image segmentation based on modified fractional allen–cahn...

Research ArticleImage Segmentation Based on Modified FractionalAllenndashCahn Equation

Dongsun Lee1 and Seunggyu Lee 2

1Department of Mathematical Sciences Ulsan National Institute of Science and Technology Ulsan 689-798 Republic of Korea2National Institute for Mathematical Sciences Daejeon 34047 Republic of Korea

Correspondence should be addressed to Seunggyu Lee sglee89nimsrekr

Received 31 October 2018 Accepted 14 January 2019 Published 30 January 2019

Academic Editor John D Clayton

Copyright copy 2019 Dongsun Lee and Seunggyu Lee This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We present the image segmentation model using the modified AllenndashCahn equation with a fractional Laplacian The motion ofthe interface for the classical AllenndashCahn equation is known as the mean curvature flows whereas its dynamics is changed to themacroscopic limit of Levy process by replacing the Laplacian operator with the fractional one To numerical implementation weprove the unconditionally unique solvability and energy stability of the numerical scheme for the proposed model The effect ofa fractional Laplacian operator in our own and in the AllenndashCahn equation is checked by numerical simulations Finally we givesome image segmentation results with different fractional order including the standard Laplacian operator

1 Introduction

Image segmentation is a process of image partitioning intononintersection parts with similar properties such as graylevel color texture brightness and contrast [1] The medicalimage segmentation is important to study anatomical struc-tures to identify region of interest such as tumor lesion andother abnormalities to measure tissue volume of tumor orarea of lesion and to help in treatment planning [2]

One of the most widely used methods for image seg-mentation is the MumfordndashShah model [3] and it has beenextensively studied and extended in many works [4ndash6]

This paper is organized as follows In Section 2 wedescribe themathematical model for the image segmentationusing a fractional Laplacian operator In Section 3 simulationresults are shown for effect of a fractional operator and imagesegmentations Finally the conclusion is drawn in Section 4

2 Mathematical Model

21 Fractional AllenndashCahn Equation The AllenndashCahn equa-tion which was first introduced to describe coarsening inbinary alloys [7] is a gradient flow under 1198712-inner product

space with the following GinzburgndashLandau free energy func-tional

E119860119862 (120601) = int

Ω(119865 (120601)1205982 + 12 1003816100381610038161003816nabla12060110038161003816100381610038162)119889x (1)

and it has the following form

120597120601 (x 119905)120597119905 = minusgradE119860119862 (120601) = minus1198651015840 (120601)1205982 + Δ120601 (2)

Here 120601(x 119905) isin [minus1 1] is the concentration defined in abounded domain Ω isin R2 120598 is the coefficient related to aninterfacial energy and 119865(120601) = 025(1minus1206012)2 is the double-wellpotential energy function

In recent years the fractional AllenndashCahn equation hasbeen researched in some literature to study the competingstable phases having an identical Lyapunov functional density[8ndash10] 120597120601120597119905 = minus1198651015840 (120601)1205982 + Δ119904x120601 (3)

where Δ119904x is the fractional Laplacian obtained as the macro-scopic limit of Levy process [11] with fractional order 0 lt

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3980181 6 pageshttpsdoiorg10115520193980181

2 Mathematical Problems in Engineering

119904 le 1 Here the macroscopic limit in space is computed byconsidering that the microscopic spatial scale is very smallIndeed when the random walk involves correlations non-Gaussian or non-Markovian memory effects the classicaldiffusion equation fails to describe the macroscopic limitHowever a generalization of the Brownian random walk(Levy process) model allows us to have the incorporationof nondiffusive effects It eventually leads to the fractionalLaplacian instead of the classical Laplacian in the macro-scopic limit Note that similar models have been studied inphysical literature in a very different context such as a barriercrossing of a particle driven by white symmetric Levy noise[12ndash16]

