tomographic inverse problem with estimating missing...

Research ArticleTomographic Inverse Problem with EstimatingMissing Projections

Masashi Kimura 1 Yusaku Yamaguchi 2

Omar M Abou Al-Ola 3 and Tetsuya Yoshinaga 4

1Graduate School of Health Sciences Tokushima University 3-18-15 Kuramoto Tokushima 770-8509 Japan2National Hospital Organization Shikoku Medical Center for Children and Adults 2-1-1 Senyu Zentsuji 765-8507 Japan3Faculty of Science Tanta University El-Giesh St Tanta Gharbia 31527 Egypt4Institute of Biomedical Sciences Tokushima University 3-18-15 Kuramoto Tokushima 770-8509 Japan

Correspondence should be addressed to Tetsuya Yoshinaga yosinagamedscitokushima-uacjp

Received 17 December 2018 Accepted 10 February 2019 Published 13 March 2019

Academic Editor Higinio Ramos

Copyright copy 2019 Masashi Kimura et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Image reconstruction in computed tomography can be treated as an inverse problem namely obtaining pixel values ofa tomographic image from measured projections However a seriously degraded image with artifacts is produced whena certain part of the projections is inaccurate or missing A novel method for simultaneously obtaining a reconstructedimage and an estimated projection by solving an initial-value problem of differential equations is proposed A systemof differential equations is constructed on the basis of optimizing a cost function of unknown variables for an imageand a projection Three systems described by nonlinear differential equations are constructed and the stability of aset of equilibria corresponding to an optimized solution for each system is proved by using the Lyapunov stabilitytheorem To validate the theoretical result given by the proposed method metal artifact reduction was numericallyperformed

1 Introduction

In the field of computed tomography (CT) [1ndash4] imagereconstruction is generally considered an inverse problemnamely obtaining pixel values of an image from measuredprojections and a known projection operator In X-ray CThowever an image is seriously degraded by for examplemetal and ring artifacts [5ndash7] when a certain part of theprojections received by the detector is inaccurate or miss-ing Interpolation projection is a well-known method forreducing such artifacts [5 7] that is inaccurate projectionsare interpolated from their neighboring data and replacedby synthesized values Another method for reducing metalartifacts completes inaccurate projections by subtracting theprojections that cause the artifacts [6 7] in particular thesubtraction process uses synthesized projections reprojected

as an image including an estimated and corrected metalportion

As for formulating the inverse problem 119901 isin 119877119868+ and 119860 isin119877119868times119869+ denote respectively projection data and a projectionoperator where 119877+ represents a set of non-negative realnumbers and 119868 and 119869 respectively indicate the numbers ofprojections and pixels in a reconstructed image Here thecase in which a part of projection 119901 is given by inaccurateor missing measurements (for example due to the existenceof a metal object in the X-ray CT scan) is considered It isassumed that without loss of generality the elements of 119901 aresorted and divided into two parts as follows

119901 = (119902119903) (1)

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 7932318 11 pageshttpsdoiorg10115520197932318

2 Mathematical Problems in Engineering

where 119902 isin 119877119872+ and 119903 isin 119877119868minus119872+ correspond to the inaccurate(119868 gt 119872 gt 0) and the accurate projection data respectively Itis also assumed that 119860 = (119861119862) (2)

where 119861 isin 119877119872times119869+ and 119862 isin 119877(119868minus119872)times119869+ with rows correspondingto elements of 119901

To reconstruct an image with pixel values and to estimatecertain parts of projections instead of inaccurate part 119902unknown variables 119909 isin 119877119869+ and 119910 isin 119877119872+ are treated respec-tively It is assumed that pixel values correspond to forexample in the case of X-ray CT actual values of attenuationcoefficients including the minimum value zero correspond-ing to attenuation by air It is also assumed that some elementsof vector 119909 satisfying 119903 = 119862119909 are undetermined becauserank119862 lt 119869 due to missing measurements Accordingly theundetermined elements should be permitted to have arbitraryvalues The problem considered in this paper is thereforedefined as follows

Definition 1 The tomographic inverse problem with an inac-curate projection is consistent if the set

E = 119890 isin 119877119869+ 119903 = 119862119890 (3)

is not empty

It follows that the problem is to obtain unknown variables119909 and119910 byminimizing an appropriate cost function that takeszero at 119909 = 119890 and 119910 = 119861119890 for some 119890 isin E respectively Tosolve this problem the following cost function of 119909 isin 119877119869+ and119910 isin 119877119872+ with 119890 isin E is considered119881 (119909 119910) = 119869sum

119895=1

120582minus1119895 KL (119890119895 119909119895) + 120572minus1KL (119861119890 119910)= 119869sum119895=1

120582minus1119895 (119890119895 log 119890119895119909119895 + 119909119895 minus 119890119895)+ 120572minus1 119872sum119894=1

(119861119890)119894 log (119861119890)119894119910119894 + 119910119894 minus (119861119890)119894(4)

where 120572 gt 0 is a weight parameter (V)ℓ denotes theℓth element of vector V KL indicates the generalizedKullbackndashLeibler divergence [8 9] and 120582119895 is defined as

120582119895 = ( 119868sum119894=1

119860 119894119895)minus1 (5)

for 119895 = 1 2 119869 Note that the function in Eq (4) isnonnegative for arbitrary 119909 isin 119877119869+ and 119910 isin 119877119872+ and zero ifand only if 119909 = 119890 and 119910 = 119861119890

This paper is organized as follows In Section 2 a novelmethod of simultaneously obtaining a reconstructed imageand an estimated projection is proposed It solves an initial-value problem of differential equations consisting of state

variables including parts of projections as well as imagepixel values A system of differential equations based on anextension of dynamical methods [10ndash20] is constructed Itis an approach that optimizes a cost function of unknownvariables consisting of an image and a projection Threesystems described by nonlinear differential equations withdifferent vector fields among each other are presented InSection 3 for each system the stability of a set of equilibriacorresponding to an optimized solution is proved by using theLyapunov stability theorem [21] In Section 4 the intentionof the proposed dynamical system is described in detailand how it works is shown by using a simple toy modelAdditionally to validate the theoretical results given by theproposed method in comparison with results obtained bylinear interpolation and a reduced system reduction of metalartifacts is numerically simulated by using a large-sizedmodel

