Image Segmentation Based on Modified Fractional Allen–Cahn Equation

Dongsun Lee,Seunggyu Lee

doi:10.1155/2019/3980181

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.

Highlights

Image segmentation is a process of image partitioning into nonintersection parts with similar properties such as gray level, color, texture, brightness, and contrast [1]
We present the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian
This paper is organized as follows: In Section 2, we describe the mathematical model for the image segmentation using a fractional Laplacian operator

Summary

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 [4,5,6]. This paper is organized as follows: In Section 2, we describe the mathematical model for the image segmentation using a fractional Laplacian operator.

Mathematical Model

Numerical Experiments

Conclusions