22 Modification to Image Segmentation Let 1198910(x) Ω 997888rarr[0 1] be a grayscale given image Then the MumfordndashShahmodel which is one of the most widely used models forimage segmentation [3] minimizes the following functionalto perform image segmentation

E119872119878 (119906 119862) = ]1Length (119862) + ]2 int

Ω

10038161003816100381610038161198910 minus 11990610038161003816100381610038162 119889x+ ]3 int

Ωminus119862|nabla119906|2 119889x (4)

where 119906 is the piecewise smooth function approximating 1198910119862 is the segmenting curve representing the set of edges in thegiven image 119906 and ]1 ]2 ]3 are the positive constants Let usmodify the energy functional (4) as a phase-field formulationFirst it should be noted that 119862 can be considered as the zero-contour of 120601 Then the following equation holds

Length (119862) = int1198621119889x asymp int

Ω

119865 (120601)1205982 119889x (5)

Next if we replace 119906 with (1 + 120601)2 (4) can be written asfollows

E119872119878 (120601) = ]1 int

Ω

119865 (120601)1205982 119889x + ]2 intΩ

100381610038161003816100381610038161003816100381610038161198910 minus 1 + 1206012 100381610038161003816100381610038161003816100381610038162 119889x

+ ]3 intΩminus119862

10038161003816100381610038161003816100381610038161003816nabla (1 + 1206012 )100381610038161003816100381610038161003816100381610038162 119889x (6)

Since nabla120601 = 0 at x isin 119862 intΩ|nabla120601|2119889x = int

Ωminus119862|nabla120601|2119889x

Therefore the fractional analogue phase-field approach ofthe MumfordndashShah model is considered as minimizing thefollowing energy functional

E (120601) = intΩ(119865 (120601)1205982 + 12 10038161003816100381610038161003816(minusΔ)1199042 120601100381610038161003816100381610038162)119889x

+ 120583intΩ

100381610038161003816100381610038161003816100381610038161198910 minus 1 + 1206012 100381610038161003816100381610038161003816100381610038162 119889x (7)

and the governing equation can be written as follows

120597120601120597119905 = minus1206013 minus 1206011205982 + Δ119904x120601 + 120583(1198910 minus 1 + 1206012 ) (8)

23 Numerical Solution We consider the Fourier spectralmethod in space and the linear convex splitting schemewhich is known as stable and uniquely solvable one [17] intime First the temporal discretization of (8) is written asfollows

120601119899+1 minus 120601119899Δ119905 = minus(120601119899)3 minus 3120601119899 + 2120601119899+11205982 + Δ119904x120601119899+1+ 120583(1198910 minus 1 + 120601119899+12 ) for 119899 = 1 119899119905 (9)

Remark that (9) has the bounded solution 120601119899infin le 1 for any119899 = 1 119899119905 if 1198910infin le 1 and 1206010infin le 1 where 120601infin =max119909120601 is the 119897infin-norm It comes from the fact that minusΔ119904x hasnonnegative eigenvalues

Theorem 1 The numerical scheme (9) is uniquely solvable forany time step Δ119905 gt 0Proof Weconsider following functional defined on a 1198972-innerproduct space

119866 (120601) = 12 1003817100381710038171003817120601100381710038171003817100381722 + Δ119905E119888 (120601) minus (119878119899 120601)2 (10)

where (120601 120595)2 = intΩ120601120595119889x is the 1198972-inner product 12060122 =(120601 120601)2 is the 1198972-normE119888(120601) = (12060121205982 1)2 + 05(minusΔ)1199042x 12060122 +1205831198910 minus 05(1 + 120601)22 119878119899 = 120601119899 minusΔ119905((120601119899)3 minus 3120601119899)1205982 Note that it

can be solved if and only if 119866 has the unique minimizer 120601119899+1For any 120595 and scalar 120572

119889119889120572E119888 (120601 + 120572120595)10038161003816100381610038161003816100381610038161003816120572=0= (21206011205982 minus Δ119904x120601 minus 120583(1198910 minus 1 + 1206012 ) 120595)

