a) objects with higher absorption in the upper layers (b) the LOI of the phantom

3 Software Design and Implementation

3D tomosynthesis simulator was written in C++ An object-oriented programming language and Net framework wasused to design the graphical user interface (GUI) of thesimulator on Visual StudioNet 2010 which was run on apersonal computer with Intel Core i7 200GHz processor and6GB RAM memory Unlike the procedural programminglanguages that separate data from operations object-orientedC++ programming language is capable of considering acollection of classes that combine data and operations ondata

Figure 2 shows the flow diagram of the simulator wherethree main parts of the simulator and related operators areshown As shown in this figure the simulator consists of threemain classes configuration (parameters) projection andreconstruction classesThe first class includes the parametersof all the system parts and functions to readwrite data fromXML files Three main parts of DBT system are defined asthree different subclasses which refer to X-ray source phan-tom and detector The projection class includes methods toreceive the system parameters from the configuration partand to find the projection images of a particular phantomwhich could be used by the reconstruction class to rundifferent reconstruction methods The simulator includes agraphical user interface (GUI) which facilitates design andediting of a phantom executes the projection and reconstruc-tion method and saves the results

Figure 3 shows the configuration interface of the simula-tor that allows one to insert or select the parameters of the X-ray source phantom and detector such as their location anddimensionsmanually in the specified places or by loading thexml files that include the desired data in a predefined formatOne can load the xml file by pressing the load button andinserting the path for the desired file

It is possible to choose a set of small 3D objects such asrectangular parallelepiped sphere or ellipsoid for both orig-inal and initial objects to start the reconstruction procedure

Figure 4 shows the projection interface of the simulatorit is possible to insert and edit a set of small 3D objects suchas rectangular parallelepiped sphere or ellipsoid in the phan-tom which are displayed sequentially in the object list part ofthe form The user is required to insert the characteristics ofeach object to generate the desired phantom After generatingthe phantom the projection task can be performed to getthe projection images of the phantom The results of theprojection on the detector could be displayed by choosing thedesired angle of projection

Figure 5 exhibits the reconstruction interface where theuser can choose one of the reconstructionmethods includingART SART ART with 3DTV and SART with 3DTV andcan insert the number of iterations (NOI) of the iterativemethod with the layer of interest (LOI) number of the 3Dphantom It is also possible to revise the initial object listfor the reconstruction After running the chosen method thereconstruction results such as the image of all layers includingthe LOI of the object and also the structural similarity (SSIM)contrast-to-noise-ratio (CNR) and root mean square error(RMSE) diagramswill be shown as the output of the program

A special xml file format is designed to store all of thesystem characteristics using the different tags such as X-raysource detector phantom and reconstruction A user caninsert and edit the contents of the file and load it automaticallyto update the parameters of the system

4 Results

In order to exhibit the performance of the different recon-structionmethods run by the simulator a 3D phantommodel

6 Computational and Mathematical Methods in Medicine

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(a)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(b)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(c)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(d)

Figure 7 Images of the LOI of the reconstructed phantom (11th layer of the phantom) (a) reconstructed LOI image using ART method (b)reconstructed LOI image using SART method (c) reconstructed LOI image using ART+TV 3D method and (d) reconstructed LOI imageusing SART+TV 3D method

was designed with resolution of 128 times 128 voxels in 16 layersThis phantom was created to imitate the overlapping tissueproblem of the breast imaging The phantom includes somesmaller objects where objects with the low X-ray absorptionare obscured by the objects with higher X-ray absorption(Figure 6) Measured projections were generated from thephantom for the range of scan angles from minus25∘ to +25∘ and11 projections

Parameters of the simulator and phantom are listed inTable 1

Reconstructed images of the LOI for ART SARTART+TV 3D and SART+TV 3D are shown in Figure 7(a) toFigure 7(d) respectively

One of the mostly used image quality metrics is the meansquare error (MSE) because of its simplicity in calculating

Table 1 Simulation parameters

Parameter ValueSource to detector distance 300 pixelsObject to detector distance 100 pixelsScan angle 50∘ degrees (minus25∘ to +25∘)Number of projections 11 projectionsTV regularization parameter 08Phantom size 128 times 128 times 16

Detector size 160 times 160 times 1

and clear physical meaning but this is not a very appropriatemetric to exhibit the visual quality of the images [20ndash22] Anumber of quality assessment methods are developed that

Computational and Mathematical Methods in Medicine 7

0 5 10 1504

05

06

07

08

09

1

Iteration

ART SART

MSS

IM

ART + TV3D SART + TV3D

Figure 8 MSSIM value comparison of ART ART+3D TV SARTand SART+3D TV for the layer of interest

implement the characteristics of the human visual system(HVS) One of the well-known quality assessment methodsis the measure of structural similarity (SSIM) that compareslocal patterns of pixel values which are normalized foramount of luminance and contrast [23]

The SSIM index is shown

SSIM (119909 119910) =(2120583119909120583119910+ 1198621) (120590119909119910+ 1198622)

(1205832119909+ 1205832119910+ 1198621) (1205902119909+ 1205902119910+ 1198622) (12)

where 120583119909and 120583119910refer to the mean of the intensities of signals

119909 and119910 respectively and120590119909and120590119910are the standard deviation

of them 1198621and 119862

2are given

119862119898= (119870119898119871)2 119898 = 1 2 (13)

where 119871 is the dynamic range of the pixel values and119870119898≪ 1

for 119896 = 1 2 are small constantsPractically we need a single overall quality measure of the

entire image In this study we used a mean SSIM (MSSIM)index to evaluate the overall image quality

MSSIM (119883 119884) = 1

119872

119879

sum

119895=1

SSIM (119909119895+ 119910119895) (14)

where 119883 and 119884 refer to original and reconstructed imagesrespectively 119909

119895and 119910

119895are the image contents at the 119895th local

window and 119879 is the number of local windows of the imageMSSIM indexes of the reconstruction methods tested

with the simulator are given in Figure 8 Compressed sensedmethods implemented as ART+TV 3D and SART+TV 3Dprovided improved results compared with the results of ARTand SART while ART and SART performed similarly

The time needed to perform a simulation study dependson the complexity of the phantom and detector size In this

study the average time required to complete each iterationof ART on the proposed simulator is 128 seconds which isobtained after measuring the first fifteen iterations The sameproblem takes 28800 seconds on the MATLAB which wasonly implemented by us for comparison

5 Conclusion

A new simulator was designed for 3D DBT studies Oursimulator is capable of implementing several reconstructiontechniques including recently proposed compressed sensingbased methods A user friendly graphical user interface(GUI) helps users to select and run the desired methods onthe designed phantom models or real data sets The simu-lator was implemented for 3D limited view angle imagingproblems using C++ programming language whereas theformer simulation software was typically developed usingMATLAB scripts for 2D data We tested the simulator byrunning different reconstruction methods with a specific 3Dphantommodel which was created to imitate the overlappingtissue problem of the breast imaging We also compared themethods in the simulator by demonstrating reconstructedimages of LOI and evaluating their performances usingRMSE and MSSIM metrics The simulator can be extendedby including new reconstruction methods

Conflict of Interests

The authors declared no conflict of interests

Acknowledgment

This work has been supported by TUBITAK The Scientificand Research Council of Turkey under Grant 111E086

References

[1] R Siegel DNaishadham andA Jemal ldquoCancer statistics 2012rdquoCA Cancer Journal for Clinicians vol 62 no 1 pp 10ndash29 2012

