tomographic image reconstruction based on minimization of...

Research ArticleTomographic Image Reconstruction Based on Minimization ofSymmetrized Kullback-Leibler Divergence

Ryosuke Kasai 1 Yusaku Yamaguchi 2 Takeshi Kojima 3 and Tetsuya Yoshinaga 3

1Graduate School of Health Sciences Tokushima University 3-18-15 Kuramoto Tokushima 770-8509 Japan2Shikoku Medical Center for Children and Adults 2-1-1 Senyu Zentsuji 765-8507 Japan3Institute of Biomedical Sciences Tokushima University 3-18-15 Kuramoto Tokushima 770-8509 Japan

Correspondence should be addressed to Tetsuya Yoshinaga yosinagamedscitokushima-uacjp

Received 3 April 2018 Revised 8 June 2018 Accepted 26 June 2018 Published 17 July 2018

Academic Editor Elena Zaitseva

Copyright copy 2018 Ryosuke Kasai 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

Iterative reconstruction (IR) algorithms based on the principle of optimization are known for producing better reconstructedimages in computed tomography In this paper we present an IR algorithm based on minimizing a symmetrized Kullback-Leibler divergence (SKLD) that is called Jeffreysrsquo 119869-divergence The SKLD with iterative steps is guaranteed to decrease inconvergence monotonically using a continuous dynamical method for consistent inverse problems Specifically we constructan autonomous differential equation for which the proposed iterative formula gives a first-order numerical discretization anddemonstrate the stability of a desired solution using Lyapunovrsquos theorem We describe a hybrid Euler method combined withadditive and multiplicative calculus for constructing an effective and robust discretization method thereby enabling us to obtainan approximate solution to the differential equation We performed experiments and found that the IR algorithm derived from thehybrid discretization achieved high performance

1 Introduction

Various kinds of iterative reconstruction (IR) [1ndash4] algo-rithms based on the principle of optimization are known forproducing better reconstructed images in computed tomog-raphy (CT) In accordance with the base objective functioneach IR method has intrinsic properties in the quality ofimages and in the convergence of iterative solutions Themaximum-likelihood expectation-maximization (ML-EM)[4] algorithm decreases the generalized Kullback-Leibler(KL) divergence [5 6] between the measured projection andthe forward projection along with the iteration Howeverthe objective function in which the multiplicative algebraicreconstruction technique (MART) [7ndash10] forces a decrease inthe iterative process is an alternative KL-divergence done byexchanging themeasured and forward projectionsDue to theasymmetry of the KL-divergence both objective functionshave generally different values

In this paper we present an IR algorithm based onthe minimization of a symmetrized KL-divergence (SKLD)

which is formulated using the mean of the two mutuallyalternativeKL-divergences and is called Jeffreysrsquo 119869-divergence[11 12] Because the base optimization function is a symmet-ric premetric measure and it gives an upper bound of theJensen-Shannon divergence [13 14] one can expect a betterperformance while preserving good properties of ML-EMandMART algorithmsThe convergence to an exact solutionand themonotonic decreasing of the SKLDwith each iterativestep for a consistent inverse problem are guaranteed usingthe approach of the continuous dynamical method [15ndash20] Specifically we construct an autonomous differentialequation for which the proposed iterative formula gives afirst-order numerical discretization with some step size anddemonstrate the stability of an equilibrium in a continuous-time system using Lyapunovrsquos stability theorem [21] Thetheory is extended to a case where the Lyapunov function canbe an 120572-skew 119869-divergence with a parameter 120572 isin [0 1] whichis the SKLD when 120572 = 05

We also propose a novel method for constructing aneffective and robust method in the discretization thereby

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 8973131 9 pageshttpsdoiorg10115520188973131

2 Mathematical Problems in Engineering

enabling us to obtain an approximate solution to the differ-ential equation That is a hybrid Euler discretization com-bined with additive and multiplicative calculus works wellhowever neither additive nor multiplicative Euler methoddoes

We conducted experiments and found that the IR algo-rithm derived from the hybrid Euler discretization of thecontinuous-time dynamical system achieved high perfor-mance

2 ML-EM and MART Algorithms

The problem of image reconstruction frommeasured projec-tions in CT is formulated as solving unknown variable 119909 isin 119877119869+for pixel values such that

119910 = 119860119909 + 120590 (1)

where 119910 isin 119877119868++ 119860 isin 119877119868times119869+ and 120590 isin 119877119868 represent the measuredprojection projection operator and noise respectively with119877+ and119877++ respectively denoting the sets of nonnegative andpositive real numbers If the system119910 = 119860119909 has a nonnegativesolution it is consistent otherwise it is inconsistent

We introduce the following definitions and notationsThevariable denoted by the symbol120582119895 for 119895 = 1 2 119869 is definedby

120582119895 fl ( 119868sum119894=1

119860 119894119895)minus1

(2)

where 119860 119894119895 is the element in the 119894th row and 119895th column of119860 The notations 119911119895 and 119860 119894 indicate the 119895th element and the119894th row vector of the vector 119911 and the matrix 119860 respectivelyThe function KL(119901 119902) of two nonnegative vectors 119901 and 119902represents the generalized KL-divergence [5]