2

(11)

11988921198891205722E119888 (120601 + 120572120595)100381610038161003816100381610038161003816100381610038161003816120572=0 = 21205982 + 10038171003817100381710038171003817(minusΔ)1199042x 1205951003817100381710038171003817100381722 + 1205832 ge 0 (12)

Therefore E119888(120601) is convex which implies that 119866(120601) is alsoconvex Note that the unique minimizer 120601119899+1 makes the firstvariation of 119866(120601) zeros ie

(120601119899+1 minus 120601119899Δ119905 + (120601119899)3 minus 3120601119899 + 2120601119899+11205982 minus Δ119904x120601119899+1minus 120583(1198910 minus 1 + 120601119899+12 ) 120595)

2

= 0 (13)

and it holds if and only if (9) holds

Theorem 2 The numerical scheme (9) is unconditionallyenergy stable ie E(120601119899+1) le E(120601119899) for any time step Δ119905 gt 0Proof Note that we already prove thatE119888(120601) is convex in theproof ofTheorem 1 LetE119890(120601) = E119888(120601)minusE(120601)Then it is clear

Mathematical Problems in Engineering 3

that E119890(120601) is concave since 120601 isin [minus1 1] and the followingshold for any 120601 and 120595

E119888 (120601) minusE119888 (120595)le (21206011205982 minus Δ119904x120601 minus 120583(1198910 minus 1 + 1206012 ) 120601 minus 120595)

2

(14)

E119890 (120601) minusE119890 (120595) ge (minus1205953 minus 31205951205982 120601 minus 120595)2

(15)

Then

E (120601) minusE (120595) = (E119888 (120601) minusE119888 (120595)) minus (E119890 (120601)minusE119890 (120595)) le (21206011205982 minus Δ119904x120601 minus 120583(1198910 minus 1 + 1206012 )+ 1205953 minus 31205951205982 120601 minus 120595)

2

(16)

We respectively replace 120601 and 120595 with 120601119899+1 and 120601119899 Then (9)says that

E (120601119899+1) minusE (120601119899) le ((120601119899)3 minus 3120601119899 + 2120601119899+11205982minus Δ119904x120601119899+1 minus 120583(1198910 minus 1 + 120601119899+12 ) 120601119899+1 minus 120601119899)

2

= (minus120601119899+1 minus 120601Δ119905 120601119899+1 minus 120601)

2

= minus 1Δ119905 10038171003817100381710038171003817120601119899+1 minus 1206011003817100381710038171003817100381722le 0

(17)

Before applying the Fourier spectral method we presentthe Fourier definition of the fractional Laplacian operator

Definition 3 The Fourier definition of Δ119904x is [19]F (Δ119904x120601) (120585) = minus 10038161003816100381610038161205851003816100381610038161003816119904F120601 (120585) (18)

where F(sdot) is a Fourier transformation Note that theFourier transformation of the standard Laplacian operator isF(Δ120601)(120585) = minus|120585|2F120601(120585)

By the definition (9) can be written as followsusing the discrete Fourier transformation Fℎ120601119899 asympsum1198731199092minus1119901=minus1198731199092

sum1198731199102minus1119902=minus1198731199102

1206011198991199011199021198902120587(119901119909119871119909+119902119910119871119910)(119873119909119873119910)120601119899+1119901119902 minus 120601119899119901119902Δ119905 = minus(120601119899119901119902)3 minus 3120601119899119901119902 + 2120601119899+11199011199021205982 minus 1003816100381610038161003816100381612058511990111990210038161003816100381610038161003816119904 120601119899+1119901119902