2 Dynamical Systems

Three kinds of continuous-time dynamical systems with statevariables (119909⊤ 119910⊤)⊤ isin 119877119870++ are proposed where 119870 fl 119869 + 119872and 119877++ denotes the set of positive real numbers In thesesystems 119881(119909(119905) 119910(119905)) is expected to decrease in time 119905 alongtheir solutions The three dynamical systems are described byautonomous nonlinear differential equationsThe first systemis defined as119889119909119889119905 = 119883Λ119861⊤ (119910 minus 119861119909) + 119883Λ119862⊤ (119903 minus 119862119909) 119889119910119889119905 = 120572119884 (119861119909 minus 119910) (6)

with initial states 119909(0) = 1199090 isin 119877119869++ and 119910(0) = 1199100 isin119877119872++ where 119883 119884 and Λ indicate the diagonal matrices ofvectors 119909 119910 and 120582 consisting of 120582119895 in Eq (5) respectivelyNote that variables 119909(119905) and 119910(119905) are mutually coupled in thevector field The derivative 119889119909119889119905 for the dynamics of imagereconstruction in the case 119872 = 0 has the same form as thecontinuous-time image reconstruction system presented inRef [17]

The second and third systems which are respectivelyinspired by continuous analogs [19 22] of the multiplica-tive algebraic reconstruction technique [23 24] and themaximum-likelihood expectation-maximization [3 25] algo-rithm are given by119889119909119889119905 = 119883Λ119861⊤ (Log (119910) minus Log (119861119909))+ 119883Λ119862⊤ (Log (119903) minus Log (119862119909)) 119889119910119889119905 = 120572119884 (Log (119861119909) minus Log (119910))

(7)

and 119889119909119889119905 = 119883 Log (Λ119861⊤Exp (Log (119910) minus Log (119861119909))

Mathematical Problems in Engineering 3+ Λ119862⊤Exp (Log (119903) minus Log (119862119909))) 119889119910119889119905 = 120572119884 (Log (119861119909) minus Log (119910)) (8)

with the same initial states as in Eq (6) Here Log(120573) andExp(120573) denote vector-valued functions Log(120573) fl (log(1205731)log(1205732) log(120573119871))⊤ and Exp(120573) fl (exp(1205731) exp(1205732) exp(120573119871))⊤ of each element in vector 120573 = (1205731 1205732 120573119871)⊤ respectively3 Theoretical Analysis

Theoretical results for the common behavior of the solutionsto the three dynamical systems are presented hereafter Itis first shown that any solution to each dynamical systemis positive regardless of whether the inverse problem isconsistent or not

Proposition2 If initial value (1199090⊤ 1199100⊤)⊤ isin 119877119870++ in each of thedynamical systems described by Eqs (6) (7) and (8) is takensolution 120593(119905 1199090 1199100) stays in 119877119870++ for all 119905 gt 0Proof Since the dynamics of the 119896th elements of 119908 fl(119909⊤ 119910⊤)⊤ can be written as 119889119908119896119889119905 = 119908119896119891119896(119908) with afunction 119891119896 it follows that on the subspace restricted to119908119896 =0 the solution satisfies 119889120593119896119889119905 equiv 0 for any 119896 = 1 2 119870Therefore according to the uniqueness of solutions for theinitial value problem [26] the subspace is invariant andtrajectories cannot pass through every invariant subspaceThis restriction leads to any solution 120593(119905 1199090 1199100) of any systemwith initial value (1199090⊤ 1199100⊤)⊤ isin 119877119870++ at 119905 = 0 being in 119877119870++ forall 119905 gt 0

Under the assumption that the tomographic inverseproblem minimizing cost function 119881 in Eq (4) is consistentequilibrium (119890⊤ (119861119890)⊤)⊤ isin 119877119870++ for each of the differentialequations in Eqs (6) (7) and (8) is asymptotically stableaccording to the Lyapunov theorem [21]

Theorem 3 If the set E in Eq (3) is not empty the solution(119909⊤ 119910⊤)⊤ to the dynamical system described by Eq (6) withpositive initial values asymptotically converges to (119890⊤ (119861119890)⊤)⊤for some 119890 isin E 13e same property holds for the dynamicalsystems in Eqs (7) and (8)

Proof Consider the function 119881(119909 119910) ge 0 in Eq (4)as a Lyapunov-candidate-function which is well-definedbecause fromProposition 2 the state variable (119909(119905)⊤ 119910(119905)⊤)⊤of the dynamical system in Eq (6) belongs to119877119872++ for any 119905 bychoosing positive initial valuesThe function can be rewrittenas 119881 (119909 119910) = 119869sum

119895=1

120582minus1119895 int119909119895119890119895

119906 minus 119890119895119906 119889119906+ 120572minus1 119872sum119894=1

int119910119894(119861119890)119894

V minus (119861119890)119894V

119889V (9)

so the derivative along the solution to the system in Eq (6) isgiven by119889119889119905119881 (119909 119910)10038161003816100381610038161003816100381610038161003816(6) = minus 119869sum119895=1 (119909119895 minus 119890119895) (119861⊤)119895 (119861119909 minus 119910)minus 119869sum

119895=1

(119909119895 minus 119890119895) (119862⊤)119895 (119862119909 minus 119903)minus 119872sum119894=1

(119910119894 minus (119861119890)119894) (119910119894 minus (119861119909)119894)= minus (119861119909 minus 119861119890)⊤ (119861119909 minus 119910)minus (119862119909 minus 119903)⊤ (119862119909 minus 119903)minus (119861119890 minus 119910)⊤ (119861119909 minus 119910)= minus 119862119909 minus 11990322 + 1003817100381710038171003817B119909 minus 119910100381710038171003817100381722 le 0

(10)

with equality if and only if 119909 = 119890 and 119910 = 119861119890 Thusthe asymptotic convergence of solutions to some 119890 in E isguaranteed by using the Lyapunov theorem

In the same way we see that 119881 is also Lyapunov functionsfor the systems described by Eqs (7) and (8) according to119889119889119905119881 (119909 119910)10038161003816100381610038161003816100381610038161003816(7)= minus 119869sum

119895=1

(119909119895 minus 119890119895) (119861⊤)119895 (Log (119861119909) minus Log (119910))minus 119869sum119895=1


(119910119894 minus (119861119890)119894) (log (119910119894) minus log ((119861119909)119894)) = minus (119861119909minus 119861119890)⊤ (Log (119861119909) minus Log (119910)) minus (119862119909 minus 119903)⊤ (Log (119862119909)minus Log (119903)) minus (119861119890 minus 119910)⊤ (Log (119861119909) minus Log (119910))= minus KL (119862119909 119903) + KL (119903 119862119909) + KL (119861119909 119910)+ KL (119910 119861119909) le 0

(11)

and 119889119889119905119881 (119909 119910)10038161003816100381610038161003816100381610038161003816(8) = minus 119869sum119895=1 (119890119895 minus 119909119895) 120582minus1119895sdot log (120582119895 (119861⊤)119895 Exp (Log (119910) minus Log (119861119909))+ 120582119895 (119862⊤)119895 Exp (Log (119903) minus Log (119862119909))) minus 119872sum119894=1

(119910119894minus (119861119890)119894) (log (119910119894) minus log ((119861119909)119894)) le minus 119869sum