KL (119901 119902) = 119868sum119894=1

119901119894 log 119901119894119902119894 + 119902119894 minus 119901119894 (3)

Known methods of reconstructing tomographic imagesin clinical CT are filtered back projection (FBP) as atransform method and IR as an optimization process KL-divergence is a common measure for deriving IR algorithmsand it plays an important role in accordance with theaxiom that minimizing the KL-divergence is equivalent tomaximizing the likelihood function that is considered suit-able for reconstruction modeled with a probability distribu-tionThe sequences KL(119910 119860119911(119899))infin119899=0 and KL(119860119911(119899) 119910)infin119899=0decrease for the respective iterative sequences 119911(119899)infin119899=0 of theML-EM and the (simultaneous) MART algorithms definedby

119911119895 (119899 + 1) = 119911119895 (119899) 120582119895 119868sum119894=1

119860 119894119895 119910119894119860 119894119911 (119899) (4)

and

119911119895 (119899 + 1) = 119911119895 (119899) exp(120582119895 119868sum119894=1

119860 119894119895 log 119910119894119860 119894119911 (119899)) (5)

for 119895 = 1 2 119869 with 119911(0) = 1199090 isin 119877119869++

3 Proposed System

The IR algorithm proposed in this paper can be describedusing the following difference equation

119911119895 (119899 + 1) = 119911119895 (119899) (1 + 120575119891119895 (119911 (119899))) exp (120575119892119895 (119911 (119899))) (6)

where 120575 gt 0 is a parameter and where the functions 119891119895 and119892119895 are respectively defined by

119891119895 (119911 (119899)) = (1 minus 120572) 120582119895 119868sum119894=1

119860 119894119895 ( 119910119894119860 119894119911 (119899) minus 1) (7)

and

119892119895 (119911 (119899)) = 120572120582119895 119868sum119894=1

119860 119894119895 log 119910119894119860 119894119911 (119899) (8)

for 119895 = 1 2 119869 with 120572 isin [0 1] and 119911(0) = 1199090 isin 119877119869++For investigating the dynamics of the difference system

in (6) we give its continuous analog described using thedifferential equation

119889119909119895 (119905)119889119905 = 120582119895119909119895 (119905) 119868sum119894=1

119860 119894119895sdot ((1 minus 120572) ( 119910119894119860 119894119909 (119905) minus 1) + 120572 log

119910119894119860 119894119909 (119905))(9)

for 119895 = 1 2 119869 and 119905 ge 0 with 119909(0) = 1199090 isin 119877119869++ The termldquocontinuous analogrdquo means that a numerical discretizationof the differential equation is the same as the differenceequation The idea is based on the approach for solving aconstrained tomographic inverse problem using nonlineardynamical methods [22ndash25]

31 Theoretical Analysis The theoretical results on thecontinuous-time system in (9) are given First we show thata solution is possible to ensure positive values remain

Proposition 1 If we choose an initial value 1199090 isin 119877119869++ for thesystem in (9) the solution 120601(119905 1199090) remains in the subspace 119877119869++for all 119905 ge 0Proof The vector field of the 119895th element of the system ismultiplied by 119909119895 therefore the derivative 119889120601119895119889119905 equiv 0 holdsfor any 119895 on the subspace 119909119895 equiv 0 which means the subspaceis invariant and which means any flow cannot pass throughinvariant subspace on the basis of the uniqueness property ofsolutions to the initial value problem Hence the trajectoryemanating from the initial value in 119877++ behaves in the sub-state space

Let us define119869120572 (119909) fl (1 minus 120572)KL (119910 119860119909) + 120572KL (119860119909 119910)

= 119868sum119894=1

(1 minus 120572) (119910119894 log 119910119894119860 119894119909 + 119860 119894119909 minus 119910119894)+ 120572(119860 119894119909 log 119860 119894119909119910119894 + 119910119894 minus 119860 119894119909)

(10)

Mathematical Problems in Engineering 3

with 120572 in the interval [0 1] as an 120572-skew 119869-divergence Thefunction is essential for a premetric measure of differencebetween119910 and119860119909 in accordancewith 119869120572(119909) ge 0 and 119869120572(119909) = 0if and only if 119910 = 119860119909 When 120572 = 05 the divergence 119869120572becomes a symmetrical form of KL-divergence by averagingKL(119910 119860119909) and KL(119860119909 119910) which we abbreviate as SKLD

Then under the assumption that the inverse problem isconsistent the following shows that the solution converges toa desired equilibrium and that the nonnegative function 119869120572monotonically decreases along solution trajectories

Theorem 2 If a unique solution 119890 isin 119877119869++ exists to the system119910 = 119860119909 the equilibrium 119890 of the dynamical system in (9) withany 120572 isin [0 1] is asymptotically stable

Proof A Lyapunov candidate function 119869120572 is defined forapplying Lyapunovrsquos stability theorem in (10)