[2] J Baker J Lo A Hebecker Th Mertelmeier and J OrmanldquoDigital breast tomosynthesisrdquo Medical Solutions RSNA pp84ndash89 2006

[3] J T Dobbins III andD J Godfrey ldquoDigital x-ray tomosynthesiscurrent state of the art and clinical potentialrdquo Physics inMedicine and Biology vol 48 no 19 pp R65ndashR106 2003

[4] B Plantes ldquoEine neue methode zur differenzierung in derroentgenographie (planigraphie)rdquo Acta Radiologica vol 13 pp182ndash192 1932

[5] J B Garrison D G Grant W H Guier and R J JohnsldquoThree dimensional roentgenographyrdquoTheAmerican Journal ofRoentgenology RadiumTherapy and Nuclear Medicine vol 105no 4 pp 903ndash908 1969

[6] A G Richards ldquoVariable depth laminagraphyrdquo BiomedicalSciences Instrumentation vol 6 pp 194ndash199 1969

[7] E R Miller E M McCurry and B Hruska ldquoAn infinitenumber of laminagrams from a finite number of radiographsrdquoRadiology vol 98 no 2 pp 249ndash255 1971

8 Computational and Mathematical Methods in Medicine

[8] D G Grant ldquoTomosynthesis a three-dimensional radiographicimaging techniquerdquo IEEE Transactions on Biomedical Engineer-ing vol 19 no 1 pp 20ndash28 1972

[9] S Kaczmarz ldquoAngenaherteauflosung von systemenlinearergle-ichungenrdquo Bulletin de LrsquoAcademie Polonaise des Sciences etLet-tres vol 35 pp 355ndash357 1937

[10] G-H Chen J Tang and S Leng ldquoPrior image constrainedcompressed sensing (PICCS) a method to accurately recon-struct dynamic CT images from highly undersampled projec-tion data setsrdquoMedical Physics vol 35 no 2 pp 660ndash663 2008

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

[12] L T Niklason B T Christian L E Niklason et al ldquoDigitaltomosynthesis in breast imagingrdquo Radiology vol 205 no 2 pp399ndash406 1997

[13] R Gordon R Bender and G T Herman ldquoAlgebraic Recon-struction Techniques (ART) for three-dimensional electronmicroscopy and X-ray photographyrdquo Journal of TheoreticalBiology vol 29 no 3 pp 471ndash481 1970

[14] P Ch Hansen and M S Hansen ldquoAIR toolsmdasha MATLABpackage of algebraic iterative reconstruction methodsrdquo Journalof Computational and Applied Mathematics vol 236 no 8 pp2167ndash2178 2012

[15] D Raparia J Alessi and A Kponou ldquoThe algebraic reconstruc-tion technique (Art)rdquo in Proceedings of the 17th Particle Accel-erator Conference (PAC rsquo97) vol 2 pp 2023ndash2025 VancouverCanada May 1997

[16] T Nikazad Algebraic reconstruction methods [PhD thesis]Mathematics Department Linkoping Univerity 2008

[17] A H Andersen and A C Kak ldquoSimultaneous AlgebraicReconstruction Technique (SART) a superior implementationof the art algorithmrdquoUltrasonic Imaging vol 6 no 1 pp 81ndash941984

[18] M Jiang andGWang ldquoConvergence of the Simultaneous Alge-braic Reconstruction Technique (SART)rdquo IEEE Transactions onImage Processing vol 12 no 8 pp 957ndash961 2003

[19] T Hobiger T Kondo and Y Koyama ldquoConstrained Simulta-neous Algebraic Reconstruction Technique (C-SART)mdasha newand simple algorithm applied to ionospheric tomographyrdquoEarth Planets and Space vol 60 no 7 pp 727ndash735 2008

[20] B Girod ldquoWhatrsquos wrong with mean-squared errorrdquo in DigitalImages and Human Vision A BWatson Ed pp 207ndash220 MITPress Cambridge Mass USA 1993

[21] P C Teo and D J Heeger ldquoPerceptual image distortionrdquoin Proceedings of the IEEE International Conference on ImageProcessing (ICIP rsquo94) vol 2 pp 982ndash986 Austin Tex USA1994

[22] A M Eskicioglu and P S Fisher ldquoImage quality measures andtheir performancerdquo IEEE Transactions on Communications vol43 no 12 pp 2959ndash2965 1995

[23] Zh Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment from error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004

4 Computational and Mathematical Methods in Medicine

Figure 5 Simulatorrsquos reconstruction part user can choose areconstruction method insert the layer of interest and number ofiterations and change the initial objects characteristics then run thedesired method

23 Compressed Sensing (CS) Compressed sensing (CS)image reconstruction is used to reconstruct a sparse image byminimizing the 1198971 norm of the sparse image There are somesignificant factors in original CS method to be considered(1) the image must be sparse (2) reconstruction of the imagemust be done using a nonlinear method and (3) the standardlinear reconstruction method should generate incoherentview aliasing artifacts by applying the sparsifying transformin (7) [10 11] The image can be sparsified using sparsifyingtransform (Ψ) which is a linear transform operator and isused to transform nonsparse version of image 119883 to thesparsified version Equation (5) shows the constrained min-imization problem which CS image reconstruction theorytries to solve iteratively

min Ψ1198831 (5)

st 119860119883 = 119884 (6)

24 Prior Image Constrained Compressed Sensing (PICCS)PICCS method considers a high quality prior image 119883

119875to

reconstruct the image 119883 from an undersampled data set bysolving the following constrained minimization problem

min [1205721003817100381710038171003817Ψ1 (119883 minus 119883119884)10038171003817100381710038171 + (1 minus 120572)

1003817100381710038171003817Ψ2 (119883)10038171003817100381710038171] (7)

where 119860119883 = 119884 is assumed and Ψ1and Ψ

2can be any

transform like those used in CS and they can be the sameor different transforms and 120572 is the regularization parameterthat can be selected between 0 and 1 for 120572 = 0 the PICCSalgorithm is equivalent to the known CS method [10 11]

The constrainedminimization problemof PICCSmethodis numerically implemented using arithmetic reconstruction

technique (ART) and the total variation (TV) regularizationmethods respectively ART is used to reconstruct the image119883 by considering the consistency condition 119860119883 = 119884 andTV regularization of 119883 is defined as 1198971 norm of the discretegradient of the image Equation (8) shows the 2D TV of pixel119883(119894 119895) in the image

TV2D (119883 (119894 119895)) =

119873

sum

119894119895=1

10038161003816100381610038161003816nabla119894119895(119883 (119894 119895))

100381610038161003816100381610038161 (8)

The discrete gradient of image in pixel (119894 119895) is defined as

10038161003816100381610038161003816nabla119894119895119883(119894 119895)

10038161003816100381610038161003816= radic(119863

119909119883)2+ (119863119910119883)2

(9)

where 119883(119894 119895) is the intensity value at pixel (119894 119895) 119863119909119883 =

119883(119894 119895) minus 119883(119894 + 1 119895) and119863119910119883 = 119883(119894 119895) minus 119883(119894 119895 + 1)

TV regularization can be assumed in 3D objects whereit shows better performance in the 119911-axis neighborhood oraxial direction of the object (10) shows the 3DTVof the voxel119883(119894 119895 119896) in 3D object

TV3D (119883 (119894 119895 119896)) =

119873

sum

119894119895119896=1

10038161003816100381610038161003816nabla119894119895119896(119883 (119894 119895 119896))

100381610038161003816100381610038161 (10)

The discrete gradient of image in voxel (119894 119895 119896) is shown in

10038161003816100381610038161003816nabla119894119895119896119883(119894 119895 119896)