+ 120583(1198910119901119902 minus 1 + 120601119899+11199011199022 ) (19)

where 120601 is the discrete Fourier coefficient 119871119909 and 119871119910 arerespectively the length of the domain in 119909- and 119910-axis 119873119909and119873119910 are respectively the number of grid points in 119909- and119910minusaxis Δ119905 is the temporal step size and 1205852119901119902 = (2120587119901119871119909)2 +(2120587119902119871119909)2 Therefore we can get 120601119899+1 as follows

120601119899+1 asymp F1ℎ120601119901119902

= Fminus1ℎ (120601119901119902 + Δ119905 ((3120601119901119902 minus (120601119901119902)3) 1205982 + 120583 (1198910119901119902 minus 12))1 + Δ119905 (21205982 + 1205852119901119902 + 1205832) ) (20)

3 Numerical Experiments

31 Effect of the Fractional Laplacian Operator To observethe only difference between the fractional Laplacian Δ119904x andclassical Laplacian Δ x in image segmentation let us select120598 ≫ 1 large enough such that

120601119899+1 minus 120601119899Δ119905 asymp 120598Δ119904x120601119899+1 + 120583(1198910 minus 1 + 120601119899+12 ) (21)

where 120598 is the rescaled parameter Since scheme (9) hasbounded and unconditionally stable solutions it is easy to seethat (21) has the same properties as well For the numericalillustration we consider original and noise 512 times 512 images1198911199051199031199061198900 1198911198991199001198941199041198900 We take the image 1198911198991199001198941199041198900 as the initial data Thefractional orders and parameters for both operators Δ119904x andΔ x are given by 119904 = 05 120598 = 147 and 119904 = 1 120598 = 001respectively Note that there are no criteria of choosing thebest parameters for image processing while these parametersheuristically give the best resultThe other parameters120583 = 20Δ119905 = 00001 119899119905 = 40 are used For our convenience wedenote the final solution from the operator of fractional order119904 by 120601119904

With initial data 1198911198991199001198941199041198900 and its error 1198911199051199031199061198900 minus 1198911198991199001198941199041198900 119891119903119900 asymp3366 we perform numerical tests Comparisons are madefor the numerical results which we have 1198911199051199031199061198900 minus 12060105119891119903119900 asymp1198911199051199031199061198900 minus 1206011119891119903119900 asymp 2983 If large noise is imposed upon theoriginal image 1198911199051199031199061198900 the value of 1198911199051199031199061198900 minus 1198911198991199001198941199041198900 119891119903119900 is alsolarge If the other way around there is no noise then1198911198991199001198941199041198900 isequal to1198911199051199031199061198900 ie 1198911199051199031199061198900 minus1198911198991199001198941199041198900 119891119903119900 is zero In our setting theerror 3366 implies the difference between initial and noisedimages Note that ldquoBarbarardquo image is comprised of 512 times 512pixels and each pixel belongs to [0 1] We can observe thatcartoon and texture are kept in Figure 1(c) However there isnoticeable blur at texture region in Figure 1(d)

32 Fractional AllenndashCahn Equation It is known that themotion of interfaces for the classical AllenndashCahn equationfollows the mean curvature flow [20] In this section weperformnumerical simulations to compare the fractional andclassical AllenndashCahn equations Figure 2 shows the evolutionof zero-contours of 120601 solving the fractional AllenndashCahnequation with different 119904 values with circular and squareinitial conditions respectively Here we used the followingparameters 119873119909 = 128 Ω = (0 1)2 Δ119905 = 01 the final time


(a) (b) (c) (d)

Figure 1 (a) Original image 1198911199051199031199061198900 (b) noise image 1198911198991199001198941199041198900 (c) close-up image at 12060105 with order 119904 = 05 and (d) close-up image at 1206011 with order119904 = 10

(a) 119904 = 1 (b) 119904 = 05 (c) 119904 = 025 (d) 119904 = 01

Figure 2 Evolution of zero-contours of 120601 solving fractional AllenndashCahn equation with different 119904 values and circular (top) and square(bottom) initial conditions

