Abstract
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.
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 [1]. 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 [2].
One of the most widely used methods for image segmentation is the Mumford–Shah model [3] and it has been extensively studied and extended in many works [4–6].
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 [7], is a gradient flow under -inner product space with the following Ginzburg–Landau free energy functional:and it has the following form: Here is the concentration defined in a bounded domain , is the coefficient related to an interfacial energy, and 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 [8–10]: where is the fractional Laplacian, obtained as the macroscopic limit of Lévy process [11], with fractional order . 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 [12–16].
2.2. Modification to Image Segmentation
Let be a grayscale given image. Then, the Mumford–Shah model, which is one of the most widely used models for image segmentation [3], minimizes the following functional to perform image segmentation:where is the piecewise smooth function approximating , is the segmenting curve representing the set of edges in the given image , and , , are the positive constants. Let us modify the energy functional (4) as a phase-field formulation. First, it should be noted that can be considered as the zero-contour of . Then, the following equation holds: Next, if we replace with , (4) can be written as follows: Since at , . Therefore, the fractional analogue phase-field approach of the Mumford–Shah model is considered as minimizing the following energy functional: and the governing equation can be written as follows:
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 [17], in time. First, the temporal discretization of (8) is written as follows:Remark that (9) has the bounded solution for any if and where is the -norm. It comes from the fact that has nonnegative eigenvalues.
Theorem 1. The numerical scheme (9) is uniquely solvable for any time step .
Proof. We consider following functional defined on a -inner product space: where is the -inner product, is the -norm, , . Note that it can be solved if and only if has the unique minimizer . For any and scalar ,Therefore is convex, which implies that is also convex. Note that the unique minimizer makes the first variation of zeros, i.e.,and it holds if and only if (9) holds.
Theorem 2. The numerical scheme (9) is unconditionally energy stable, i.e., for any time step .
Proof. Note that we already prove that is convex in the proof of Theorem 1. Let . Then, it is clear that is concave since and the followings hold for any and : Then, We, respectively, replace and with and . Then (9) says that
Before applying the Fourier spectral method, we present the Fourier definition of the fractional Laplacian operator.
Definition 3. The Fourier definition of is [19] where is a Fourier transformation. Note that the Fourier transformation of the standard Laplacian operator is .
By the definition, (9) can be written as follows using the discrete Fourier transformation :where is the discrete Fourier coefficient, and are, respectively, the length of the domain in - and -axis, and are, respectively, the number of grid points in - and axis, is the temporal step size, and . Therefore, we can get as follows:
3. Numerical Experiments
3.1. Effect of the Fractional Laplacian Operator
To observe the only difference between the fractional Laplacian and classical Laplacian in image segmentation, let us select large enough such that where 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 images , . We take the image as the initial data. The fractional orders and parameters for both operators and are given by , and , , 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 , , are used. For our convenience, we denote the final solution from the operator of fractional order by .
With initial data and its error , we perform numerical tests. Comparisons are made for the numerical results which we have . If large noise is imposed upon the original image , the value of is also large. If, the other way around, there is no noise, then is equal to , i.e., is zero. In our setting, the error implies the difference between initial and noised images. Note that “Barbara” image is comprised of pixels, and each pixel belongs to . 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)
(b)
(c)
(d)
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 [20]. 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 solving the fractional Allen–Cahn equation with different values with circular and square initial conditions, respectively. Here, we used the following parameters: , , , the final time , , and the dotted lines represent the initial conditions.
(a)
(b)
(c)
(d)
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 is relatively small case, , 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 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: , , , , and , 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
(b) s = 1
(c) 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: , , , . 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 becomes too small (see Figure 4(d)).
(a) Initial
(b) s = 1
(c) s = 0.25
(d) 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 [18] when (a) and (b) using the following parameters: , , and . 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.
(a) s = 1
(b) 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).