10038161003816100381610038161003816= radic(119863

119909119883)2+ (119863119910119883)2

+ (119863119911119883)2 (11)

where 119883(119894 119895 119896) is the intensity value at voxel (119894 119895 119896) 119863119909119883 =

119883(119894 119895 119896) minus119883(119894 + 1 119895 119896)119863119910119883 = 119883(119894 119895 119896) minus119883(119894 119895 + 1 119896) and

119863119911119883 = 119883(119894 119895 119896) minus 119883(119894 119895 119896 + 1)The TV method is applied after each iteration of ART

method After applying TV the forward projection runsagain TV method can be applied to 2D or 3D data 2D TVis applied for each layer of 3D object but the 3D version ofTV regularization is applied to the whole of the 3D object atthe end of each iterationThe pseudocode of the ARTwith 3DTV or SART with 3D TV implementation is shown below

119883119875larr Prior Image

119884119875larr Forward Projection of119883

119875

119884 larrMeasured Projections

while (119884119875minus 119884 gt 120576)

for each iteration

Calculate Δ119883119896 Using ARTUpdate119883(119896)

119875(119883(119896+1)119875

= 119883(119896)

119875+ Δ119883(119896))

end for each iteration3D Total Variation Regularization119884119875larr Forward Projection of119883

119875

end while

Computational and Mathematical Methods in Medicine 5

20 40 60 80 100 120

20

40

60

80

100

1200

5

10

15

(a)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

5

(b)

Figure 6 Original 3D phantom (a) objects with higher absorption in the upper layers (b) the LOI of the phantom

3 Software Design and Implementation

3D tomosynthesis simulator was written in C++ An object-oriented programming language and Net framework wasused to design the graphical user interface (GUI) of thesimulator on Visual StudioNet 2010 which was run on apersonal computer with Intel Core i7 200GHz processor and6GB RAM memory Unlike the procedural programminglanguages that separate data from operations object-orientedC++ programming language is capable of considering acollection of classes that combine data and operations ondata

Figure 2 shows the flow diagram of the simulator wherethree main parts of the simulator and related operators areshown As shown in this figure the simulator consists of threemain classes configuration (parameters) projection andreconstruction classesThe first class includes the parametersof all the system parts and functions to readwrite data fromXML files Three main parts of DBT system are defined asthree different subclasses which refer to X-ray source phan-tom and detector The projection class includes methods toreceive the system parameters from the configuration partand to find the projection images of a particular phantomwhich could be used by the reconstruction class to rundifferent reconstruction methods The simulator includes agraphical user interface (GUI) which facilitates design andediting of a phantom executes the projection and reconstruc-tion method and saves the results

Figure 3 shows the configuration interface of the simula-tor that allows one to insert or select the parameters of the X-ray source phantom and detector such as their location anddimensionsmanually in the specified places or by loading thexml files that include the desired data in a predefined formatOne can load the xml file by pressing the load button andinserting the path for the desired file

It is possible to choose a set of small 3D objects such asrectangular parallelepiped sphere or ellipsoid for both orig-inal and initial objects to start the reconstruction procedure

Figure 4 shows the projection interface of the simulatorit is possible to insert and edit a set of small 3D objects suchas rectangular parallelepiped sphere or ellipsoid in the phan-tom which are displayed sequentially in the object list part ofthe form The user is required to insert the characteristics ofeach object to generate the desired phantom After generatingthe phantom the projection task can be performed to getthe projection images of the phantom The results of theprojection on the detector could be displayed by choosing thedesired angle of projection

Figure 5 exhibits the reconstruction interface where theuser can choose one of the reconstructionmethods includingART SART ART with 3DTV and SART with 3DTV andcan insert the number of iterations (NOI) of the iterativemethod with the layer of interest (LOI) number of the 3Dphantom It is also possible to revise the initial object listfor the reconstruction After running the chosen method thereconstruction results such as the image of all layers includingthe LOI of the object and also the structural similarity (SSIM)contrast-to-noise-ratio (CNR) and root mean square error(RMSE) diagramswill be shown as the output of the program

A special xml file format is designed to store all of thesystem characteristics using the different tags such as X-raysource detector phantom and reconstruction A user caninsert and edit the contents of the file and load it automaticallyto update the parameters of the system

4 Results

In order to exhibit the performance of the different recon-structionmethods run by the simulator a 3D phantommodel

6 Computational and Mathematical Methods in Medicine

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(a)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(b)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(c)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(d)

Figure 7 Images of the LOI of the reconstructed phantom (11th layer of the phantom) (a) reconstructed LOI image using ART method (b)reconstructed LOI image using SART method (c) reconstructed LOI image using ART+TV 3D method and (d) reconstructed LOI imageusing SART+TV 3D method

was designed with resolution of 128 times 128 voxels in 16 layersThis phantom was created to imitate the overlapping tissueproblem of the breast imaging The phantom includes somesmaller objects where objects with the low X-ray absorptionare obscured by the objects with higher X-ray absorption(Figure 6) Measured projections were generated from thephantom for the range of scan angles from minus25∘ to +25∘ and11 projections

Parameters of the simulator and phantom are listed inTable 1

Reconstructed images of the LOI for ART SARTART+TV 3D and SART+TV 3D are shown in Figure 7(a) toFigure 7(d) respectively

One of the mostly used image quality metrics is the meansquare error (MSE) because of its simplicity in calculating

Table 1 Simulation parameters

Parameter ValueSource to detector distance 300 pixelsObject to detector distance 100 pixelsScan angle 50∘ degrees (minus25∘ to +25∘)Number of projections 11 projectionsTV regularization parameter 08Phantom size 128 times 128 times 16

Detector size 160 times 160 times 1

and clear physical meaning but this is not a very appropriatemetric to exhibit the visual quality of the images [20ndash22] Anumber of quality assessment methods are developed that

Computational and Mathematical Methods in Medicine 7

0 5 10 1504

05

06

07

08

09

1

Iteration

ART SART

MSS

IM

ART + TV3D SART + TV3D

Figure 8 MSSIM value comparison of ART ART+3D TV SARTand SART+3D TV for the layer of interest

implement the characteristics of the human visual system(HVS) One of the well-known quality assessment methodsis the measure of structural similarity (SSIM) that compareslocal patterns of pixel values which are normalized foramount of luminance and contrast [23]

The SSIM index is shown

SSIM (119909 119910) =(2120583119909120583119910+ 1198621) (120590119909119910+ 1198622)

(1205832119909+ 1205832119910+ 1198621) (1205902119909+ 1205902119910+ 1198622) (12)

where 120583119909and 120583119910refer to the mean of the intensities of signals

119909 and119910 respectively and120590119909and120590119910are the standard deviation

of them 1198621and 119862

2are given

119862119898= (119870119898119871)2 119898 = 1 2 (13)

where 119871 is the dynamic range of the pixel values and119870119898≪ 1

for 119896 = 1 2 are small constantsPractically we need a single overall quality measure of the

entire image In this study we used a mean SSIM (MSSIM)index to evaluate the overall image quality

MSSIM (119883 119884) = 1

119872

119879

sum

119895=1

SSIM (119909119895+ 119910119895) (14)

where 119883 and 119884 refer to original and reconstructed imagesrespectively 119909

119895and 119910

119895are the image contents at the 119895th local

window and 119879 is the number of local windows of the imageMSSIM indexes of the reconstruction methods tested

with the simulator are given in Figure 8 Compressed sensedmethods implemented as ART+TV 3D and SART+TV 3Dprovided improved results compared with the results of ARTand SART while ART and SART performed similarly