119895=1

119890119895 (119861⊤)119895


sdot (Log (119910) minus Log (119861119909)) minus 119869sum119895=1

119890119895 (119862⊤)119895 (Log (119903)minus Log (119862119909)) + 119869sum

119895=1

119909119895 (119861⊤)119895 (Exp (Log (119910)minus Log (119861119909)) minus 119906) + 119869sum

119895=1

119909119895 (119862⊤)119895 (Exp (Log (119903)minus Log (119862119909)) minus V) minus 119872sum

119894=1

119910119894 (log (119910119894) minus log ((119861119909)119894))+ 119872sum119894=1

(119861119890)119894 (log (119910119894) minus log ((119861119909)119894)) = minus 119869sum119895=1

119890119895 (119862⊤)119895sdot (Log (119903) minus Log (119862119909))+ 119869sum119895=1

119909119895 (119862⊤)119895 (Exp (Log (119903) minus Log (119862119909)) minus V)minus 119872sum119894=1

119910119894 (log (119910119894) minus log ((119861119909)119894)) + 119872sum119894=1

(119861119909)119894sdot (exp (log (119910119894) minus log ((119861119909)119894)) minus 1) le minus119868minus119872sum

119894=1

119903119894 (1minus exp (log ((119862119909)119894) minus log (119903119894))) + 119868minus119872sum

119894=1

(119862119909)119894sdot (exp (log (119903119894) minus log ((119862119909)119894)) minus 1) minus 119872sum

119894=1

119910119894 (1minus exp (log ((119861119909)119894) minus log (119910119894))) + 119872sum

119894=1

(119861119909)119894sdot (exp (log (119910119894) minus log ((119861119909)119894)) minus 1) = 0(12)

respectively where 119906 = (1 1 1)⊤ isin 119877119872++ and V =(1 1 1)⊤ isin 119877119868minus119872++ The derivatives are zero if and only if119909 = 119890 and 119910 = 119861119890 Therefore the solutions to the dynamicalsystems in Eqs (7) and (8) also asymptotically converge tosome 119890 isin E4 Application

Numerical experiments were performed to illustrate the effi-ciency of the proposed method in applications for reducingmetal artifacts

41 Experimental Method The systems of differential equa-tions described by Eqs (6) (7) and (8) with initial states119909(0) = 1199090 isin 119877119869++ and 119910(0) = 1199100 isin 119877119872++ were used for

the numerical experiments In the experiments sinogramsconsisting of projections described as

119901lowast = (119902lowast119903 ) (13)

which resulted in the effects of beam hardening and severattenuation of the X-ray beam caused by the metal hardware[27ndash29] were synthesized Here 119902lowast isin 119877119872+ was obtained bythe following procedure Let inaccurate projections be theouter neighborhood of the metal affected projections thatare identified by thresholding [7 28 30] The inaccuratemeasured projections 119902 in Eq (1) were linearly interpolatedfrom their neighboring projections in 119903 at each common pro-jection angle and replaced by synthesized values 119902lowast Becausethe main feature of the proposed method is that inaccurateprojections are considered as variables of optimization thesynthesized sinogram is treated as an initial value for theproposed dynamical system and as a fixed projection valuefor the system to be compared Then initial state (1199090⊤ 1199100⊤)⊤of the proposed system is chosen as 1199090119895 = 1199090lowast gt 0 for119895 = 1 2 119869 and 1199100 = 119902lowast However the system describedby the following differential equations is defined as a linear-interpolated system compared to the proposed system in Eq(6) 119889119909119889119905 = 119883Λ119860⊤ (119901lowast minus 119860119909)= 119883Λ119861⊤ (119902lowast minus 119861119909) + 119883Λ119862⊤ (119903 minus 119862119909) (14)

with initial state 119909(0) = 1199090 Note that both solutions tothe differential equation in Eq (14) with 119909(0) = 1199090 and tothe equations in Eq (6) in which the time derivative of 119910 isreplaced with 119889119910119889119905 = 0 with 119909(0) = 1199090 and 119910(0) = 119902lowast areequivalent because 119910(119905) equiv 119902lowast for all 119905 ge 0 satisfies in regardto the latter system An ODE solver ode113 in MATLAB(MathWorks Natick USA) was used for solving the initial-value problem of the differential equations in Eqs (6) and(14)

For comparison with the proposed system another sys-tem called a reduced system is established Namely forreconstructing image 119911 isin 119877119869++ subject to minimizing119903 minus 1198621199112 an iterative formula by a projected Landweberalgorithm [31] is defined as119911 (119899 + 1) = (119892 (119911 (119899)))+ 119899 = 0 1 2 (15)

with 119911(0) = 1199090 where119892 (119911 (119899)) fl 119911 (119899) + 120588minus1119862⊤ (119903 minus 119862119911 (119899)) (16)

with 120588 denoting the largest eigenvalue of matrix 119862⊤119862 andvector-valued function (120573)+ of vector 120573 = (1205731 1205732 120573119871)⊤indicating (120573)+ fl (max0 1205731max0 1205732 max0 120573119871)⊤Relaxation parameter 120588minus1 is set according to Theorem 31 inRef [32] so that the iterative sequence converges to a solutionclosest to 119890 isin E

Mathematical Problems in Engineering 5

12

0

(a)

06

0

(b)

Figure 1 (a) Phantom image and (b) sinogram

12

0

(a)

12

0

(b)

12

0

(c)

Figure 2 Reconstructed images using (a) proposed (b) linear-interpolated and (c) reduced systems

To evaluate the quality of the images quantitatively adistance measure is defined by the 1198711-norm as

119880 (119909) = 119869sum119895=1119895notinM

10038161003816100381610038161003816119909119895 minus 119890lowast119895 10038161003816100381610038161003816 (17)

where M sub 1 2 119869 denotes the set of indices corre-sponding to the metal position in the true image 119890lowast isin 119877119869+42 Simple Model To explain how the proposed systemworks a simple toy model is considered as follows In themodel true image 119890lowast isin 1198779+ is defined as119890lowast = (09 1 0 0 0 0 07 0 0)⊤ (18)

A graphical image of 3times3 pixels corresponding to 119890lowast is shownin Figure 1(a) It is assumed that ametal object in the phantomis located at the two pixels colored in white Although thepixel values are undetermined due to the high attenuation ofmetal in X-ray CT-scan imaging for convenience the valuesof 119890lowast5 and 119890lowast6 were both set to zero A sinogram obtained bytaking samples at every 120 degrees in the range of 360 and7 detector bins per projection angle is shown in Figure 1(b)The white regions in the 3 times 7-sized sinogram indicatean inaccurate part 119902 isin 1198778+ The correspondence relationbetween elements 119901 and (119902⊤ 119903⊤)⊤ in Eq (1) is described by

the following format which corresponds to the sinogram inFigure 1(b)

