MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2019/3980181 3980181 Research Article Image Segmentation Based on Modified Fractional Allen–Cahn Equation Lee Dongsun 1 http://orcid.org/0000-0003-3141-6968 Lee Seunggyu 2 Clayton John D. 1 Department of Mathematical Sciences Ulsan National Institute of Science and Technology Ulsan 689-798 Republic of Korea unist.ac.kr 2 National Institute for Mathematical Sciences Daejeon 34047 Republic of Korea nims.re.kr 2019 3012019 2019 31 10 2018 14 01 2019 3012019 2019 Copyright © 2019 Dongsun Lee and Seunggyu Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian. The motion of the interface for the classical Allen–Cahn equation is known as the mean curvature flows, whereas its dynamics is changed to the macroscopic limit of Lévy process by replacing the Laplacian operator with the fractional one. To numerical implementation, we prove the unconditionally unique solvability and energy stability of the numerical scheme for the proposed model. The effect of a fractional Laplacian operator in our own and in the Allen–Cahn equation is checked by numerical simulations. Finally, we give some image segmentation results with different fractional order, including the standard Laplacian operator.

Korean Government NRF-2015R1C1A1A01054694 2017R1C1B1001937
1. Introduction

Image segmentation is a process of image partitioning into nonintersection parts with similar properties such as gray level, color, texture, brightness, and contrast . The medical image segmentation is important to study anatomical structures, to identify region of interest such as tumor, lesion, and other abnormalities, to measure tissue volume of tumor or area of lesion, and to help in treatment planning .

One of the most widely used methods for image segmentation is the Mumford–Shah model  and it has been extensively studied and extended in many works .

This paper is organized as follows: In Section 2, we describe the mathematical model for the image segmentation using a fractional Laplacian operator. In Section 3, simulation results are shown for effect of a fractional operator and image segmentations. Finally, the conclusion is drawn in Section 4.

2. Mathematical Model 2.1. Fractional Allen–Cahn Equation

The Allen–Cahn equation, which was first introduced to describe coarsening in binary alloys , is a gradient flow under L2-inner product space with the following Ginzburg–Landau free energy functional:(1)EACϕ=ΩFϕϵ2+12ϕ2dx,and it has the following form:(2)ϕx,tt=-gradEACϕ=-Fϕϵ2+Δϕ. Here ϕ(x,t)[-1,1] is the concentration defined in a bounded domain ΩR2, ϵ is the coefficient related to an interfacial energy, and F(ϕ)=0.25(1-ϕ2)2 is the double-well potential energy function.

In recent years, the fractional Allen–Cahn equation has been researched in some literature to study the competing stable phases having an identical Lyapunov functional density :(3)ϕt=-Fϕϵ2+Δxsϕ, where Δxs is the fractional Laplacian, obtained as the macroscopic limit of Lévy process , with fractional order 0<s1. Here, the macroscopic limit in space is computed by considering that the microscopic spatial scale is very small. Indeed, when the random walk involves correlations, non-Gaussian or non-Markovian memory effects, the classical diffusion equation fails to describe the macroscopic limit. However a generalization of the Brownian random walk (Lévy process) model allows us to have the incorporation of nondiffusive effects. It eventually leads to the fractional Laplacian instead of the classical Laplacian in the macroscopic limit. Note that similar models have been studied in physical literature in a very different context such as a barrier crossing of a particle driven by white symmetric Lévy noise .

2.2. Modification to Image Segmentation

Let f0(x):Ω[0,1] be a grayscale given image. Then, the Mumford–Shah model, which is one of the most widely used models for image segmentation , minimizes the following functional to perform image segmentation:(4)EMSu,C=ν1LengthC+ν2Ωf0-u2dx+ν3Ω-Cu2dx,where u is the piecewise smooth function approximating f0, C is the segmenting curve representing the set of edges in the given image u, and ν1, ν2, ν3 are the positive constants. Let us modify the energy functional (4) as a phase-field formulation. First, it should be noted that C can be considered as the zero-contour of ϕ. Then, the following equation holds:(5)LengthC=C1dxΩFϕϵ2dx. Next, if we replace u with (1+ϕ)/2, (4) can be written as follows:(6)EMSϕ=ν1ΩFϕϵ2dx+ν2Ωf0-1+ϕ22dx+ν3Ω-C1+ϕ22dx. Since ϕ=0 at xC, Ωϕ2dx=Ω-Cϕ2dx. Therefore, the fractional analogue phase-field approach of the Mumford–Shah model is considered as minimizing the following energy functional:(7)Eϕ=ΩFϕϵ2+12-Δs/2ϕ2dx+μΩf0-1+ϕ22dx, and the governing equation can be written as follows:(8)ϕt=-ϕ3-ϕϵ2+Δxsϕ+μf0-1+ϕ2.