119869120572 (119909) = 119868sum119894=1

(1 minus 120572)int119860119894119909119910119894

1 minus 119910119894119908119889119908 + 120572int119860119894119909119910119894

log 119908119910119894= 119868sum119894=1

(1 minus 120572) [119908 minus 119910119894 log119908]119860119894119909119910119894+ 120572 [119908 log 119908119910119894 minus 119908]

119860119894119909

119910119894

(11)

which is well defined via Proposition 1 when choosing aninitial value 1199090 in 119877119869++ We then calculate its derivative withrespect to the dynamical system in (9) with any 120572 isin [0 1] as

119889119869120572119889119905 (119909)10038161003816100381610038161003816100381610038161003816(9) =

119868sum119894=1

((1 minus 120572) (1 minus 119910119894119860 119894119909) + 120572 log119860 119894119909119910119894 )

sdot 119869sum119895=1

119860 119894119895 119889119909119895119889119905 = minus 119869sum119895=1

119868sum119894=1

119860 119894119895 ((1 minus 120572) ( 119910119894119860 119894119909 minus 1) + 120572

sdot log 119910119894119860 119894119909)120582119895119909119895119868sum119896=1

119860119896119895 ((1 minus 120572) ( 119910119896119860119896119909 minus 1) + 120572

sdot log 119910119896119860119896119909) = minus 119869sum119895=1

(radic120582119895119909119895 119868sum119894=1

119860 119894119895

sdot ((1 minus 120572) ( 119910119894119860 119894119909 minus 1) + 120572 log 119910119894119860 119894119909))2

le 0

(12)

The derivative equals zero at 119909 = 119890 isin 119877119869++ Consequently thesystem has a Lyapunov function so the equilibrium 119890 of thesystem is asymptotically stable

32 Hybrid Euler Discretization Consider an autonomousdifferential equation for the state variable 119909(119905) gt 0 119905 ge 0written by

119889119909 (119905)119889119905 = 119909 (119905) (119891 (119909 (119905)) + 119892 (119909 (119905))) 119909 (0) = 1199090 (13)

with sufficiently smooth functions 119891(119909) and 119892(119909) satisfying119891(119890) = 119892(119890) = 0 where 119890 gt 0 denotes an equilibrium of (13)Then we have the geometric multiplicative derivative

120587119909 (119905)120587119905 = exp (119891 (119909 (119905))) exp (119892 (119909 (119905))) 119909 (0) = 1199090 (14)

as a counterpart of the additive derivative Analogous tothe ordinary Euler method the multiplicative first-orderexpansion [26 27]

119909 ((1 + 120575) 119905) = 119909 (119905) (120587119909 (119905)120587119905 )120575 (15)

with small 120575 gt 0 leads to the iterative formula of themultiplicative Euler discretization

119911 (119899 + 1) = 119911 (119899) exp (120575119891 (119911 (119899))) exp (120575119892 (119911 (119899))) 119911 (0) = 1199090 (16)

with iteration numbers 119899 = 0 1 2 However when 119892 equiv 0 in (13) its additive Euler

discretization

119911 (119899 + 1) = 119911 (119899) (1 + 120575119891 (119911 (119899))) (17)

corresponds to the multiplicative one in (16) with 119892 equiv 0 Theterm (1 + 120575119891(119911)) is also considered as a first-order term inthe Taylor series expansion of the function exp(120575119891(119911)) with 119911in the neighborhood of the steady state 119890 By replacing a partof the multiplicative term exp(120575119891(119911)) in (16) with the term(1+120575119891(119911))while preserving themultiplication of 119911 we obtainthe formula of a hybrid Euler discretization

119911 (119899 + 1) = 119911 (119899) (1 + 120575119891 (119911 (119899))) exp (120575119892 (119911 (119899))) 119911 (0) = 1199090 (18)

for 119899 = 0 1 2 with a combination based on theadditive and multiplicative calculus for the functions 119891 and119892 respectively The hybrid Euler method is effective for apractical calculation of an initial value problem in (13) fromthe viewpoint of choosing a larger value of the step-size 120575when either an additive or multiplicative calculus is better foreach term of the partial vector fields 119909119891(119909) and 119909119892(119909)

The hybrid Euler discretization of the differential equa-tion in (9) gives the IR algorithm in (6)

4 Experimental Results and Discussion

We conducted experiments to demonstrate the effectivenessof not only the proposed IR method in (6) with 120572 = 05and 120575 = 1 say SKLD algorithm based on the minimizationof SKLD but also the hybrid Euler discretization of thecontinuous analog in (9)

We used a modified Shepp-Logan phantom image con-sisting of 119890 isin 119877119869+ shown in Figure 1(a) which is composed of128 times 128 pixels (119869 = 16 384) The projection 119910 isin 119877119868+ wassimulated by using the model 119910 = 119860119890 + 120590 with 120590 denoting


(a) (b)