The time needed to perform a simulation study dependson the complexity of the phantom and detector size In this

study the average time required to complete each iterationof ART on the proposed simulator is 128 seconds which isobtained after measuring the first fifteen iterations The sameproblem takes 28800 seconds on the MATLAB which wasonly implemented by us for comparison

5 Conclusion

A new simulator was designed for 3D DBT studies Oursimulator is capable of implementing several reconstructiontechniques including recently proposed compressed sensingbased methods A user friendly graphical user interface(GUI) helps users to select and run the desired methods onthe designed phantom models or real data sets The simu-lator was implemented for 3D limited view angle imagingproblems using C++ programming language whereas theformer simulation software was typically developed usingMATLAB scripts for 2D data We tested the simulator byrunning different reconstruction methods with a specific 3Dphantommodel which was created to imitate the overlappingtissue problem of the breast imaging We also compared themethods in the simulator by demonstrating reconstructedimages of LOI and evaluating their performances usingRMSE and MSSIM metrics The simulator can be extendedby including new reconstruction methods

Conflict of Interests

The authors declared no conflict of interests

Acknowledgment

This work has been supported by TUBITAK The Scientificand Research Council of Turkey under Grant 111E086

References

[1] R Siegel DNaishadham andA Jemal ldquoCancer statistics 2012rdquoCA Cancer Journal for Clinicians vol 62 no 1 pp 10ndash29 2012

[2] J Baker J Lo A Hebecker Th Mertelmeier and J OrmanldquoDigital breast tomosynthesisrdquo Medical Solutions RSNA pp84ndash89 2006

[3] J T Dobbins III andD J Godfrey ldquoDigital x-ray tomosynthesiscurrent state of the art and clinical potentialrdquo Physics inMedicine and Biology vol 48 no 19 pp R65ndashR106 2003

[4] B Plantes ldquoEine neue methode zur differenzierung in derroentgenographie (planigraphie)rdquo Acta Radiologica vol 13 pp182ndash192 1932

[5] J B Garrison D G Grant W H Guier and R J JohnsldquoThree dimensional roentgenographyrdquoTheAmerican Journal ofRoentgenology RadiumTherapy and Nuclear Medicine vol 105no 4 pp 903ndash908 1969

[6] A G Richards ldquoVariable depth laminagraphyrdquo BiomedicalSciences Instrumentation vol 6 pp 194ndash199 1969

[7] E R Miller E M McCurry and B Hruska ldquoAn infinitenumber of laminagrams from a finite number of radiographsrdquoRadiology vol 98 no 2 pp 249ndash255 1971

8 Computational and Mathematical Methods in Medicine

[8] D G Grant ldquoTomosynthesis a three-dimensional radiographicimaging techniquerdquo IEEE Transactions on Biomedical Engineer-ing vol 19 no 1 pp 20ndash28 1972

[9] S Kaczmarz ldquoAngenaherteauflosung von systemenlinearergle-ichungenrdquo Bulletin de LrsquoAcademie Polonaise des Sciences etLet-tres vol 35 pp 355ndash357 1937

[10] G-H Chen J Tang and S Leng ldquoPrior image constrainedcompressed sensing (PICCS) a method to accurately recon-struct dynamic CT images from highly undersampled projec-tion data setsrdquoMedical Physics vol 35 no 2 pp 660ndash663 2008

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

[12] L T Niklason B T Christian L E Niklason et al ldquoDigitaltomosynthesis in breast imagingrdquo Radiology vol 205 no 2 pp399ndash406 1997

[13] R Gordon R Bender and G T Herman ldquoAlgebraic Recon-struction Techniques (ART) for three-dimensional electronmicroscopy and X-ray photographyrdquo Journal of TheoreticalBiology vol 29 no 3 pp 471ndash481 1970

[14] P Ch Hansen and M S Hansen ldquoAIR toolsmdasha MATLABpackage of algebraic iterative reconstruction methodsrdquo Journalof Computational and Applied Mathematics vol 236 no 8 pp2167ndash2178 2012

[15] D Raparia J Alessi and A Kponou ldquoThe algebraic reconstruc-tion technique (Art)rdquo in Proceedings of the 17th Particle Accel-erator Conference (PAC rsquo97) vol 2 pp 2023ndash2025 VancouverCanada May 1997

[16] T Nikazad Algebraic reconstruction methods [PhD thesis]Mathematics Department Linkoping Univerity 2008

[17] A H Andersen and A C Kak ldquoSimultaneous AlgebraicReconstruction Technique (SART) a superior implementationof the art algorithmrdquoUltrasonic Imaging vol 6 no 1 pp 81ndash941984

[18] M Jiang andGWang ldquoConvergence of the Simultaneous Alge-braic Reconstruction Technique (SART)rdquo IEEE Transactions onImage Processing vol 12 no 8 pp 957ndash961 2003

[19] T Hobiger T Kondo and Y Koyama ldquoConstrained Simulta-neous Algebraic Reconstruction Technique (C-SART)mdasha newand simple algorithm applied to ionospheric tomographyrdquoEarth Planets and Space vol 60 no 7 pp 727ndash735 2008

[20] B Girod ldquoWhatrsquos wrong with mean-squared errorrdquo in DigitalImages and Human Vision A BWatson Ed pp 207ndash220 MITPress Cambridge Mass USA 1993

[21] P C Teo and D J Heeger ldquoPerceptual image distortionrdquoin Proceedings of the IEEE International Conference on ImageProcessing (ICIP rsquo94) vol 2 pp 982ndash986 Austin Tex USA1994

[22] A M Eskicioglu and P S Fisher ldquoImage quality measures andtheir performancerdquo IEEE Transactions on Communications vol43 no 12 pp 2959ndash2965 1995

[23] Zh Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment from error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004

Computational and Mathematical Methods in Medicine 5

20 40 60 80 100 120

20

40

60

80

100

1200

5

10

15

(a)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

5

(b)

Figure 6 Original 3D phantom (a) objects with higher absorption in the upper layers (b) the LOI of the phantom

3 Software Design and Implementation

3D tomosynthesis simulator was written in C++ An object-oriented programming language and Net framework wasused to design the graphical user interface (GUI) of thesimulator on Visual StudioNet 2010 which was run on apersonal computer with Intel Core i7 200GHz processor and6GB RAM memory Unlike the procedural programminglanguages that separate data from operations object-orientedC++ programming language is capable of considering acollection of classes that combine data and operations ondata

Figure 2 shows the flow diagram of the simulator wherethree main parts of the simulator and related operators areshown As shown in this figure the simulator consists of threemain classes configuration (parameters) projection andreconstruction classesThe first class includes the parametersof all the system parts and functions to readwrite data fromXML files Three main parts of DBT system are defined asthree different subclasses which refer to X-ray source phan-tom and detector The projection class includes methods toreceive the system parameters from the configuration partand to find the projection images of a particular phantomwhich could be used by the reconstruction class to rundifferent reconstruction methods The simulator includes agraphical user interface (GUI) which facilitates design andediting of a phantom executes the projection and reconstruc-tion method and saves the results

Figure 3 shows the configuration interface of the simula-tor that allows one to insert or select the parameters of the X-ray source phantom and detector such as their location anddimensionsmanually in the specified places or by loading thexml files that include the desired data in a predefined formatOne can load the xml file by pressing the load button andinserting the path for the desired file