(1199011 1199012 1199013 1199014 1199015 1199016 11990171199018 1199019 11990110 11990111 11990112 11990113 1199011411990115 11990116 11990117 11990118 11990119 11990120 11990121)= (1199031 1199032 1199021 1199022 1199033 1199034 11990351199036 1199023 1199024 1199025 1199037 1199038 119903911990310 11990311 1199026 1199027 1199028 11990312 11990313) š(119878(119902)1119878 (119902)2119878 (119902)3)= 119878 (119902) isin 1198773times7+

(19)

When using this notation in the case of proposed system theelements of measured projection 119903 and estimated projection119910(119905) 119905 ge 0 instead of 119902 can be arranged as matrix 119878(119910(119905)) inthe same format

Images were reconstructed by using solutions of threesystems described by Eqs (6) (14) and (15) The parameter120572 used for the proposed system uniform initial pixel values1199090lowast integration time 120591 and iteration number ] were set to01 05 105 and 105 respectively unless otherwise specifiedAn example of reconstructed images is shown in Figure 2Values of distance measures 119880(119909(120591)) and 119880(119911(])) defined inEq (17) with M = 5 6 for reconstructed images (a) (b)and (c) in Figure 2 are 02887 13773 and 24881 respectivelyIt can be seen that the proposed reconstruction has betterimage quality Moreover as indicted in Table 1 compared to


Table 1 119880(119909(120591)) and 119880(119911(])) for images reconstructed using theproposedmethodwith120572 = 001 and 01 linear-interpolatedmethodand reduced method with variation of initial pixel values1199090lowast proposed method interpolated reduced120572 = 001 120572 = 01 method method01 02255 02300 13773 0722705 02308 02887 13773 248811 02331 03200 13773 50240

the reduced method the proposedmethod gives more robustperformance in terms of the variation of initial values

For each of the proposed and reduced systems it wasnumerically confirmed that the distance between 119903 and119862119909(120591)

(or 119862119911(])) is zero that is convergent values 119909(120591) and 119911(])belong to set E in Eq (3) An explicit description of this setis introduced as follows Neglecting any trivial element 119903119894 of119903 isin 11987713+ and the corresponding row 119862119894 of 119862 isin 11987713times9+ such that119903119894 = 0 and 119862119894119895 = 0 for any 119895 (excluding the case that 119903119894 = 0but 119862119894119895 = 0 for some 119895) obtains

(11990321199036119903711990312)=(0548601413002573) (20)

and

119862lowast fl(11986221198626119862711986212)=(03429 0 0 03429 0 0 03429 0 001066 00453 00075 0 0 0 0 0 00 0 0 0 0 0 0 0 0004300573 0 0 01887 0 0 02940 00131 0 ) (21)

The set defined in Eq (3) can therefore be explicitly describedas

E = 119890lowast + 119873119904 119904 isin 1198775 (22)

where 119873 is the matrix whose column vectors are theorthonormal basis for the null space of matrix 119862lowast ie

119873 =((((((((((((((

0 0 minus02766 minus00011 003820 0 06562 00039 007400 0 minus00302 minus00080 minus099180 0 06068 01247 minus008491 0 0 0 00 1 0 0 00 0 minus03302 minus01236 004670 0 minus01215 09844 000820 0 0 0 0

))))))))))))))

(23)

Approximate values of coefficients 119904 in Eq (22) used for theproposed and reduced systems are respectively given as

(((

0004401407minus0016300014minus01194)))

and (((

05000050000076105737minus05958)))

(24)

It is clear that the convergent value obtained by the proposedsystem gives smaller absolute values of 1199043 1199044 and 1199045

A graph of the distance measures 119880(119909(120591)) and 119880(119911(]))for comparing the three methods is shown in Figure 3 Thedistance given by the proposed method varies with differentvalues of 120572 The figure shows that there exists a range of 120572at least in interval (0 01] in which the quantitative measurehas a minimum value Note that the system of Eq (6) as 120572tends to zero is similar to the interpolated system of Eq (14)A smaller value of 120572 results in not only a larger weight ofterm KL(119861119890 119910) in Eq (4) but also a slower dynamics of statevariable 119910(119905) On the other hand when parameter 120572 tendsto infinity the system of Eq (6) is equivalent to a reducedsystem optimizing the function sum119869119895=1 120582minus1119895 KL(119890119895 119909119895) which isa part of the cost function 119881 in Eq (4) without informationabout 119902lowast The dynamics of solutions to the proposed systemare explained as follows In Figure 4 the dynamics expressedby the time evolution of state variables obtained by the threedynamical systems is illustrated With regard to the proposedsystem the behavior of 119861119909(119905) is restricted by the dynamics of119910(119905) emanating from interpolated value 119902lowast and converges tocoincide with 119910(119905) after the passage of time However in theinterpolated and reduced systems 119861119909(119905) attempts to reach 119902lowastwithout reaching it and is independent of 119902lowast respectivelyThemutual dynamical coupling of 119861119909(119905) and 119910(119905) in the proposedsystem gives rise to a better convergence performance ofthe state 119909(119905) in regard to approaching ideal value 119890lowast Thedynamics of 119910(119905) that eventually converges to a limit set inthe neighborhood of 119902lowast in the state space is effective forestimating the inaccurate projection and produces a betterquality image through 119861119909(119905) which converges to 119910(119905)43 Large-Sized Model The proposed method was exper-imentally demonstrated by applying it to reducing metalartifacts by using a larger sized model A phantom image with


0 0100800600400202

05

1

2

U(x

())

U(z(]))

0 cup

Figure 3 Distances (at 120591 = 105 and ] = 105) for the proposed method with variation of 120572 interpolatedmethod and reduced method whichis independent of 120572 indicated by blue points a green point and a magenta line respectively

0

02

04

06

08

1

t

t

xj(t)

yi(t) (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(a)

0

02

04

06

08

1

t

t

xj(t)

qlowasti (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(b)

0

02

04

06

08

1

n

n

zj(n) elowastj

100 101 102 103 104 105

(Bz(n))i

100

100

101 102 103 104 105

10minus1

(c)

Figure 4 Time evolution of state variable for pixel values 119909 (upper panel) and projection values (lower panel) obtained by using (a) proposed(b) linear-interpolated and (c) reduced systems In the upper graph trajectories 119909119895 and 119911119895 with lines and 119890lowast119895 marked with colored dots at theright edge are indicated by the same color as the corresponding pixel in the phantom image In the lower graph blue andmagenta lines denote119910 (or 119902lowast) and 119861119909 (or 119861119911) respectively192 times 192 pixels so 119869 = 36 864 was simulated as shownin Figure 5(a) An image was reconstructed by using 275detector bins per projection angle and 360-degree scanningwith sampling every two degrees which corresponds tonumber of projections 119868 = 49 500 For simulating reductionof metal artifacts it was assumed that three small metalobjects exist in the phantom The white regions in the gray-scaled sinogram shown inFigure 5(b) indicate inaccurate part119902 in the sinogram or projection 119901 in Eq (1)