Figure 1 (a) Phantom image and (b) sinogram with noisy projection

the white Gaussian noise such that the signal-to-noise ratio(SNR) was 30 dB unless otherwise specified and by settingthe number of view angles and detector bins to 180 and 184respectively (119868 = 33 120) as illustrated in Figure 1(b) for-matted into a sinogram A standard MATLAB (MathWorksNatick USA) ODE solver ode113 implementing a variablestep-size variable order Adams-Bashforth-Moulton methodwas used for a numerical calculation of solving an initial valueproblem of the differential equation in (9)

Let us first consider which is the most robust discretiza-tion method for numerical approximation Figure 2 showsthe time course of the objective function 119869120572(119909(119905)) with 120572 =05 which we denote as 11986905(119909(119905)) for the continuous-timesystem with 120572 = 05 at 119905 and the discrete-time systems at119905 = 120575119899 which are discretizations of the differential equationusing the additive multiplicative and hybrid Euler methodswith the fixed step sizes 120575 of 001 01 and 1 We see that thehybrid Euler discretization gives a similar good convergenceto that of a continuous-time system however in the case120575 = 1 the additive Euler method causes as time goes on anerror with oscillation and solutions using the multiplicativemethod divergence

Next we consider the property of the SKLD algorithmcompared with the algorithms using theML-EMmethod andMART Note that the iterative formula in (6) with 120575 = 1becomes the ML-EM SKLD and MART algorithms whenthe parameter 120572 is equal to 0 05 and 1 respectively Thevalues of the root mean squared (RMS) distance or 1198712-normof difference between the phantom image 119890 and iterativestate 119911(119873) obtained using the formula in (6) at the 119873thiteration starting from the common initial value 119911119895(0) =sum119868119894=1 119910119894sum119868119894=1sum1198691198951015840=1 119860 1198941198951015840 119895 = 1 2 119869 and variations inthe parameter 120572 are plotted in Figure 3(a) We observed avalue of 120572 exists in the open interval (0 1) minimizing the

quantitative measure when the number119873 is relatively smallIn particular the measure of SKLD is less than those of ML-EM and MART at eg 119873 = 10 and 12 The presence ofnoise is required for this property which is supported bythe fact that an experiment from another simulation withnoise-free projection data gave different results as shownin Figure 3(b) such that we had the minimum value of themeasure at 120572 = 1 This was due to characteristics of the ML-EMandMART algorithms being disadvantageous As a resultof these characteristics ML-EM has a slow convergence [28ndash31] which means that a relatively large number of iterationsis required to obtain an acceptable value of the objectivefunction and MART is more susceptible to noise [32 33]which may result in an increase of the convergence ratewith increasing iterative steps due to the noisy projection astypically seen at 120572 = 0 and 1 in Figure 3 respectively

We carried out other experiments to examine whetherthere exists an 120572 not equal to 0 or 1 in which the value ofan objective function obtained by using the algorithm in (6)with 120575 = 1 takes a minimum A set of SPECT projection datafrom a publicly accessible database [34] was examined firstThe sinogram of a brain scan and an image reconstructed bythe SKLDalgorithmare shown in Figure 4 where the numberof projections and pixels of the image are respectively 119868 =7 680 (128 acquisition bins and 60 projection directions in180 degrees) and 119869 = 7 569 (87 times 87 pixels) The projectiondata for the second example were acquired from an X-rayCT scanner (Toshiba Medical Systems Tochigi Japan) witha body phantom [35] (Kyoto Kagaku Kyoto Japan) usinga tube voltage of 120 kVp and a tube current of 300 mAFigure 5 represents the sinogram with 119868 = 86 130 (957acquisition bins and 90 projection directions in 180 degrees)and a reconstructed image with 119869 = 454 276 (674 times 674pixels) The value of the objective function 11986905 for the imagereconstructed by SKLD algorithm at 30th iteration is 286 times


Additive Multiplicative Hybrid

001

0 10 20 30t

103

102

J05

Continuous-timeDiscrete-time

01

0 10 20 30t

103

102

J05


1

0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


Figure 2 Time course of 11986905(119909(119905)) as a function of t for continuous-time (magenta) and discrete-time (blue) systems using additivemultiplicative and hybrid Euler methods with 120575 = 001 01 and 1

104 which is less than 461 times 104 obtained using FBP with aShepp-Logan filter

In the physical experiments as well as a feature of theobject to be reconstructed and also an overdetermined orunderdetermined inverse problem there exists a nonemptyset of the iteration number 119873 and the parameter 120572 isin (0 1)(or especially 120572 = 05) in which there is a minimum valueof the objective function 11986905 as shown in Figure 6 FromFigure 6(a) we observe that the algorithm with 120572 ge 01

sufficiently converges to a local minimum at a small iter-ation number whereas the ML-EM algorithm has a slowconvergence rate when the state variable is far away fromthe local minimum in the state space Considering the noisynature of themeasured projection and large datasets requiredfor reconstructing large sized images in clinical X-ray CTthe conditions suitable for practical use include a relativelysmall number of iterations exclusively used in IR systems [36]underwhich our IRmethod is effective as seen in Figure 6(b)


d

1

0 058

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(a)

d

1

0 05

8

6

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(b)