It is possible to choose a set of small 3D objects such asrectangular parallelepiped sphere or ellipsoid for both orig-inal and initial objects to start the reconstruction procedure

Figure 4 shows the projection interface of the simulatorit is possible to insert and edit a set of small 3D objects suchas rectangular parallelepiped sphere or ellipsoid in the phan-tom which are displayed sequentially in the object list part ofthe form The user is required to insert the characteristics ofeach object to generate the desired phantom After generatingthe phantom the projection task can be performed to getthe projection images of the phantom The results of theprojection on the detector could be displayed by choosing thedesired angle of projection

Figure 5 exhibits the reconstruction interface where theuser can choose one of the reconstructionmethods includingART SART ART with 3DTV and SART with 3DTV andcan insert the number of iterations (NOI) of the iterativemethod with the layer of interest (LOI) number of the 3Dphantom It is also possible to revise the initial object listfor the reconstruction After running the chosen method thereconstruction results such as the image of all layers includingthe LOI of the object and also the structural similarity (SSIM)contrast-to-noise-ratio (CNR) and root mean square error(RMSE) diagramswill be shown as the output of the program

A special xml file format is designed to store all of thesystem characteristics using the different tags such as X-raysource detector phantom and reconstruction A user caninsert and edit the contents of the file and load it automaticallyto update the parameters of the system

4 Results

In order to exhibit the performance of the different recon-structionmethods run by the simulator a 3D phantommodel

6 Computational and Mathematical Methods in Medicine

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(a)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(b)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(c)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(d)

Figure 7 Images of the LOI of the reconstructed phantom (11th layer of the phantom) (a) reconstructed LOI image using ART method (b)reconstructed LOI image using SART method (c) reconstructed LOI image using ART+TV 3D method and (d) reconstructed LOI imageusing SART+TV 3D method

was designed with resolution of 128 times 128 voxels in 16 layersThis phantom was created to imitate the overlapping tissueproblem of the breast imaging The phantom includes somesmaller objects where objects with the low X-ray absorptionare obscured by the objects with higher X-ray absorption(Figure 6) Measured projections were generated from thephantom for the range of scan angles from minus25∘ to +25∘ and11 projections

Parameters of the simulator and phantom are listed inTable 1

Reconstructed images of the LOI for ART SARTART+TV 3D and SART+TV 3D are shown in Figure 7(a) toFigure 7(d) respectively

One of the mostly used image quality metrics is the meansquare error (MSE) because of its simplicity in calculating

Table 1 Simulation parameters

Parameter ValueSource to detector distance 300 pixelsObject to detector distance 100 pixelsScan angle 50∘ degrees (minus25∘ to +25∘)Number of projections 11 projectionsTV regularization parameter 08Phantom size 128 times 128 times 16

Detector size 160 times 160 times 1

and clear physical meaning but this is not a very appropriatemetric to exhibit the visual quality of the images [20ndash22] Anumber of quality assessment methods are developed that

Computational and Mathematical Methods in Medicine 7

0 5 10 1504

05

06

07

08

09

1

Iteration

ART SART

MSS

IM

ART + TV3D SART + TV3D

Figure 8 MSSIM value comparison of ART ART+3D TV SARTand SART+3D TV for the layer of interest

implement the characteristics of the human visual system(HVS) One of the well-known quality assessment methodsis the measure of structural similarity (SSIM) that compareslocal patterns of pixel values which are normalized foramount of luminance and contrast [23]

The SSIM index is shown

SSIM (119909 119910) =(2120583119909120583119910+ 1198621) (120590119909119910+ 1198622)

(1205832119909+ 1205832119910+ 1198621) (1205902119909+ 1205902119910+ 1198622) (12)

where 120583119909and 120583119910refer to the mean of the intensities of signals

119909 and119910 respectively and120590119909and120590119910are the standard deviation

of them 1198621and 119862

2are given

119862119898= (119870119898119871)2 119898 = 1 2 (13)

where 119871 is the dynamic range of the pixel values and119870119898≪ 1

for 119896 = 1 2 are small constantsPractically we need a single overall quality measure of the

entire image In this study we used a mean SSIM (MSSIM)index to evaluate the overall image quality

MSSIM (119883 119884) = 1

119872

119879

sum

119895=1

SSIM (119909119895+ 119910119895) (14)

where 119883 and 119884 refer to original and reconstructed imagesrespectively 119909

119895and 119910

119895are the image contents at the 119895th local

window and 119879 is the number of local windows of the imageMSSIM indexes of the reconstruction methods tested

with the simulator are given in Figure 8 Compressed sensedmethods implemented as ART+TV 3D and SART+TV 3Dprovided improved results compared with the results of ARTand SART while ART and SART performed similarly

The time needed to perform a simulation study dependson the complexity of the phantom and detector size In this

study the average time required to complete each iterationof ART on the proposed simulator is 128 seconds which isobtained after measuring the first fifteen iterations The sameproblem takes 28800 seconds on the MATLAB which wasonly implemented by us for comparison

5 Conclusion

A new simulator was designed for 3D DBT studies Oursimulator is capable of implementing several reconstructiontechniques including recently proposed compressed sensingbased methods A user friendly graphical user interface(GUI) helps users to select and run the desired methods onthe designed phantom models or real data sets The simu-lator was implemented for 3D limited view angle imagingproblems using C++ programming language whereas theformer simulation software was typically developed usingMATLAB scripts for 2D data We tested the simulator byrunning different reconstruction methods with a specific 3Dphantommodel which was created to imitate the overlappingtissue problem of the breast imaging We also compared themethods in the simulator by demonstrating reconstructedimages of LOI and evaluating their performances usingRMSE and MSSIM metrics The simulator can be extendedby including new reconstruction methods

Conflict of Interests

The authors declared no conflict of interests

Acknowledgment

This work has been supported by TUBITAK The Scientificand Research Council of Turkey under Grant 111E086

References

[1] R Siegel DNaishadham andA Jemal ldquoCancer statistics 2012rdquoCA Cancer Journal for Clinicians vol 62 no 1 pp 10ndash29 2012

[2] J Baker J Lo A Hebecker Th Mertelmeier and J OrmanldquoDigital breast tomosynthesisrdquo Medical Solutions RSNA pp84ndash89 2006

[3] J T Dobbins III andD J Godfrey ldquoDigital x-ray tomosynthesiscurrent state of the art and clinical potentialrdquo Physics inMedicine and Biology vol 48 no 19 pp R65ndashR106 2003

[4] B Plantes ldquoEine neue methode zur differenzierung in derroentgenographie (planigraphie)rdquo Acta Radiologica vol 13 pp182ndash192 1932

[5] J B Garrison D G Grant W H Guier and R J JohnsldquoThree dimensional roentgenographyrdquoTheAmerican Journal ofRoentgenology RadiumTherapy and Nuclear Medicine vol 105no 4 pp 903ndash908 1969

[6] A G Richards ldquoVariable depth laminagraphyrdquo BiomedicalSciences Instrumentation vol 6 pp 194ndash199 1969

[7] E R Miller E M McCurry and B Hruska ldquoAn infinitenumber of laminagrams from a finite number of radiographsrdquoRadiology vol 98 no 2 pp 249ndash255 1971

8 Computational and Mathematical Methods in Medicine

[8] D G Grant ldquoTomosynthesis a three-dimensional radiographicimaging techniquerdquo IEEE Transactions on Biomedical Engineer-ing vol 19 no 1 pp 20ndash28 1972

[9] S Kaczmarz ldquoAngenaherteauflosung von systemenlinearergle-ichungenrdquo Bulletin de LrsquoAcademie Polonaise des Sciences etLet-tres vol 35 pp 355ndash357 1937