The experimental results obtained by using the proposedsystem described by Eq (6) with 120572 = 001 the linear-interpolated system described by Eq (14) and the reducedsystem described by Eq (15) are presented as follows Recon-structed images obtained from solutions 119909(120591) at 120591 = 100for the continuous-time system and 119911(]) at ] = 3 500for the discrete-time system which are emanating from theinitial value with 1199090lowast = 05 are shown in Figure 6 Thevalues of 120591 and ] for adjusting the difference between the


12

0

(a)

08

0

(b)

Figure 5 (a) Phantom image with three metal positions (small black regions) indicated by white arrows and (b) its sinogram

12

0

(a)

12

0

(b)

12

0

(c)

Figure 6 Reconstructed images by (a) proposed (b) interpolated and (c) reduced methods

0 35003000250020001500100050035tn

100

101

102

103

104

rminusCx(t)

1

rminusCz(n

)1

Figure 7 Time evolutions of objective function by the proposedinterpolated and reduced methods indicated by blue green andmagenta lines respectively

continuous-time and discrete-time systems describing theproposed and reduced methods respectively were chosen sothat the values of objective functions defined by 119903 minus 119862119909(119905)1

010010002430

440

450

460

470

480

U(x

())

U(z(]))

Figure 8 Distances (at 120591 = 100 and ] = 3 500) for theproposed method with variation of 120572 and the reduced methodbeing independent of 120572 indicated by blue points and magenta linesrespectively

and 119903 minus 119862119911(119899)1 coincide at 35120591 = ] as shown in Figure 7Under the same conditions for reconstruction a qualitativereduction of artifacts in the reconstructed image obtained by


010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)

Figure 9 Distances (at ] = 3 500) for proposed method given (a) by Eq (25) and (b) Eq (26) with variation of 120572 along with reducedmethodbeing independent of 120572 indicated by blue points and magenta lines respectively

the proposed method was compared with that by the othertwo methods From Figure 8 which compares quantitativemeasures 119880(119909(120591)) for the proposed method and 119880(119911(])) forthe reduced method in addition to the observation of themeasure 8306 for the interpolated method it is clear thatthe proposed method with 120572 isin [0002 01] has the smallestmeasure The advantage of the proposed method is due tothe simultaneous dynamical behavior of a pair of variables119909(119905) and 119910(119905) which is based on mutual dynamical couplingdescribed by Eq (6)

Discretization of the differential equations is requiredfor low-cost computation in practical use Iterative formulaederived from a multiplicative Euler discretization [33] ofdifferential equations in Eqs (7) and (8) are respectivelygiven by

119911 (119899 + 1) = 119885 (119899)sdot Exp (Λ119861⊤ (Log (119908 (119899)) minus Log (119861119911 (119899)))+ Λ119862⊤ (Log (119903) minus Log (119862119911 (119899)))) 119908 (119899 + 1) = 119882 (119899)Exp (120572 (Log (119861119911 (119899))minus Log (119908 (119899)))) (25)

and 119911 (119899 + 1) = 119885 (119899)sdot (Λ119861⊤Exp (Log (119908 (119899)) minus Log (119861119911 (119899)))+ Λ119862⊤Exp (Log (119903) minus Log (119862119911 (119899)))) 119908 (119899 + 1) = 119882 (119899)sdot Exp (120572 (Log (119861119911 (119899)) minus Log (119908 (119899)))) (26)

for 119899 = 0 1 2 with initial states (119911(0)⊤ 119908(0)⊤)⊤ =(1199090⊤ 1199100⊤)⊤ where 119885 and 119882 indicate the diagonal matricesof 119911 and 119908 respectively Graphs of the distances measured by119880(119911) for reconstructed images obtained after 3 500 iterationsof the proposed systems in Eqs (25) and (26) and the reducedsystem in Eq (15) are shown in Figure 9 The multiplicativeEuler discretization approximates solutions to the differentialequations of Eqs (7) and (8) at the discrete points with a stepsize equal to one It is noted that the course of the distanceswith variation 120572 for the proposed method given by Eqs (25)and (26) as well as Eq (6) is qualitatively consistent and thevalues of these distances in the range of 120572 are lower than thatgiven by the reduced method The system of Eq (25) yieldsthe best performance with respect to minimizing distance119880(119911) under the same conditions

5 Conclusion

An image-reconstruction method for optimizing a costfunction of unknown variables corresponding to missingprojections as well as image densities was proposed Threedynamical systems described by nonlinear differential equa-tions were constructed and the stability of their equilibriawas proved by using the Lyapunov theorem The efficiencyof the proposed method was evaluated through numericalexperiments simulating the reduction of metal artifactsQualitative and quantitative evaluations using image qualityand a distance measure indicate that the proposed methodperforms better than the conventional linear-interpolationand reduced methods This paper shows a novel approach tooptimizing the cost function of unknown variables consistingof both image and projection However not only furtheranalysis of convergence rates and investigation on the influ-ence of measured noise but also experimental evaluationand detailed comparison with other existing methods [7 29]of artifact reduction are required to extend the proposedapproach for practical use


Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

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

Acknowledgments

This research was partially supported by JSPS KAKENHIGrant Number 18K04169

References

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

[2] H Stark Image Recorvery 13eorem And Application AcademicPress Orlando Fla USA 1987

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

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

[5] W A Kalender R Hebel and J Ebersberger ldquoReduction of CTartifacts caused by metallic implantsrdquo Radiology vol 164 no 2Article ID 3602406 pp 576-577 1987

[6] C R Crawford J G Colsher N J Pelc and A H R LonnldquoHigh speed reprojection and its applicationsrdquo Proceedings ofSPIE -13e International Society for Optical Engineering vol 914pp 311ndash318 1988

[7] L Gjesteby B De Man Y Jin et al ldquoMetal artifact reduction inCT where are we after four decadesrdquo IEEE Access vol 4 pp5826ndash5849 2016

[8] S Kullback and R A Leibler ldquoOn information and sufficiencyrdquoAnnals of Mathematical Statistics vol 22 pp 79ndash86 1951

[9] I Csiszar ldquoWhy least squares and maximum entropy anaxiomatic approach to inference for linear inverse problemsrdquo13e Annals of Statistics vol 19 no 4 pp 2032ndash2066 1991

[10] M K Gavurin ldquoNonlinear functional equations and con-tinuous analogues of iteration methodsrdquo Izvestiya VysshikhUchebnykh Zavedenii Matematika vol 5 no 6 pp 18ndash31 1958

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

[12] 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

[13] R Airapetyan ldquoContinuous Newton method and its modifica-tionrdquo Applicable Analysis An International Journal vol 73 no3-4 pp 463ndash484 1999