119879 = 130 120598 = 00075 and the dotted lines represent the initialconditions

In Figure 2(a) the interfaces evolve as motions by meancurvature since it is the classical AllenndashCahn equation casewhereas we can observe that the dynamics is different fromthe classical one especially at the tip of the interfaces inFigures 2(b) and 2(c) When 119904 is relatively small case 119904 =01 the results give same shapes as the initial conditions(see Figure 2(d)) From the results we can assume that itis better to capture a sharpen interface using the fractionalAllenndashCahn equation with the proper 119904 value and then usingthe classical AllenndashCahn equation

33 Basic Figures In this section we consider the segmen-tation with basic figures First the segmentation resultsusing classical and fractional AllenndashCahn equation with thefollowing parameters are shown in Figure 3 119873119909 = 256 Ω =(0 1)2 Δ119905 = 10 120598 = 001 and 120583 = 10000 and star-shaped

initial condition At the tip of the star we can observe that thefractional AllenndashCahn equation gives better performance

Next we consider other segmenting simulations for twosimple figures circle and square with salt and pepper noiseusing the following parameters 119873119909 = 256 Ω = (0 1)2Δ119905 = 10 120598 = 00038 120583 = 10000 As shown in Figure 4 thecase using the fractional AllenndashCahn equation (Figure 4(c))gives better result than the case using classical AllenndashCahnequation (Figure 4(b)) especially at the corner of the squareHowever the segmentation is interrupted by the noise when119904 becomes too small (see Figure 4(d))

34 Medical Images Here we perform the numerical sim-ulations applying the proposed segmentation algorithm tomedical images Figure 5 shows the segmentation results forbrain CT image with and without injury [18] when (a) 119904 = 1and (b) 119904 = 005 using the following parameters Δ119905 = 10120598 = 01 and 120583 = 10000 As opposed to the case with classical


(a) Initial (b) s = 1 (c) s = 015

Figure 3 (a) Initial condition and segmentation results with (b) classical and (c) fractional AllenndashCahn equations

(a) Initial (b) s = 1 (c) s = 025 (d) s = 015

Figure 4 (a) Initial condition and (b)ndash(d) segmentation results with different 119904 values

(a) s = 1 (b) s = 005

Figure 5 Segmentation results for brain CT images with (left) and without (right) injury [18] when (a) 119904 = 1 and (b) 119904 = 005AllenndashCahn equation the injured part can be segmented inthe case of Figure 5(b) which uses the fractional Laplacianoperators

4 Conclusions

We proposed the image segmentation model using the mod-ified AllenndashCahn equation with a fractional Laplacian basedon the MumfordndashShah energy functional The fractionalorder obtained as the macroscopic limit of Levy processwas expected to change the dynamics of the AllenndashCahnequation Based on the convex splitting method we provedthe unconditionally unique solvability and energy stability ofthe numerical scheme The segmentation results show thatthe fractional Laplacian operator has a better performancewhen the original image has sharp tips and corners andthe abnormalities are close to each other Note that our

approach requires parameter tuning for image segmentationThebest segmentation-parameter combination including thefractional order ldquosrdquo depends on the original image Theminimizer of our proposed functional is different for eachof the initial images so that we have to select the bestparameter combination within all possible combinations ofthe parameters

Data Availability

All the data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper


Acknowledgments

The author D Lee was supported by the National ResearchFoundation of Korea (NRF) grant funded by the KoreanGov-ernment (MSIP) (NRF-2015R1C1A1A01054694) The authorS Lee was supported by the National Institute for Mathe-matical Sciences (NIMS) grant funded by the Korean Gov-ernment and the National Research Foundation of Korea(NRF) grant funded by the Korean Government (MSIP) (No2017R1C1B1001937)

References

[1] N R Pal and S K Pal ldquoA review on image segmentationtechniquesrdquo Pattern Recognition vol 26 no 9 pp 1277ndash12941993