[10] G-H Chen J Tang and S Leng ldquoPrior image constrainedcompressed sensing (PICCS) a method to accurately recon-struct dynamic CT images from highly undersampled projec-tion data setsrdquoMedical Physics vol 35 no 2 pp 660ndash663 2008

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

[12] L T Niklason B T Christian L E Niklason et al ldquoDigitaltomosynthesis in breast imagingrdquo Radiology vol 205 no 2 pp399ndash406 1997

[13] R Gordon R Bender and G T Herman ldquoAlgebraic Recon-struction Techniques (ART) for three-dimensional electronmicroscopy and X-ray photographyrdquo Journal of TheoreticalBiology vol 29 no 3 pp 471ndash481 1970

[14] P Ch Hansen and M S Hansen ldquoAIR toolsmdasha MATLABpackage of algebraic iterative reconstruction methodsrdquo Journalof Computational and Applied Mathematics vol 236 no 8 pp2167ndash2178 2012

[15] D Raparia J Alessi and A Kponou ldquoThe algebraic reconstruc-tion technique (Art)rdquo in Proceedings of the 17th Particle Accel-erator Conference (PAC rsquo97) vol 2 pp 2023ndash2025 VancouverCanada May 1997

[16] T Nikazad Algebraic reconstruction methods [PhD thesis]Mathematics Department Linkoping Univerity 2008

[17] A H Andersen and A C Kak ldquoSimultaneous AlgebraicReconstruction Technique (SART) a superior implementationof the art algorithmrdquoUltrasonic Imaging vol 6 no 1 pp 81ndash941984

[18] M Jiang andGWang ldquoConvergence of the Simultaneous Alge-braic Reconstruction Technique (SART)rdquo IEEE Transactions onImage Processing vol 12 no 8 pp 957ndash961 2003

[19] T Hobiger T Kondo and Y Koyama ldquoConstrained Simulta-neous Algebraic Reconstruction Technique (C-SART)mdasha newand simple algorithm applied to ionospheric tomographyrdquoEarth Planets and Space vol 60 no 7 pp 727ndash735 2008

[20] B Girod ldquoWhatrsquos wrong with mean-squared errorrdquo in DigitalImages and Human Vision A BWatson Ed pp 207ndash220 MITPress Cambridge Mass USA 1993

[21] P C Teo and D J Heeger ldquoPerceptual image distortionrdquoin Proceedings of the IEEE International Conference on ImageProcessing (ICIP rsquo94) vol 2 pp 982ndash986 Austin Tex USA1994

[22] A M Eskicioglu and P S Fisher ldquoImage quality measures andtheir performancerdquo IEEE Transactions on Communications vol43 no 12 pp 2959ndash2965 1995

[23] Zh Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment from error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004

6 Computational and Mathematical Methods in Medicine

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(a)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(b)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(c)

20 40 60 80 100 120

20

40

60

80

100

1200

05

1

15

2

25

3

35

4

45

(d)

Figure 7 Images of the LOI of the reconstructed phantom (11th layer of the phantom) (a) reconstructed LOI image using ART method (b)reconstructed LOI image using SART method (c) reconstructed LOI image using ART+TV 3D method and (d) reconstructed LOI imageusing SART+TV 3D method

was designed with resolution of 128 times 128 voxels in 16 layersThis phantom was created to imitate the overlapping tissueproblem of the breast imaging The phantom includes somesmaller objects where objects with the low X-ray absorptionare obscured by the objects with higher X-ray absorption(Figure 6) Measured projections were generated from thephantom for the range of scan angles from minus25∘ to +25∘ and11 projections

Parameters of the simulator and phantom are listed inTable 1

Reconstructed images of the LOI for ART SARTART+TV 3D and SART+TV 3D are shown in Figure 7(a) toFigure 7(d) respectively

One of the mostly used image quality metrics is the meansquare error (MSE) because of its simplicity in calculating

Table 1 Simulation parameters

Parameter ValueSource to detector distance 300 pixelsObject to detector distance 100 pixelsScan angle 50∘ degrees (minus25∘ to +25∘)Number of projections 11 projectionsTV regularization parameter 08Phantom size 128 times 128 times 16

Detector size 160 times 160 times 1

and clear physical meaning but this is not a very appropriatemetric to exhibit the visual quality of the images [20ndash22] Anumber of quality assessment methods are developed that

Computational and Mathematical Methods in Medicine 7

0 5 10 1504

05

06

07

08

09

1

Iteration

ART SART

MSS

IM

ART + TV3D SART + TV3D

Figure 8 MSSIM value comparison of ART ART+3D TV SARTand SART+3D TV for the layer of interest

implement the characteristics of the human visual system(HVS) One of the well-known quality assessment methodsis the measure of structural similarity (SSIM) that compareslocal patterns of pixel values which are normalized foramount of luminance and contrast [23]

The SSIM index is shown

SSIM (119909 119910) =(2120583119909120583119910+ 1198621) (120590119909119910+ 1198622)

(1205832119909+ 1205832119910+ 1198621) (1205902119909+ 1205902119910+ 1198622) (12)

where 120583119909and 120583119910refer to the mean of the intensities of signals

119909 and119910 respectively and120590119909and120590119910are the standard deviation

of them 1198621and 119862

2are given

119862119898= (119870119898119871)2 119898 = 1 2 (13)

where 119871 is the dynamic range of the pixel values and119870119898≪ 1

for 119896 = 1 2 are small constantsPractically we need a single overall quality measure of the

entire image In this study we used a mean SSIM (MSSIM)index to evaluate the overall image quality

MSSIM (119883 119884) = 1

119872

119879

sum

119895=1

SSIM (119909119895+ 119910119895) (14)

where 119883 and 119884 refer to original and reconstructed imagesrespectively 119909

119895and 119910

119895are the image contents at the 119895th local

window and 119879 is the number of local windows of the imageMSSIM indexes of the reconstruction methods tested

with the simulator are given in Figure 8 Compressed sensedmethods implemented as ART+TV 3D and SART+TV 3Dprovided improved results compared with the results of ARTand SART while ART and SART performed similarly

The time needed to perform a simulation study dependson the complexity of the phantom and detector size In this

study the average time required to complete each iterationof ART on the proposed simulator is 128 seconds which isobtained after measuring the first fifteen iterations The sameproblem takes 28800 seconds on the MATLAB which wasonly implemented by us for comparison

5 Conclusion

A new simulator was designed for 3D DBT studies Oursimulator is capable of implementing several reconstructiontechniques including recently proposed compressed sensingbased methods A user friendly graphical user interface(GUI) helps users to select and run the desired methods onthe designed phantom models or real data sets The simu-lator was implemented for 3D limited view angle imagingproblems using C++ programming language whereas theformer simulation software was typically developed usingMATLAB scripts for 2D data We tested the simulator byrunning different reconstruction methods with a specific 3Dphantommodel which was created to imitate the overlappingtissue problem of the breast imaging We also compared themethods in the simulator by demonstrating reconstructedimages of LOI and evaluating their performances usingRMSE and MSSIM metrics The simulator can be extendedby including new reconstruction methods

Conflict of Interests

The authors declared no conflict of interests

Acknowledgment

This work has been supported by TUBITAK The Scientificand Research Council of Turkey under Grant 111E086

References

[1] R Siegel DNaishadham andA Jemal ldquoCancer statistics 2012rdquoCA Cancer Journal for Clinicians vol 62 no 1 pp 10ndash29 2012