Figure 3 Values of RMS distance 119889 using proposed algorithm in (6) at Nth iteration while varying 120572 with (a) noisy and (b) noise-freeprojections

(a) (b)

Figure 4 (a) Sinogram and (b) reconstructed image using SKLD algorithm at 30th iteration in SPECT example

5 Concluding Remarks

We proposed the IR algorithm based on the minimizationof 120572-skew 119869-divergence between the measured and forwardprojections The objective functions to be minimized withiterative process are KL(119910 119860119911) KL(119860119911 119910) and 119869-divergenceafter setting the value of parameter 120572 to 0 1 and 05 in

the IR algorithm respectively We used Lyapunovrsquos theoremto construct a differential equation for which the IR algo-rithm is equivalent to the hybrid Euler discretization anddemonstrated the monotonically decreasing convergence ofthe 119869-divergence with iterative steps to a desired solutionof the continuous-time system The hybrid Euler was foundto have more robust performance than the additive and


(a) (b)

Figure 5 (a) Sinogram and (b) reconstructed image using SKLD algorithm at 30th iteration in X-ray CT example

4000

5000

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(a)

3

35

times104

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(b)

Figure 6 Values of 11986905 using proposed algorithm in (6) at Nth iteration while varying 120572 in (a) SPECT and (b) X-ray CT examples

multiplicative Euler methods The numerical and physicalexperiments revealed that the SKLDmethod is advantageouswith respect to the minimization of a distance measurewhen the projection data are noisy and when the maximum

iteration number is relatively small Further investigationis required to clarify the performance of the proposed IRalgorithm from the viewpoint of image quality and noiseevaluation


Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Acknowledgments

This research was partially supported by JSPS KAKENHIGrant no 18K04169

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[2] A C Kak and M Slaney Principles of Computerized Tomo-graphic Imaging IEEE Service Center Piscataway NJ USA1988

[3] H Stark Ed Image Recorvery Theory and Application Aca-demic Press Orlando Fla USA 1987

[4] L A Shepp and Y Vardi ldquoMaximum likelihood reconstructionfor emission tomographyrdquo IEEE Transactions on Medical Imag-ing vol 1 no 2 pp 113ndash122 1982

[5] S Kullback and R A Leibler ldquoOn information and sufficiencyrdquoAnnals of Mathematical Statistics vol 22 no 1 pp 79ndash86 1951

[6] C L Byrne ldquoIterative image reconstruction algorithms basedon cross-entropy minimizationrdquo IEEE Transactions on ImageProcessing vol 2 no 1 pp 96ndash103 1993

[7] J N Darroch and D Ratcliff ldquoGeneralized iterative scaling forlog-linearmodelsrdquoAnnals ofMathematical Statistics vol 43 no5 pp 1470ndash1480 1972

[8] P Schmidlin ldquoIterative separation of sections in tomographicscintigramsrdquo Nuklearmedizin Nuclear Medicine vol 11 no 1pp 1ndash16 1972

[9] C Badea and R Gordon ldquoExperiments with the nonlinear andchaotic behaviour of the multiplicative algebraic reconstruc-tion technique (MART) algorithm for computed tomographyrdquoPhysics in Medicine and Biology vol 49 no 8 pp 1455ndash14742004

[10] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Problemsvol 20 no 1 pp 103ndash120 2004

[11] H Jeffreys Theory of Probability Oxford University PressOxford 1939

[12] H Jeffreys ldquoAn invariant form for the prior probability in esti-mation problemsrdquo Proceedings of the Royal Society of LondonSeries A Mathematical and Physical Sciences vol 186 no 1007pp 453ndash461 1946

[13] J Lin ldquoDivergence measures based on the Shannon entropyrdquoIEEE Transactions on InformationTheory vol 37 no 1 pp 145ndash151 1991

[14] G E Crooks ldquoInequalities between the Jensen-Shannon andJeffreys divergencesrdquo Technical Note vol 004 2008

[15] J Schropp ldquoUsing dynamical systems methods to solve mini-mization problemsrdquoAppliedNumericalMathematics vol 18 no1-3 pp 321ndash335 1995

[16] R G Airapetyan A G Ramm and A B Smirnova ldquoContinu-ous analog of theGauss-NewtonmethodrdquoMathematicalModelsandMethods in Applied Sciences vol 9 no 3 pp 463ndash474 1999

[17] R G Airapetyan and A G Ramm ldquoDynamical systems anddiscrete methods for solving nonlinear ill-posed problemsrdquo inApplied mathematics reviews G A Anastassiou Ed vol 1pp 491ndash536 World Scientific Publishing Company Singapore2000

[18] R G Airapetyan A G Ramm and A B Smirnova ldquoCon-tinuous methods for solving nonlinear ill-posed problemsrdquo inOperator theory and its applications A G Ramm ShivakumarP N and A V Strauss Eds vol 25 of Fields Inst Comm pp111ndash136 American Mathematical Society Providence RI USA2000