[14] 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

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

[16] 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

[17] O M Abou Al-Ola K Fujimoto and T Yoshinaga ldquoCommonLyapunov function based on KullbackndashLeibler divergence fora switched nonlinear systemrdquo Mathematical Problems in Engi-neering vol 2011 Article ID 723509 12 pages 2011

[18] 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

[19] K Tateishi Y Yamaguchi OMAbouAl-Ola andT YoshinagaldquoContinuous analog of accelerated OS-EM algorithm for com-puted tomographyrdquoMathematical Problems in Engineering vol2017 Article ID 1564123 8 pages 2017

[20] R Kasai Y Yamaguchi T Kojima and T Yoshinaga ldquoTomo-graphic image reconstruction based on minimization of sym-metrized Kullback-Leibler divergencerdquoMathematical Problemsin Engineering vol 2018 Article ID 8973131 9 pages 2018

[21] AM Lyapunov Stability of Motion Academic Press NewYorkNY USA 1966

[22] K Tateishi Y Yamaguchi O M Abou Al-Ola T Kojima andT Yoshinaga ldquoContinuous analog of multiplicative algebraicreconstruction technique for computed tomographyrdquo in Pro-ceedings of the SPIE Medical Imaging 2016 March 2016

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

[24] 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

[25] S Vishampayan J Stamos R Mayans K Kenneth NClinthorne and W Leslie Rogers ldquoMaximum likelihood imagereconstruction for SPECTrdquo Journal of Nuclear Medicine (AbsBook) vol 26 no 5 p 20 1985

[26] L S Pontryagin Ordinary Differential Equations Addison-Wesley Pergamon Turkey 1962

[27] B De Man J Nuyts P Dupont G Marchai and P SuetensldquoReduction of metal streak artifacts in x-ray computed tomog-raphy using a transmission maximum a posteriori algorithmrdquoIEEE Transactions on Nuclear Science vol 47 no 3 pp 977ndash9812000

[28] S Karimi P Cosman CWald and H Martz ldquoSegmentation ofartifacts and anatomy in CT metal artifact reductionrdquo MedicalPhysics vol 39 no 10 pp 5857ndash5868 2012

[29] J Y Huang J R Kerns J L Nute et al ldquoAn evaluation of threecommercially availablemetal artifact reductionmethods for CTimagingrdquo Physics in Medicine and Biology vol 60 no 3 pp1047ndash1067 2015

[30] H S Park J K Choi K-R Park et al ldquoMetal artifact reductionin CT by identifying missing data hidden in metalsrdquo Journal ofX-Ray Science and Technology vol 21 no 3 pp 357ndash372 2013

[31] C L Byrne ldquoIterative Algorithms in Inverse Problemsrdquo Lecturenote 2006


[32] C Byrne ldquoBlock-iterative algorithmsrdquo International Transac-tions in Operational Research vol 16 no 4 pp 427ndash463 2009

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

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


where 119902 isin 119877119872+ and 119903 isin 119877119868minus119872+ correspond to the inaccurate(119868 gt 119872 gt 0) and the accurate projection data respectively Itis also assumed that 119860 = (119861119862) (2)

where 119861 isin 119877119872times119869+ and 119862 isin 119877(119868minus119872)times119869+ with rows correspondingto elements of 119901

To reconstruct an image with pixel values and to estimatecertain parts of projections instead of inaccurate part 119902unknown variables 119909 isin 119877119869+ and 119910 isin 119877119872+ are treated respec-tively It is assumed that pixel values correspond to forexample in the case of X-ray CT actual values of attenuationcoefficients including the minimum value zero correspond-ing to attenuation by air It is also assumed that some elementsof vector 119909 satisfying 119903 = 119862119909 are undetermined becauserank119862 lt 119869 due to missing measurements Accordingly theundetermined elements should be permitted to have arbitraryvalues The problem considered in this paper is thereforedefined as follows

Definition 1 The tomographic inverse problem with an inac-curate projection is consistent if the set

E = 119890 isin 119877119869+ 119903 = 119862119890 (3)

is not empty

It follows that the problem is to obtain unknown variables119909 and119910 byminimizing an appropriate cost function that takeszero at 119909 = 119890 and 119910 = 119861119890 for some 119890 isin E respectively Tosolve this problem the following cost function of 119909 isin 119877119869+ and119910 isin 119877119872+ with 119890 isin E is considered119881 (119909 119910) = 119869sum

119895=1

120582minus1119895 KL (119890119895 119909119895) + 120572minus1KL (119861119890 119910)= 119869sum119895=1

120582minus1119895 (119890119895 log 119890119895119909119895 + 119909119895 minus 119890119895)+ 120572minus1 119872sum119894=1

(119861119890)119894 log (119861119890)119894119910119894 + 119910119894 minus (119861119890)119894(4)

where 120572 gt 0 is a weight parameter (V)ℓ denotes theℓth element of vector V KL indicates the generalizedKullbackndashLeibler divergence [8 9] and 120582119895 is defined as

120582119895 = ( 119868sum119894=1

119860 119894119895)minus1 (5)

for 119895 = 1 2 119869 Note that the function in Eq (4) isnonnegative for arbitrary 119909 isin 119877119869+ and 119910 isin 119877119872+ and zero ifand only if 119909 = 119890 and 119910 = 119861119890

This paper is organized as follows In Section 2 a novelmethod of simultaneously obtaining a reconstructed imageand an estimated projection is proposed It solves an initial-value problem of differential equations consisting of state

variables including parts of projections as well as imagepixel values A system of differential equations based on anextension of dynamical methods [10ndash20] is constructed Itis an approach that optimizes a cost function of unknownvariables consisting of an image and a projection Threesystems described by nonlinear differential equations withdifferent vector fields among each other are presented InSection 3 for each system the stability of a set of equilibriacorresponding to an optimized solution is proved by using theLyapunov stability theorem [21] In Section 4 the intentionof the proposed dynamical system is described in detailand how it works is shown by using a simple toy modelAdditionally to validate the theoretical results given by theproposed method in comparison with results obtained bylinear interpolation and a reduced system reduction of metalartifacts is numerically simulated by using a large-sizedmodel

2 Dynamical Systems

Three kinds of continuous-time dynamical systems with statevariables (119909⊤ 119910⊤)⊤ isin 119877119870++ are proposed where 119870 fl 119869 + 119872and 119877++ denotes the set of positive real numbers In thesesystems 119881(119909(119905) 119910(119905)) is expected to decrease in time 119905 alongtheir solutions The three dynamical systems are described byautonomous nonlinear differential equationsThe first systemis defined as119889119909119889119905 = 119883Λ119861⊤ (119910 minus 119861119909) + 119883Λ119862⊤ (119903 minus 119862119909) 119889119910119889119905 = 120572119884 (119861119909 minus 119910) (6)

with initial states 119909(0) = 1199090 isin 119877119869++ and 119910(0) = 1199100 isin119877119872++ where 119883 119884 and Λ indicate the diagonal matrices ofvectors 119909 119910 and 120582 consisting of 120582119895 in Eq (5) respectivelyNote that variables 119909(119905) and 119910(119905) are mutually coupled in thevector field The derivative 119889119909119889119905 for the dynamics of imagereconstruction in the case 119872 = 0 has the same form as thecontinuous-time image reconstruction system presented inRef [17]