[2] J Baker J Lo A Hebecker Th Mertelmeier and J OrmanldquoDigital breast tomosynthesisrdquo Medical Solutions RSNA pp84ndash89 2006

[3] J T Dobbins III andD J Godfrey ldquoDigital x-ray tomosynthesiscurrent state of the art and clinical potentialrdquo Physics inMedicine and Biology vol 48 no 19 pp R65ndashR106 2003

[4] B Plantes ldquoEine neue methode zur differenzierung in derroentgenographie (planigraphie)rdquo Acta Radiologica vol 13 pp182ndash192 1932

[5] J B Garrison D G Grant W H Guier and R J JohnsldquoThree dimensional roentgenographyrdquoTheAmerican Journal ofRoentgenology RadiumTherapy and Nuclear Medicine vol 105no 4 pp 903ndash908 1969

[6] A G Richards ldquoVariable depth laminagraphyrdquo BiomedicalSciences Instrumentation vol 6 pp 194ndash199 1969

[7] E R Miller E M McCurry and B Hruska ldquoAn infinitenumber of laminagrams from a finite number of radiographsrdquoRadiology vol 98 no 2 pp 249ndash255 1971

8 Computational and Mathematical Methods in Medicine

[8] D G Grant ldquoTomosynthesis a three-dimensional radiographicimaging techniquerdquo IEEE Transactions on Biomedical Engineer-ing vol 19 no 1 pp 20ndash28 1972

[9] S Kaczmarz ldquoAngenaherteauflosung von systemenlinearergle-ichungenrdquo Bulletin de LrsquoAcademie Polonaise des Sciences etLet-tres vol 35 pp 355ndash357 1937

[10] G-H Chen J Tang and S Leng ldquoPrior image constrainedcompressed sensing (PICCS) a method to accurately recon-struct dynamic CT images from highly undersampled projec-tion data setsrdquoMedical Physics vol 35 no 2 pp 660ndash663 2008

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

[12] L T Niklason B T Christian L E Niklason et al ldquoDigitaltomosynthesis in breast imagingrdquo Radiology vol 205 no 2 pp399ndash406 1997

[13] R Gordon R Bender and G T Herman ldquoAlgebraic Recon-struction Techniques (ART) for three-dimensional electronmicroscopy and X-ray photographyrdquo Journal of TheoreticalBiology vol 29 no 3 pp 471ndash481 1970

[14] P Ch Hansen and M S Hansen ldquoAIR toolsmdasha MATLABpackage of algebraic iterative reconstruction methodsrdquo Journalof Computational and Applied Mathematics vol 236 no 8 pp2167ndash2178 2012

[15] D Raparia J Alessi and A Kponou ldquoThe algebraic reconstruc-tion technique (Art)rdquo in Proceedings of the 17th Particle Accel-erator Conference (PAC rsquo97) vol 2 pp 2023ndash2025 VancouverCanada May 1997

[16] T Nikazad Algebraic reconstruction methods [PhD thesis]Mathematics Department Linkoping Univerity 2008

[17] A H Andersen and A C Kak ldquoSimultaneous AlgebraicReconstruction Technique (SART) a superior implementationof the art algorithmrdquoUltrasonic Imaging vol 6 no 1 pp 81ndash941984

[18] M Jiang andGWang ldquoConvergence of the Simultaneous Alge-braic Reconstruction Technique (SART)rdquo IEEE Transactions onImage Processing vol 12 no 8 pp 957ndash961 2003

[19] T Hobiger T Kondo and Y Koyama ldquoConstrained Simulta-neous Algebraic Reconstruction Technique (C-SART)mdasha newand simple algorithm applied to ionospheric tomographyrdquoEarth Planets and Space vol 60 no 7 pp 727ndash735 2008

[20] B Girod ldquoWhatrsquos wrong with mean-squared errorrdquo in DigitalImages and Human Vision A BWatson Ed pp 207ndash220 MITPress Cambridge Mass USA 1993

[21] P C Teo and D J Heeger ldquoPerceptual image distortionrdquoin Proceedings of the IEEE International Conference on ImageProcessing (ICIP rsquo94) vol 2 pp 982ndash986 Austin Tex USA1994

[22] A M Eskicioglu and P S Fisher ldquoImage quality measures andtheir performancerdquo IEEE Transactions on Communications vol43 no 12 pp 2959ndash2965 1995

[23] Zh Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment from error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004

Computational and Mathematical Methods in Medicine 7

0 5 10 1504

05

06

07

08

09

1

Iteration

ART SART

MSS

IM

ART + TV3D SART + TV3D

Figure 8 MSSIM value comparison of ART ART+3D TV SARTand SART+3D TV for the layer of interest

implement the characteristics of the human visual system(HVS) One of the well-known quality assessment methodsis the measure of structural similarity (SSIM) that compareslocal patterns of pixel values which are normalized foramount of luminance and contrast [23]

The SSIM index is shown

SSIM (119909 119910) =(2120583119909120583119910+ 1198621) (120590119909119910+ 1198622)

(1205832119909+ 1205832119910+ 1198621) (1205902119909+ 1205902119910+ 1198622) (12)

where 120583119909and 120583119910refer to the mean of the intensities of signals

119909 and119910 respectively and120590119909and120590119910are the standard deviation

of them 1198621and 119862

2are given

119862119898= (119870119898119871)2 119898 = 1 2 (13)

where 119871 is the dynamic range of the pixel values and119870119898≪ 1

for 119896 = 1 2 are small constantsPractically we need a single overall quality measure of the

entire image In this study we used a mean SSIM (MSSIM)index to evaluate the overall image quality

MSSIM (119883 119884) = 1

119872

119879

sum

119895=1

SSIM (119909119895+ 119910119895) (14)

where 119883 and 119884 refer to original and reconstructed imagesrespectively 119909

119895and 119910

119895are the image contents at the 119895th local

window and 119879 is the number of local windows of the imageMSSIM indexes of the reconstruction methods tested

with the simulator are given in Figure 8 Compressed sensedmethods implemented as ART+TV 3D and SART+TV 3Dprovided improved results compared with the results of ARTand SART while ART and SART performed similarly

The time needed to perform a simulation study dependson the complexity of the phantom and detector size In this

study the average time required to complete each iterationof ART on the proposed simulator is 128 seconds which isobtained after measuring the first fifteen iterations The sameproblem takes 28800 seconds on the MATLAB which wasonly implemented by us for comparison

5 Conclusion

A new simulator was designed for 3D DBT studies Oursimulator is capable of implementing several reconstructiontechniques including recently proposed compressed sensingbased methods A user friendly graphical user interface(GUI) helps users to select and run the desired methods onthe designed phantom models or real data sets The simu-lator was implemented for 3D limited view angle imagingproblems using C++ programming language whereas theformer simulation software was typically developed usingMATLAB scripts for 2D data We tested the simulator byrunning different reconstruction methods with a specific 3Dphantommodel which was created to imitate the overlappingtissue problem of the breast imaging We also compared themethods in the simulator by demonstrating reconstructedimages of LOI and evaluating their performances usingRMSE and MSSIM metrics The simulator can be extendedby including new reconstruction methods

Conflict of Interests

The authors declared no conflict of interests

Acknowledgment

This work has been supported by TUBITAK The Scientificand Research Council of Turkey under Grant 111E086

References

[1] R Siegel DNaishadham andA Jemal ldquoCancer statistics 2012rdquoCA Cancer Journal for Clinicians vol 62 no 1 pp 10ndash29 2012