[19] A G Ramm ldquoDynamical systems method for solving operatorequationsrdquo Communications in Nonlinear Science and Numeri-cal Simulation vol 9 no 4 pp 383ndash402 2004

[20] L Li and B Han ldquoA dynamical system method for solvingnonlinear ill-posed problemsrdquo Applied Mathematics and Com-putation vol 197 no 1 pp 399ndash406 2008

[21] AM Lyapunov Stability ofMotion Academic Press New YorkNY USA 1966

[22] K Fujimoto O Abou Al-Ola and T Yoshinaga ldquoContinuous-time image reconstruction using differential equations forcomputed tomographyrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 6 pp 1648ndash1654 2010

[23] O M Abou Al-Ola K Fujimoto and T Yoshinaga ldquoCommonLyapunov Function Based on KullbackndashLeibler Divergencefor a Switched Nonlinear Systemrdquo Mathematical Problems inEngineering vol 2011 pp 1ndash12 2011

[24] Y Yamaguchi K Fujimoto O M Abou Al-Ola and TYoshinaga ldquoContinuous-time image reconstruction for binarytomographyrdquoCommunications inNonlinear Science andNumer-ical Simulation vol 18 no 8 pp 2081ndash2087 2013

[25] K Tateishi Y Yamaguchi OMAbouAl-Ola andT YoshinagaldquoContinuous Analog of Accelerated OS-EM Algorithm forComputed Tomographyrdquo Mathematical Problems in Engineer-ing vol 2017 pp 1ndash8 2017

[26] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007

[27] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[28] H M Hudson and R S Larkin ldquoAccelerated image reconstruc-tion using ordered subsets of projection datardquo IEEE Transac-tions on Medical Imaging vol 13 no 4 pp 601ndash609 1994

[29] J A Fessler and A O Hero ldquoPenalized Maximum-LikelihoodImage Reconstruction Using Space-Alternating GeneralizedEM Algorithmsrdquo IEEE Transactions on Image Processing vol 4no 10 pp 1417ndash1429 1995

[30] C L Byrne ldquoAccelerating the EMML algorithm and relatediterative algorithms by rescaled block-iterative methodsrdquo IEEETransactions on Image Processing vol 7 no 1 pp 100ndash109 1998

[31] D Hwang andG L Zeng ldquoConvergence study of an acceleratedML-EM algorithm using bigger step sizerdquo Physics in Medicineand Biology vol 51 no 2 pp 237ndash252 2006

[32] B Gustavsson ldquoTomographic inversion for ALIS noise andresolutionrdquo Journal of Geophysical Research Space Physics vol103 no 11 pp 26621ndash26632 1998


[33] W Jiang and X Zhang ldquoRelaxation Factor Optimization forCommon Iterative Algorithms in Optical Computed Tomogra-phyrdquo Mathematical Problems in Engineering vol 2017 pp 1ndash82017

[34] I Castiglioni I Buvat G Rizzo M C Gilardi J Feuardentand F Fazio ldquoA publicly accessible Monte Carlo database forvalidation purposes in emission tomographyrdquo European Journalof Nuclear Medicine and Molecular Imaging vol 32 no 10 pp1234ndash1239 2005

[35] Kyoto Kagaku Co Ltd ldquoCT Whole Body Phantom PBU-60rdquohttpswwwkyotokagakucomproductsdetail03ph-2bhtml

[36] L L Geyer U J Schoepf F G Meinel et al ldquoState of the artiterative CT reconstruction techniquesrdquo Radiology vol 276 no2 pp 339ndash357 2015

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enabling us to obtain an approximate solution to the differ-ential equation That is a hybrid Euler discretization com-bined with additive and multiplicative calculus works wellhowever neither additive nor multiplicative Euler methoddoes

We conducted experiments and found that the IR algo-rithm derived from the hybrid Euler discretization of thecontinuous-time dynamical system achieved high perfor-mance

2 ML-EM and MART Algorithms

The problem of image reconstruction frommeasured projec-tions in CT is formulated as solving unknown variable 119909 isin 119877119869+for pixel values such that

119910 = 119860119909 + 120590 (1)

where 119910 isin 119877119868++ 119860 isin 119877119868times119869+ and 120590 isin 119877119868 represent the measuredprojection projection operator and noise respectively with119877+ and119877++ respectively denoting the sets of nonnegative andpositive real numbers If the system119910 = 119860119909 has a nonnegativesolution it is consistent otherwise it is inconsistent

We introduce the following definitions and notationsThevariable denoted by the symbol120582119895 for 119895 = 1 2 119869 is definedby

120582119895 fl ( 119868sum119894=1

119860 119894119895)minus1

(2)

where 119860 119894119895 is the element in the 119894th row and 119895th column of119860 The notations 119911119895 and 119860 119894 indicate the 119895th element and the119894th row vector of the vector 119911 and the matrix 119860 respectivelyThe function KL(119901 119902) of two nonnegative vectors 119901 and 119902represents the generalized KL-divergence [5]