The second and third systems which are respectivelyinspired by continuous analogs [19 22] of the multiplica-tive algebraic reconstruction technique [23 24] and themaximum-likelihood expectation-maximization [3 25] algo-rithm are given by119889119909119889119905 = 119883Λ119861⊤ (Log (119910) minus Log (119861119909))+ 119883Λ119862⊤ (Log (119903) minus Log (119862119909)) 119889119910119889119905 = 120572119884 (Log (119861119909) minus Log (119910))

(7)

and 119889119909119889119905 = 119883 Log (Λ119861⊤Exp (Log (119910) minus Log (119861119909))








119895=1

120582minus1119895 int119909119895119890119895

119906 minus 119890119895119906 119889119906+ 120572minus1 119872sum119894=1

int119910119894(119861119890)119894

V minus (119861119890)119894V

119889V (9)


119895=1



(10)



119895=1




(11)



119895=1

119890119895 (119861⊤)119895



119890119895 (119862⊤)119895 (Log (119903)minus Log (119862119909)) + 119869sum

119895=1


119895=1


119894=1

119910119894 (log (119910119894) minus log ((119861119909)119894))+ 119872sum119894=1




119910119894 (log (119910119894) minus log ((119861119909)119894)) + 119872sum119894=1


119894=1


119894=1


119894=1


119894=1






119901lowast = (119902lowast119903 ) (13)







12

0

(a)

06

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)



119880 (119909) = 119869sum119895=1119895notinM

10038161003816100381610038161003816119909119895 minus 119890lowast119895 10038161003816100381610038161003816 (17)




(1199011 1199012 1199013 1199014 1199015 1199016 11990171199018 1199019 11990110 11990111 11990112 11990113 1199011411990115 11990116 11990117 11990118 11990119 11990120 11990121)= (1199031 1199032 1199021 1199022 1199033 1199034 11990351199036 1199023 1199024 1199025 1199037 1199038 119903911990310 11990311 1199026 1199027 1199028 11990312 11990313) š(119878(119902)1119878 (119902)2119878 (119902)3)= 119878 (119902) isin 1198773times7+

(19)








(11990321199036119903711990312)=(0548601413002573) (20)

and

119862lowast fl(11986221198626119862711986212)=(03429 0 0 03429 0 0 03429 0 001066 00453 00075 0 0 0 0 0 00 0 0 0 0 0 0 0 0004300573 0 0 01887 0 0 02940 00131 0 ) (21)


E = 119890lowast + 119873119904 119904 isin 1198775 (22)


119873 =((((((((((((((


))))))))))))))

(23)


(((

0004401407minus0016300014minus01194)))

and (((

05000050000076105737minus05958)))

(24)




0 0100800600400202

05

1

2

U(x

())

U(z(]))

0 cup


0

02

04

06

08

1

t

t

xj(t)

yi(t) (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(a)

0

02

04

06

08

1

t

t

xj(t)

qlowasti (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(b)

0

02

04

06

08

1

n

n

zj(n) elowastj

100 101 102 103 104 105

(Bz(n))i

100

100

101 102 103 104 105

10minus1

(c)




12

0

(a)

08

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)


0 35003000250020001500100050035tn

100

101

102

103

104

rminusCx(t)

1

rminusCz(n

)1



010010002430

440

450

460

470

480

U(x

())

U(z(]))




010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)







5 Conclusion



Data Availability




Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018















119895=1

120582minus1119895 int119909119895119890119895

119906 minus 119890119895119906 119889119906+ 120572minus1 119872sum119894=1

int119910119894(119861119890)119894

V minus (119861119890)119894V

119889V (9)


119895=1



(10)



119895=1




(11)



119895=1

119890119895 (119861⊤)119895



119890119895 (119862⊤)119895 (Log (119903)minus Log (119862119909)) + 119869sum

119895=1


119895=1


119894=1

119910119894 (log (119910119894) minus log ((119861119909)119894))+ 119872sum119894=1




119910119894 (log (119910119894) minus log ((119861119909)119894)) + 119872sum119894=1


119894=1


119894=1


119894=1


119894=1






119901lowast = (119902lowast119903 ) (13)







12

0

(a)

06

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)



119880 (119909) = 119869sum119895=1119895notinM

10038161003816100381610038161003816119909119895 minus 119890lowast119895 10038161003816100381610038161003816 (17)




(1199011 1199012 1199013 1199014 1199015 1199016 11990171199018 1199019 11990110 11990111 11990112 11990113 1199011411990115 11990116 11990117 11990118 11990119 11990120 11990121)= (1199031 1199032 1199021 1199022 1199033 1199034 11990351199036 1199023 1199024 1199025 1199037 1199038 119903911990310 11990311 1199026 1199027 1199028 11990312 11990313) š(119878(119902)1119878 (119902)2119878 (119902)3)= 119878 (119902) isin 1198773times7+

(19)








(11990321199036119903711990312)=(0548601413002573) (20)

and

119862lowast fl(11986221198626119862711986212)=(03429 0 0 03429 0 0 03429 0 001066 00453 00075 0 0 0 0 0 00 0 0 0 0 0 0 0 0004300573 0 0 01887 0 0 02940 00131 0 ) (21)


E = 119890lowast + 119873119904 119904 isin 1198775 (22)


119873 =((((((((((((((


))))))))))))))

(23)


(((

0004401407minus0016300014minus01194)))

and (((

05000050000076105737minus05958)))

(24)




0 0100800600400202

05

1

2

U(x

())

U(z(]))

0 cup


0

02

04

06

08

1

t

t

xj(t)

yi(t) (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(a)

0

02

04

06

08

1

t

t

xj(t)

qlowasti (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(b)

0

02

04

06

08

1

n

n

zj(n) elowastj

100 101 102 103 104 105

(Bz(n))i

100

100

101 102 103 104 105

10minus1

(c)




12

0

(a)

08

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)


0 35003000250020001500100050035tn

100

101

102

103

104

rminusCx(t)

1

rminusCz(n

)1



010010002430

440

450

460

470

480

U(x

())

U(z(]))




010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)







5 Conclusion



Data Availability




Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018










119890119895 (119862⊤)119895 (Log (119903)minus Log (119862119909)) + 119869sum

119895=1


119895=1


119894=1

119910119894 (log (119910119894) minus log ((119861119909)119894))+ 119872sum119894=1