[2] J Baker J Lo A Hebecker Th Mertelmeier and J OrmanldquoDigital breast tomosynthesisrdquo Medical Solutions RSNA pp84ndash89 2006

[3] J T Dobbins III andD J Godfrey ldquoDigital x-ray tomosynthesiscurrent state of the art and clinical potentialrdquo Physics inMedicine and Biology vol 48 no 19 pp R65ndashR106 2003

[4] B Plantes ldquoEine neue methode zur differenzierung in derroentgenographie (planigraphie)rdquo Acta Radiologica vol 13 pp182ndash192 1932

[5] J B Garrison D G Grant W H Guier and R J JohnsldquoThree dimensional roentgenographyrdquoTheAmerican Journal ofRoentgenology RadiumTherapy and Nuclear Medicine vol 105no 4 pp 903ndash908 1969

[6] A G Richards ldquoVariable depth laminagraphyrdquo BiomedicalSciences Instrumentation vol 6 pp 194ndash199 1969

[7] E R Miller E M McCurry and B Hruska ldquoAn infinitenumber of laminagrams from a finite number of radiographsrdquoRadiology vol 98 no 2 pp 249ndash255 1971

8 Computational and Mathematical Methods in Medicine

[8] D G Grant ldquoTomosynthesis a three-dimensional radiographicimaging techniquerdquo IEEE Transactions on Biomedical Engineer-ing vol 19 no 1 pp 20ndash28 1972

[9] S Kaczmarz ldquoAngenaherteauflosung von systemenlinearergle-ichungenrdquo Bulletin de LrsquoAcademie Polonaise des Sciences etLet-tres vol 35 pp 355ndash357 1937

[10] G-H Chen J Tang and S Leng ldquoPrior image constrainedcompressed sensing (PICCS) a method to accurately recon-struct dynamic CT images from highly undersampled projec-tion data setsrdquoMedical Physics vol 35 no 2 pp 660ndash663 2008

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

[12] L T Niklason B T Christian L E Niklason et al ldquoDigitaltomosynthesis in breast imagingrdquo Radiology vol 205 no 2 pp399ndash406 1997

[13] R Gordon R Bender and G T Herman ldquoAlgebraic Recon-struction Techniques (ART) for three-dimensional electronmicroscopy and X-ray photographyrdquo Journal of TheoreticalBiology vol 29 no 3 pp 471ndash481 1970

[14] P Ch Hansen and M S Hansen ldquoAIR toolsmdasha MATLABpackage of algebraic iterative reconstruction methodsrdquo Journalof Computational and Applied Mathematics vol 236 no 8 pp2167ndash2178 2012

[15] D Raparia J Alessi and A Kponou ldquoThe algebraic reconstruc-tion technique (Art)rdquo in Proceedings of the 17th Particle Accel-erator Conference (PAC rsquo97) vol 2 pp 2023ndash2025 VancouverCanada May 1997

[16] T Nikazad Algebraic reconstruction methods [PhD thesis]Mathematics Department Linkoping Univerity 2008

[17] A H Andersen and A C Kak ldquoSimultaneous AlgebraicReconstruction Technique (SART) a superior implementationof the art algorithmrdquoUltrasonic Imaging vol 6 no 1 pp 81ndash941984

[18] M Jiang andGWang ldquoConvergence of the Simultaneous Alge-braic Reconstruction Technique (SART)rdquo IEEE Transactions onImage Processing vol 12 no 8 pp 957ndash961 2003

[19] T Hobiger T Kondo and Y Koyama ldquoConstrained Simulta-neous Algebraic Reconstruction Technique (C-SART)mdasha newand simple algorithm applied to ionospheric tomographyrdquoEarth Planets and Space vol 60 no 7 pp 727ndash735 2008

[20] B Girod ldquoWhatrsquos wrong with mean-squared errorrdquo in DigitalImages and Human Vision A BWatson Ed pp 207ndash220 MITPress Cambridge Mass USA 1993

[21] P C Teo and D J Heeger ldquoPerceptual image distortionrdquoin Proceedings of the IEEE International Conference on ImageProcessing (ICIP rsquo94) vol 2 pp 982ndash986 Austin Tex USA1994

[22] A M Eskicioglu and P S Fisher ldquoImage quality measures andtheir performancerdquo IEEE Transactions on Communications vol43 no 12 pp 2959ndash2965 1995

[23] Zh Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment from error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004

8 Computational and Mathematical Methods in Medicine

[8] D G Grant ldquoTomosynthesis a three-dimensional radiographicimaging techniquerdquo IEEE Transactions on Biomedical Engineer-ing vol 19 no 1 pp 20ndash28 1972

[9] S Kaczmarz ldquoAngenaherteauflosung von systemenlinearergle-ichungenrdquo Bulletin de LrsquoAcademie Polonaise des Sciences etLet-tres vol 35 pp 355ndash357 1937

[10] G-H Chen J Tang and S Leng ldquoPrior image constrainedcompressed sensing (PICCS) a method to accurately recon-struct dynamic CT images from highly undersampled projec-tion data setsrdquoMedical Physics vol 35 no 2 pp 660ndash663 2008

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

[12] L T Niklason B T Christian L E Niklason et al ldquoDigitaltomosynthesis in breast imagingrdquo Radiology vol 205 no 2 pp399ndash406 1997

[13] R Gordon R Bender and G T Herman ldquoAlgebraic Recon-struction Techniques (ART) for three-dimensional electronmicroscopy and X-ray photographyrdquo Journal of TheoreticalBiology vol 29 no 3 pp 471ndash481 1970

[14] P Ch Hansen and M S Hansen ldquoAIR toolsmdasha MATLABpackage of algebraic iterative reconstruction methodsrdquo Journalof Computational and Applied Mathematics vol 236 no 8 pp2167ndash2178 2012

[15] D Raparia J Alessi and A Kponou ldquoThe algebraic reconstruc-tion technique (Art)rdquo in Proceedings of the 17th Particle Accel-erator Conference (PAC rsquo97) vol 2 pp 2023ndash2025 VancouverCanada May 1997

[16] T Nikazad Algebraic reconstruction methods [PhD thesis]Mathematics Department Linkoping Univerity 2008

[17] A H Andersen and A C Kak ldquoSimultaneous AlgebraicReconstruction Technique (SART) a superior implementationof the art algorithmrdquoUltrasonic Imaging vol 6 no 1 pp 81ndash941984

[18] M Jiang andGWang ldquoConvergence of the Simultaneous Alge-braic Reconstruction Technique (SART)rdquo IEEE Transactions onImage Processing vol 12 no 8 pp 957ndash961 2003

[19] T Hobiger T Kondo and Y Koyama ldquoConstrained Simulta-neous Algebraic Reconstruction Technique (C-SART)mdasha newand simple algorithm applied to ionospheric tomographyrdquoEarth Planets and Space vol 60 no 7 pp 727ndash735 2008

[20] B Girod ldquoWhatrsquos wrong with mean-squared errorrdquo in DigitalImages and Human Vision A BWatson Ed pp 207ndash220 MITPress Cambridge Mass USA 1993

[21] P C Teo and D J Heeger ldquoPerceptual image distortionrdquoin Proceedings of the IEEE International Conference on ImageProcessing (ICIP rsquo94) vol 2 pp 982ndash986 Austin Tex USA1994

[22] A M Eskicioglu and P S Fisher ldquoImage quality measures andtheir performancerdquo IEEE Transactions on Communications vol43 no 12 pp 2959ndash2965 1995

[23] Zh Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment from error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004