KL (119901 119902) = 119868sum119894=1

119901119894 log 119901119894119902119894 + 119902119894 minus 119901119894 (3)

Known methods of reconstructing tomographic imagesin clinical CT are filtered back projection (FBP) as atransform method and IR as an optimization process KL-divergence is a common measure for deriving IR algorithmsand it plays an important role in accordance with theaxiom that minimizing the KL-divergence is equivalent tomaximizing the likelihood function that is considered suit-able for reconstruction modeled with a probability distribu-tionThe sequences KL(119910 119860119911(119899))infin119899=0 and KL(119860119911(119899) 119910)infin119899=0decrease for the respective iterative sequences 119911(119899)infin119899=0 of theML-EM and the (simultaneous) MART algorithms definedby

119911119895 (119899 + 1) = 119911119895 (119899) 120582119895 119868sum119894=1

119860 119894119895 119910119894119860 119894119911 (119899) (4)

and

119911119895 (119899 + 1) = 119911119895 (119899) exp(120582119895 119868sum119894=1

119860 119894119895 log 119910119894119860 119894119911 (119899)) (5)

for 119895 = 1 2 119869 with 119911(0) = 1199090 isin 119877119869++

3 Proposed System

The IR algorithm proposed in this paper can be describedusing the following difference equation

119911119895 (119899 + 1) = 119911119895 (119899) (1 + 120575119891119895 (119911 (119899))) exp (120575119892119895 (119911 (119899))) (6)

where 120575 gt 0 is a parameter and where the functions 119891119895 and119892119895 are respectively defined by

119891119895 (119911 (119899)) = (1 minus 120572) 120582119895 119868sum119894=1

119860 119894119895 ( 119910119894119860 119894119911 (119899) minus 1) (7)

and

119892119895 (119911 (119899)) = 120572120582119895 119868sum119894=1

119860 119894119895 log 119910119894119860 119894119911 (119899) (8)

for 119895 = 1 2 119869 with 120572 isin [0 1] and 119911(0) = 1199090 isin 119877119869++For investigating the dynamics of the difference system

in (6) we give its continuous analog described using thedifferential equation

119889119909119895 (119905)119889119905 = 120582119895119909119895 (119905) 119868sum119894=1

119860 119894119895sdot ((1 minus 120572) ( 119910119894119860 119894119909 (119905) minus 1) + 120572 log

119910119894119860 119894119909 (119905))(9)

for 119895 = 1 2 119869 and 119905 ge 0 with 119909(0) = 1199090 isin 119877119869++ The termldquocontinuous analogrdquo means that a numerical discretizationof the differential equation is the same as the differenceequation The idea is based on the approach for solving aconstrained tomographic inverse problem using nonlineardynamical methods [22ndash25]

31 Theoretical Analysis The theoretical results on thecontinuous-time system in (9) are given First we show thata solution is possible to ensure positive values remain

Proposition 1 If we choose an initial value 1199090 isin 119877119869++ for thesystem in (9) the solution 120601(119905 1199090) remains in the subspace 119877119869++for all 119905 ge 0Proof The vector field of the 119895th element of the system ismultiplied by 119909119895 therefore the derivative 119889120601119895119889119905 equiv 0 holdsfor any 119895 on the subspace 119909119895 equiv 0 which means the subspaceis invariant and which means any flow cannot pass throughinvariant subspace on the basis of the uniqueness property ofsolutions to the initial value problem Hence the trajectoryemanating from the initial value in 119877++ behaves in the sub-state space

Let us define119869120572 (119909) fl (1 minus 120572)KL (119910 119860119909) + 120572KL (119860119909 119910)

= 119868sum119894=1

(1 minus 120572) (119910119894 log 119910119894119860 119894119909 + 119860 119894119909 minus 119910119894)+ 120572(119860 119894119909 log 119860 119894119909119910119894 + 119910119894 minus 119860 119894119909)

(10)






119869120572 (119909) = 119868sum119894=1

(1 minus 120572)int119860119894119909119910119894

1 minus 119910119894119908119889119908 + 120572int119860119894119909119910119894

log 119908119910119894= 119868sum119894=1


119860119894119909

119910119894

(11)


119889119869120572119889119905 (119909)10038161003816100381610038161003816100381610038161003816(9) =

119868sum119894=1

((1 minus 120572) (1 minus 119910119894119860 119894119909) + 120572 log119860 119894119909119910119894 )

sdot 119869sum119895=1

119860 119894119895 119889119909119895119889119905 = minus 119869sum119895=1

119868sum119894=1

119860 119894119895 ((1 minus 120572) ( 119910119894119860 119894119909 minus 1) + 120572

sdot log 119910119894119860 119894119909)120582119895119909119895119868sum119896=1

119860119896119895 ((1 minus 120572) ( 119910119896119860119896119909 minus 1) + 120572


(radic120582119895119909119895 119868sum119894=1

119860 119894119895


le 0

(12)



119889119909 (119905)119889119905 = 119909 (119905) (119891 (119909 (119905)) + 119892 (119909 (119905))) 119909 (0) = 1199090 (13)