2.3. Numerical Solution

We consider the Fourier spectral method in space and the linear convex splitting scheme, which is known as stable and uniquely solvable one , in time. First, the temporal discretization of (8) is written as follows:(9)ϕn+1-ϕnΔt=-ϕn3-3ϕn+2ϕn+1ϵ2+Δxsϕn+1+μf0-1+ϕn+12for  n=1,,nt.Remark that (9) has the bounded solution ϕn1 for any n=1,,nt if f01 and ϕ01 where ϕ=maxxϕ is the l-norm. It comes from the fact that -Δxs has nonnegative eigenvalues.

Theorem 1.

The numerical scheme (9) is uniquely solvable for any time step Δt>0.

Proof.

We consider following functional defined on a l2-inner product space:(10)Gϕ=12ϕ22+ΔtEcϕ-Sn,ϕ2, where (ϕ,ψ)2=Ωϕψdx is the l2-inner product, ϕ22=(ϕ,ϕ)2 is the l2-norm, Ec(ϕ)=ϕ2/ϵ2,12+0.5(-Δ)xs/2ϕ22+μf0-0.5(1+ϕ)22, Sn=ϕn-Δtϕn3-3ϕn/ϵ2. Note that it can be solved if and only if G has the unique minimizer ϕn+1. For any ψ and scalar α,(11)ddαEcϕ+αψα=0=2ϕϵ2-Δxsϕ-μf0-1+ϕ2,ψ2,(12)d2dα2Ecϕ+αψα=0=2ϵ2+-Δxs/2ψ22+μ20.Therefore Ec(ϕ) is convex, which implies that G(ϕ) is also convex. Note that the unique minimizer ϕn+1 makes the first variation of G(ϕ) zeros, i.e.,(13)ϕn+1-ϕnΔt+ϕn3-3ϕn+2ϕn+1ϵ2-Δxsϕn+1-μf0-1+ϕn+12,ψ2=0,and it holds if and only if (9) holds.

Theorem 2.

The numerical scheme (9) is unconditionally energy stable, i.e., E(ϕn+1)E(ϕn) for any time step Δt>0.

Proof.

Note that we already prove that Ec(ϕ) is convex in the proof of Theorem 1. Let Ee(ϕ)=Ec(ϕ)-E(ϕ). Then, it is clear that Ee(ϕ) is concave since ϕ[-1,1] and the followings hold for any ϕ and ψ:(14)Ecϕ-Ecψ2ϕϵ2-Δxsϕ-μf0-1+ϕ2,ϕ-ψ2,(15)Eeϕ-Eeψ-ψ3-3ψϵ2,ϕ-ψ2. Then,(16)Eϕ-Eψ=Ecϕ-Ecψ-Eeϕ-Eeψ2ϕϵ2-Δxsϕ-μf0-1+ϕ2+ψ3-3ψϵ2,ϕ-ψ2. We, respectively, replace ϕ and ψ with ϕn+1 and ϕn. Then (9) says that(17)Eϕn+1-Eϕnϕn3-3ϕn+2ϕn+1ϵ2-Δxsϕn+1-μf0-1+ϕn+12,ϕn+1-ϕn2,=-ϕn+1-ϕΔt,ϕn+1-ϕ2=-1Δtϕn+1-ϕ220.

Before applying the Fourier spectral method, we present the Fourier definition of the fractional Laplacian operator.

Definition 3.

The Fourier definition of Δxs is (18)FΔxsϕξ=-ξsFϕξ, where F(·) is a Fourier transformation. Note that the Fourier transformation of the standard Laplacian operator is F(Δϕ)(ξ)=-ξ2Fϕ(ξ).

By the definition, (9) can be written as follows using the discrete Fourier transformation Fhϕnp=-Nx/2Nx/2-1q=-Ny/2Ny/2-1ϕ^pqne2π(px/Lx+qy/Ly)/(NxNy):(19)ϕ^pqn+1-ϕ^pqnΔt=-ϕ^pqn3-3ϕ^pqn+2ϕ^pqn+1ϵ2-ξpqsϕ^pqn+1+μf^0,pq-1+ϕ^pqn+12,where ϕ^ is the discrete Fourier coefficient, Lx and Ly are, respectively, the length of the domain in x- and y-axis, Nx and Ny are, respectively, the number of grid points in x- and y-axis, Δt is the temporal step size, and ξpq2=(2πp/Lx)2+(2πq/Lx)2. Therefore, we can get ϕn+1 as follows:(20)ϕn+1Fh1ϕ^pq=Fh-1ϕ^pq+Δt3ϕ^pq-ϕ^pq3/ϵ2+μf^0,pq-1/21+Δt2/ϵ2+ξpq2+μ/2.

3. Numerical Experiments 3.1. Effect of the Fractional Laplacian Operator

To observe the only difference between the fractional Laplacian Δxs and classical Laplacian Δx in image segmentation, let us select ϵ1 large enough such that (21)ϕn+1-ϕnΔtϵ^Δxsϕn+1+μf0-1+ϕn+12,where ϵ^ is the rescaled parameter. Since scheme (9) has bounded and unconditionally stable solutions, it is easy to see that (21) has the same properties, as well. For the numerical illustration, we consider original and noise 512×512 images f0true, f0noise. We take the image f0noise as the initial data. The fractional orders and parameters for both operators Δxs and Δx are given by s=0.5, ϵ^=1.47 and s=1, ϵ^=0.01, respectively. Note that there are no criteria of choosing the best parameters for image processing, while these parameters heuristically give the best result. The other parameters μ=20, Δt=0.0001, nt=40 are used. For our convenience, we denote the final solution from the operator of fractional order s by ϕ~s.