119910119894 (log (119910119894) minus log ((119861119909)119894)) + 119872sum119894=1


119894=1


119894=1


119894=1


119894=1






119901lowast = (119902lowast119903 ) (13)







12

0

(a)

06

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)



119880 (119909) = 119869sum119895=1119895notinM

10038161003816100381610038161003816119909119895 minus 119890lowast119895 10038161003816100381610038161003816 (17)




(1199011 1199012 1199013 1199014 1199015 1199016 11990171199018 1199019 11990110 11990111 11990112 11990113 1199011411990115 11990116 11990117 11990118 11990119 11990120 11990121)= (1199031 1199032 1199021 1199022 1199033 1199034 11990351199036 1199023 1199024 1199025 1199037 1199038 119903911990310 11990311 1199026 1199027 1199028 11990312 11990313) š(119878(119902)1119878 (119902)2119878 (119902)3)= 119878 (119902) isin 1198773times7+

(19)








(11990321199036119903711990312)=(0548601413002573) (20)

and

119862lowast fl(11986221198626119862711986212)=(03429 0 0 03429 0 0 03429 0 001066 00453 00075 0 0 0 0 0 00 0 0 0 0 0 0 0 0004300573 0 0 01887 0 0 02940 00131 0 ) (21)


E = 119890lowast + 119873119904 119904 isin 1198775 (22)


119873 =((((((((((((((


))))))))))))))

(23)


(((

0004401407minus0016300014minus01194)))

and (((

05000050000076105737minus05958)))

(24)




0 0100800600400202

05

1

2

U(x

())

U(z(]))

0 cup


0

02

04

06

08

1

t

t

xj(t)

yi(t) (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(a)

0

02

04

06

08

1

t

t

xj(t)

qlowasti (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(b)

0

02

04

06

08

1

n

n

zj(n) elowastj

100 101 102 103 104 105

(Bz(n))i

100

100

101 102 103 104 105

10minus1

(c)




12

0

(a)

08

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)


0 35003000250020001500100050035tn

100

101

102

103

104

rminusCx(t)

1

rminusCz(n

)1



010010002430

440

450

460

470

480

U(x

())

U(z(]))




010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)







5 Conclusion



Data Availability




Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









12

0

(a)

06

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)



119880 (119909) = 119869sum119895=1119895notinM

10038161003816100381610038161003816119909119895 minus 119890lowast119895 10038161003816100381610038161003816 (17)




(1199011 1199012 1199013 1199014 1199015 1199016 11990171199018 1199019 11990110 11990111 11990112 11990113 1199011411990115 11990116 11990117 11990118 11990119 11990120 11990121)= (1199031 1199032 1199021 1199022 1199033 1199034 11990351199036 1199023 1199024 1199025 1199037 1199038 119903911990310 11990311 1199026 1199027 1199028 11990312 11990313) š(119878(119902)1119878 (119902)2119878 (119902)3)= 119878 (119902) isin 1198773times7+

(19)








(11990321199036119903711990312)=(0548601413002573) (20)

and

119862lowast fl(11986221198626119862711986212)=(03429 0 0 03429 0 0 03429 0 001066 00453 00075 0 0 0 0 0 00 0 0 0 0 0 0 0 0004300573 0 0 01887 0 0 02940 00131 0 ) (21)


E = 119890lowast + 119873119904 119904 isin 1198775 (22)


119873 =((((((((((((((


))))))))))))))

(23)


(((

0004401407minus0016300014minus01194)))

and (((

05000050000076105737minus05958)))

(24)




0 0100800600400202

05

1

2

U(x

())

U(z(]))

0 cup


0

02

04

06

08

1

t

t

xj(t)

yi(t) (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(a)

0

02

04

06

08

1

t

t

xj(t)

qlowasti (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(b)

0

02

04

06

08

1

n

n

zj(n) elowastj

100 101 102 103 104 105

(Bz(n))i

100

100

101 102 103 104 105

10minus1

(c)




12

0

(a)

08

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)


0 35003000250020001500100050035tn

100

101

102

103

104

rminusCx(t)

1

rminusCz(n

)1



010010002430

440

450

460

470

480

U(x

())

U(z(]))




010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)







5 Conclusion



Data Availability




Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018













(11990321199036119903711990312)=(0548601413002573) (20)

and

119862lowast fl(11986221198626119862711986212)=(03429 0 0 03429 0 0 03429 0 001066 00453 00075 0 0 0 0 0 00 0 0 0 0 0 0 0 0004300573 0 0 01887 0 0 02940 00131 0 ) (21)


E = 119890lowast + 119873119904 119904 isin 1198775 (22)


119873 =((((((((((((((


))))))))))))))

(23)


(((

0004401407minus0016300014minus01194)))

and (((

05000050000076105737minus05958)))

(24)




0 0100800600400202

05

1

2

U(x

())

U(z(]))

0 cup


0

02

04

06

08

1

t

t

xj(t)

yi(t) (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(a)

0

02

04

06

08

1

t

t

xj(t)

qlowasti (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(b)

0

02

04

06

08

1

n

n

zj(n) elowastj

100 101 102 103 104 105

(Bz(n))i

100

100

101 102 103 104 105

10minus1

(c)




12

0

(a)

08

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)


0 35003000250020001500100050035tn

100

101

102

103

104

rminusCx(t)

1

rminusCz(n

)1



010010002430

440

450

460

470

480

U(x

())

U(z(]))




010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)







5 Conclusion



Data Availability




Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









0 0100800600400202

05

1

2

U(x

())

U(z(]))

0 cup


0

02

04

06

08

1

t

t

xj(t)

yi(t) (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(a)

0

02

04

06

08

1

t

t

xj(t)

qlowasti (Bx(t))i

elowastj

100 101 102 103 104 105

100 101 102 103 104 105

100

10minus1

(b)

0

02

04

06

08

1

n

n

zj(n) elowastj

100 101 102 103 104 105

(Bz(n))i

100

100

101 102 103 104 105

10minus1

(c)




12

0

(a)

08

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)


0 35003000250020001500100050035tn

100

101

102

103

104

rminusCx(t)

1

rminusCz(n

)1



010010002430

440

450

460

470

480

U(x

())

U(z(]))




010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)







5 Conclusion



Data Availability




Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









12

0

(a)

08

0

(b)


12

0

(a)

12

0

(b)

12

0

(c)


0 35003000250020001500100050035tn

100

101

102

103

104

rminusCx(t)

1

rminusCz(n

)1



010010002430

440

450

460

470

480

U(x

())

U(z(]))




010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)







5 Conclusion



Data Availability




Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









010010002380

400

420

440

460

480

U(z(]))

(a)

010010002380

400

420

440

460

480

U(z(]))

(b)







5 Conclusion



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 inverse problem with estimating missing...

Documents