[2] N Sharma A K Ray K K Shukla et al ldquoAutomated medicalimage segmentation techniquesrdquo Journal ofMedical Physics vol35 no 1 pp 3ndash14 2010

[3] DMumford and J Shah ldquoOptimal approximations by piecewisesmooth functions and associated variational problemsrdquo Com-munications on Pure and Applied Mathematics vol 42 no 5pp 577ndash685 1989

[4] L Bar T F Chan G Chung et al ldquoMumford and Shahmodel and its applications to image segmentation and imagerestorationrdquoHandbook ofMathematicalMethods in Imaging pp1ndash52 2014

[5] Y Duan H Chang W Huang J Zhou Z Lu and C Wu ldquoThe1198710 regularized Mumford-Shah model for bias correction andsegmentation of medical imagesrdquo IEEE Transactions on ImageProcessing vol 24 no 11 pp 3927ndash3938 2015

[6] S Sashida Y Okabe and H K Lee ldquoApplication of MonteCarlo simulation with block-spin transformation based onthe MumfordndashShah segmentation model to three-dimensionalbiomedical imagesrdquoComputer Vision and ImageUnderstandingvol 152 pp 176ndash189 2016

[7] S M Allen and JW Cahn ldquoAmicroscopic theory for antiphaseboundary motion and its application to antiphase domaincoarseningrdquo Acta Metallurgica et Materialia vol 27 no 6 pp1085ndash1095 1979

[8] O J J Algahtani ldquoComparing the AtanganandashBaleanu andCaputondashFabrizio derivative with fractional order Allen Cahnmodelrdquo Chaos Solitons amp Fractals vol 89 pp 552ndash559 2016

[9] Y Nec A A Nepomnyashchy and A A Golovin ldquoFront-type solutions of fractional Allen-Cahn equationrdquo Physica DNonlinear Phenomena vol 237 no 24 pp 3237ndash3251 2008

[10] S Lee and D Lee ldquoThe fractional Allen-Cahn equation withthe sextic GinzburgLandau potentialrdquoAppliedMathematics andComputation

[11] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000

[12] J Bao HWang Y Jia and Y Zhuo ldquoCancellation phenomenonof barrier escape driven by a non-Gaussian noiserdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 72no 5 Article ID 051105 2005

[13] AVChechkinV YGonchar J Klafter andRMetzler ldquoBarriercrossing of a Levy flightrdquo EPL (Europhysics Letters) vol 72 no3 pp 348ndash354 2005

[14] A V Chechkin O Y Sliusarenko R Metzler and J KlafterldquoBarrier crossing driven by Levy noise Universality and the role

of noise intensityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 75 no 4 Article ID 041101 2007

[15] P D Ditlevsen ldquoAnomalous jumping in a double-well poten-tialrdquo Physical Review E Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 60 no 1 pp 172ndash179 1999

[16] P Imkeller and I Pavlyukevich ldquoLevy flights transitions andmeta-stabilityrdquo Journal of Physics A Mathematical and Generalvol 39 no 15 pp L237ndashL246 2006

[17] D Jeong S Lee D Lee J Shin and J Kim ldquoComparison studyof numerical methods for solving the Allen-Cahn equationrdquoComputational Materials Science vol 111 pp 131ndash136 2016

[18] R Liu S Li C L Tan et al ldquoFrom hemorrhage to midline shiftA new method of tracing the deformed midline in traumaticbrain injury CT imagesrdquo in Proceedings of the 2009 IEEEInternational Conference on Image Processing ICIP 2009 pp2637ndash2640 November 2009

[19] M Kwasnicki ldquoTen equivalent definitions of the fractionalLaplace operatorrdquo Fractional Calculus and Applied Analysis vol20 no 1 pp 7ndash51 2017

[20] D S Lee and J S Kim ldquoMean curvature flow by the Allen-Cahnequationrdquo European Journal of AppliedMathematics vol 26 no4 pp 535ndash559 2015