With initial data f0noise and its error f0true-f0noisefro336.6, we perform numerical tests. Comparisons are made for the numerical results which we have f0true-ϕ~0.5frof0true-ϕ~1fro298.3. If large noise is imposed upon the original image f0true, the value of f0true-f0noisefro is also large. If, the other way around, there is no noise, then f0noise is equal to f0true, i.e., f0true-f0noisefro is zero. In our setting, the error 336.6 implies the difference between initial and noised images. Note that “Barbara” image is comprised of 512×512 pixels, and each pixel belongs to [0,1]. We can observe that cartoon and texture are kept in Figure 1(c). However, there is noticeable blur at texture region in Figure 1(d).

(a) Original image f0true, (b) noise image f0noise, (c) close-up image at ϕ~0.5 with order s=0.5, and (d) close-up image at ϕ~1 with order s=1.0.

3.2. Fractional Allen–Cahn Equation

It is known that the motion of interfaces for the classical Allen–Cahn equation follows the mean curvature flow . In this section, we perform numerical simulations to compare the fractional and classical Allen–Cahn equations. Figure 2 shows the evolution of zero-contours of ϕ solving the fractional Allen–Cahn equation with different s values with circular and square initial conditions, respectively. Here, we used the following parameters: Nx=128, Ω=(0,1)2, Δt=0.1, the final time T=130, ϵ=0.0075, and the dotted lines represent the initial conditions.

Evolution of zero-contours of ϕ solving fractional Allen–Cahn equation with different s values and circular (top) and square (bottom) initial conditions.

s = 1

s = 0.5

s = 0.25

s = 0.1

In Figure 2(a), the interfaces evolve as motions by mean curvature since it is the classical Allen–Cahn equation case, whereas we can observe that the dynamics is different from the classical one, especially at the tip of the interfaces, in Figures 2(b) and 2(c). When s is relatively small case, s=0.1, the results give same shapes as the initial conditions (see Figure 2(d)). From the results, we can assume that it is better to capture a sharpen interface using the fractional Allen–Cahn equation with the proper s value and then using the classical Allen–Cahn equation.

3.3. Basic Figures

In this section, we consider the segmentation with basic figures. First, the segmentation results using classical and fractional Allen–Cahn equation with the following parameters are shown in Figure 3: Nx=256, Ω=(0,1)2, Δt=10, ϵ=0.01, and μ=10000, and star-shaped initial condition. At the tip of the star, we can observe that the fractional Allen–Cahn equation gives better performance.

(a) Initial condition and segmentation results with (b) classical and (c) fractional Allen–Cahn equations.

Initial

s = 1

s = 0.15

Next, we consider other segmenting simulations for two simple figures, circle and square, with salt and pepper noise using the following parameters: Nx=256, Ω=(0,1)2, Δt=10, ϵ=0.0038μ=10000. As shown in Figure 4, the case using the fractional Allen–Cahn equation (Figure 4(c)) gives better result than the case using classical Allen–Cahn equation (Figure 4(b)), especially at the corner of the square. However, the segmentation is interrupted by the noise when s becomes too small (see Figure 4(d)).

(a) Initial condition and (b)–(d) segmentation results with different s values.

Initial

s = 1

s = 0.25

s = 0.15

3.4. Medical Images

Here, we perform the numerical simulations applying the proposed segmentation algorithm to medical images. Figure 5 shows the segmentation results for brain CT image with and without injury  when (a) s=1 and (b) s=0.05 using the following parameters: Δt=10, ϵ=0.1, and μ=10000. As opposed to the case with classical Allen–Cahn equation, the injured part can be segmented in the case of Figure 5(b), which uses the fractional Laplacian operators.

Segmentation results for brain CT images with (left) and without (right) injury  when (a) s=1 and (b) s=0.05.

s = 1

s = 0.05

4. Conclusions

We proposed the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian based on the Mumford–Shah energy functional. The fractional order, obtained as the macroscopic limit of Lévy process, was expected to change the dynamics of the Allen–Cahn equation. Based on the convex splitting method, we proved the unconditionally unique solvability and energy stability of the numerical scheme. The segmentation results show that the fractional Laplacian operator has a better performance when the original image has sharp tips and corners and the abnormalities are close to each other. Note that our approach requires parameter tuning for image segmentation. The best segmentation-parameter combination, including the fractional order “s”, depends on the original image. The minimizer of our proposed functional is different for each of the initial images, so that we have to select the best parameter combination within all possible combinations of the parameters.

Data Availability

All the data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author D. Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) (NRF-2015R1C1A1A01054694). The author S. Lee was supported by the National Institute for Mathematical Sciences (NIMS) grant funded by the Korean Government and the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) (No. 2017R1C1B1001937).

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