120587119909 (119905)120587119905 = exp (119891 (119909 (119905))) exp (119892 (119909 (119905))) 119909 (0) = 1199090 (14)


119909 ((1 + 120575) 119905) = 119909 (119905) (120587119909 (119905)120587119905 )120575 (15)


119911 (119899 + 1) = 119911 (119899) exp (120575119891 (119911 (119899))) exp (120575119892 (119911 (119899))) 119911 (0) = 1199090 (16)


discretization

119911 (119899 + 1) = 119911 (119899) (1 + 120575119891 (119911 (119899))) (17)


119911 (119899 + 1) = 119911 (119899) (1 + 120575119891 (119911 (119899))) exp (120575119892 (119911 (119899))) 119911 (0) = 1199090 (18)







(a) (b)









001

0 10 20 30t

103

102

J05


01

0 10 20 30t

103

102

J05


1

0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05







d

1

0 058

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(a)

d

1

0 05

8

6

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(b)


(a) (b)






(a) (b)


4000

5000

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(a)

3

35

times104

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(b)





Data Availability




Acknowledgments


References













































Journal of












Journal of







Volume 2018






Volume 2018













119869120572 (119909) = 119868sum119894=1

(1 minus 120572)int119860119894119909119910119894

1 minus 119910119894119908119889119908 + 120572int119860119894119909119910119894

log 119908119910119894= 119868sum119894=1


119860119894119909

119910119894

(11)


119889119869120572119889119905 (119909)10038161003816100381610038161003816100381610038161003816(9) =

119868sum119894=1

((1 minus 120572) (1 minus 119910119894119860 119894119909) + 120572 log119860 119894119909119910119894 )

sdot 119869sum119895=1

119860 119894119895 119889119909119895119889119905 = minus 119869sum119895=1

119868sum119894=1

119860 119894119895 ((1 minus 120572) ( 119910119894119860 119894119909 minus 1) + 120572

sdot log 119910119894119860 119894119909)120582119895119909119895119868sum119896=1

119860119896119895 ((1 minus 120572) ( 119910119896119860119896119909 minus 1) + 120572


(radic120582119895119909119895 119868sum119894=1

119860 119894119895


le 0

(12)



119889119909 (119905)119889119905 = 119909 (119905) (119891 (119909 (119905)) + 119892 (119909 (119905))) 119909 (0) = 1199090 (13)


120587119909 (119905)120587119905 = exp (119891 (119909 (119905))) exp (119892 (119909 (119905))) 119909 (0) = 1199090 (14)


119909 ((1 + 120575) 119905) = 119909 (119905) (120587119909 (119905)120587119905 )120575 (15)


119911 (119899 + 1) = 119911 (119899) exp (120575119891 (119911 (119899))) exp (120575119892 (119911 (119899))) 119911 (0) = 1199090 (16)


discretization

119911 (119899 + 1) = 119911 (119899) (1 + 120575119891 (119911 (119899))) (17)


119911 (119899 + 1) = 119911 (119899) (1 + 120575119891 (119911 (119899))) exp (120575119892 (119911 (119899))) 119911 (0) = 1199090 (18)







(a) (b)









001

0 10 20 30t

103

102

J05


01

0 10 20 30t

103

102

J05


1

0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05







d

1

0 058

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(a)

d

1

0 05

8

6

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(b)


(a) (b)






(a) (b)


4000

5000

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(a)

3

35

times104

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(b)





Data Availability




Acknowledgments


References













































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b)









001

0 10 20 30t

103

102

J05


01

0 10 20 30t

103

102

J05


1

0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05







d

1

0 058

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(a)

d

1

0 05

8

6

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(b)


(a) (b)






(a) (b)


4000

5000

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(a)

3

35

times104

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(b)





Data Availability




Acknowledgments


References













































Journal of












Journal of







Volume 2018






Volume 2018










001

0 10 20 30t

103

102

J05


01

0 10 20 30t

103

102

J05


1

0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05


0 10 20 30t

103

102

J05







d

1

0 058

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(a)

d

1

0 05

8

6

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(b)


(a) (b)






(a) (b)


4000

5000

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(a)

3

35

times104

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(b)





Data Availability




Acknowledgments


References













































Journal of












Journal of







Volume 2018






Volume 2018









d

1

0 058

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(a)

d

1

0 05

8

6

10

12

14

N = 30N = 20

N = 12N = 10

N = 15(b)


(a) (b)






(a) (b)


4000

5000

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(a)

3

35

times104

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(b)





Data Availability




Acknowledgments


References













































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b)


4000

5000

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(a)

3

35

times104

J 05

1

0 05

N = 30N = 20

N = 12N = 10

N = 15(b)





Data Availability




Acknowledgments


References













































Journal of












Journal of







Volume 2018






Volume 2018









Data Availability




Acknowledgments


References













































Journal of












Journal of







Volume 2018






Volume 2018




















Journal of












Journal of







Volume 2018






Volume 2018








tomographic image reconstruction based on minimization of...

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