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119904 le 1 Here the macroscopic limit in space is computed byconsidering that the microscopic spatial scale is very smallIndeed when the random walk involves correlations non-Gaussian or non-Markovian memory effects the classicaldiffusion equation fails to describe the macroscopic limitHowever a generalization of the Brownian random walk(Levy process) model allows us to have the incorporationof nondiffusive effects It eventually leads to the fractionalLaplacian instead of the classical Laplacian in the macro-scopic limit Note that similar models have been studied inphysical literature in a very different context such as a barriercrossing of a particle driven by white symmetric Levy noise[12ndash16]

22 Modification to Image Segmentation Let 1198910(x) Ω 997888rarr[0 1] be a grayscale given image Then the MumfordndashShahmodel which is one of the most widely used models forimage segmentation [3] minimizes the following functionalto perform image segmentation

E119872119878 (119906 119862) = ]1Length (119862) + ]2 int

Ω

10038161003816100381610038161198910 minus 11990610038161003816100381610038162 119889x+ ]3 int

Ωminus119862|nabla119906|2 119889x (4)

where 119906 is the piecewise smooth function approximating 1198910119862 is the segmenting curve representing the set of edges in thegiven image 119906 and ]1 ]2 ]3 are the positive constants Let usmodify the energy functional (4) as a phase-field formulationFirst it should be noted that 119862 can be considered as the zero-contour of 120601 Then the following equation holds

Length (119862) = int1198621119889x asymp int

Ω

119865 (120601)1205982 119889x (5)

Next if we replace 119906 with (1 + 120601)2 (4) can be written asfollows

E119872119878 (120601) = ]1 int

Ω

119865 (120601)1205982 119889x + ]2 intΩ

100381610038161003816100381610038161003816100381610038161198910 minus 1 + 1206012 100381610038161003816100381610038161003816100381610038162 119889x

+ ]3 intΩminus119862

10038161003816100381610038161003816100381610038161003816nabla (1 + 1206012 )100381610038161003816100381610038161003816100381610038162 119889x (6)

Since nabla120601 = 0 at x isin 119862 intΩ|nabla120601|2119889x = int

Ωminus119862|nabla120601|2119889x

Therefore the fractional analogue phase-field approach ofthe MumfordndashShah model is considered as minimizing thefollowing energy functional

E (120601) = intΩ(119865 (120601)1205982 + 12 10038161003816100381610038161003816(minusΔ)1199042 120601100381610038161003816100381610038162)119889x

+ 120583intΩ

100381610038161003816100381610038161003816100381610038161198910 minus 1 + 1206012 100381610038161003816100381610038161003816100381610038162 119889x (7)

and the governing equation can be written as follows

120597120601120597119905 = minus1206013 minus 1206011205982 + Δ119904x120601 + 120583(1198910 minus 1 + 1206012 ) (8)

23 Numerical Solution We consider the Fourier spectralmethod in space and the linear convex splitting schemewhich is known as stable and uniquely solvable one [17] intime First the temporal discretization of (8) is written asfollows

120601119899+1 minus 120601119899Δ119905 = minus(120601119899)3 minus 3120601119899 + 2120601119899+11205982 + Δ119904x120601119899+1+ 120583(1198910 minus 1 + 120601119899+12 ) for 119899 = 1 119899119905 (9)

Remark that (9) has the bounded solution 120601119899infin le 1 for any119899 = 1 119899119905 if 1198910infin le 1 and 1206010infin le 1 where 120601infin =max119909120601 is the 119897infin-norm It comes from the fact that minusΔ119904x hasnonnegative eigenvalues

Theorem 1 The numerical scheme (9) is uniquely solvable forany time step Δ119905 gt 0Proof Weconsider following functional defined on a 1198972-innerproduct space

119866 (120601) = 12 1003817100381710038171003817120601100381710038171003817100381722 + Δ119905E119888 (120601) minus (119878119899 120601)2 (10)

where (120601 120595)2 = intΩ120601120595119889x is the 1198972-inner product 12060122 =(120601 120601)2 is the 1198972-normE119888(120601) = (12060121205982 1)2 + 05(minusΔ)1199042x 12060122 +1205831198910 minus 05(1 + 120601)22 119878119899 = 120601119899 minusΔ119905((120601119899)3 minus 3120601119899)1205982 Note that it

can be solved if and only if 119866 has the unique minimizer 120601119899+1For any 120595 and scalar 120572

119889119889120572E119888 (120601 + 120572120595)10038161003816100381610038161003816100381610038161003816120572=0= (21206011205982 minus Δ119904x120601 minus 120583(1198910 minus 1 + 1206012 ) 120595)

2

(11)

11988921198891205722E119888 (120601 + 120572120595)100381610038161003816100381610038161003816100381610038161003816120572=0 = 21205982 + 10038171003817100381710038171003817(minusΔ)1199042x 1205951003817100381710038171003817100381722 + 1205832 ge 0 (12)

Therefore E119888(120601) is convex which implies that 119866(120601) is alsoconvex Note that the unique minimizer 120601119899+1 makes the firstvariation of 119866(120601) zeros ie

(120601119899+1 minus 120601119899Δ119905 + (120601119899)3 minus 3120601119899 + 2120601119899+11205982 minus Δ119904x120601119899+1minus 120583(1198910 minus 1 + 120601119899+12 ) 120595)

2

= 0 (13)

and it holds if and only if (9) holds

Theorem 2 The numerical scheme (9) is unconditionallyenergy stable ie E(120601119899+1) le E(120601119899) for any time step Δ119905 gt 0Proof Note that we already prove thatE119888(120601) is convex in theproof ofTheorem 1 LetE119890(120601) = E119888(120601)minusE(120601)Then it is clear




2

(14)


(15)

Then


2

(16)



2


2

= minus 1Δ119905 10038171003817100381710038171003817120601119899+1 minus 1206011003817100381710038171003817100381722le 0

(17)





sum1198731199102minus1119902=minus1198731199102


+ 120583(1198910119901119902 minus 1 + 120601119899+11199011199022 ) (19)


120601119899+1 asymp F1ℎ120601119901119902









(a) (b) (c) (d)


(a) 119904 = 1 (b) 119904 = 05 (c) 119904 = 025 (d) 119904 = 01









(a) Initial (b) s = 1 (c) s = 015


(a) Initial (b) s = 1 (c) s = 025 (d) s = 015


(a) s = 1 (b) s = 005


4 Conclusions



Data Availability





Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018











2

(14)


(15)

Then


2

(16)



2


2

= minus 1Δ119905 10038171003817100381710038171003817120601119899+1 minus 1206011003817100381710038171003817100381722le 0

(17)





sum1198731199102minus1119902=minus1198731199102


+ 120583(1198910119901119902 minus 1 + 120601119899+11199011199022 ) (19)


120601119899+1 asymp F1ℎ120601119901119902









(a) (b) (c) (d)


(a) 119904 = 1 (b) 119904 = 05 (c) 119904 = 025 (d) 119904 = 01









(a) Initial (b) s = 1 (c) s = 015


(a) Initial (b) s = 1 (c) s = 025 (d) s = 015


(a) s = 1 (b) s = 005


4 Conclusions



Data Availability





Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









(a) (b) (c) (d)


(a) 119904 = 1 (b) 119904 = 05 (c) 119904 = 025 (d) 119904 = 01









(a) Initial (b) s = 1 (c) s = 015


(a) Initial (b) s = 1 (c) s = 025 (d) s = 015


(a) s = 1 (b) s = 005


4 Conclusions



Data Availability





Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









(a) Initial (b) s = 1 (c) s = 015


(a) Initial (b) s = 1 (c) s = 025 (d) s = 015


(a) s = 1 (b) s = 005


4 Conclusions



Data Availability





Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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