Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 6098021, 10 pages

http://dx.doi.org/10.1155/2016/6098021

## Fractional Differentiation-Based Active Contour Model Driven by Local Intensity Fitting Energy

^{1}Department of Precision Instrument, Tsinghua University, Beijing 100084, China^{2}College of Computer Science and Information Technology, Zhejiang Wanli University, Ningbo 315100, China

Received 28 October 2015; Revised 28 March 2016; Accepted 19 April 2016

Academic Editor: Weizhong Dai

Copyright © 2016 Ming Gu and Renfang Wang. 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.

#### Abstract

A novel active contour model is proposed for segmentation images with inhomogeneity. Firstly, fractional order filter is defined by eight convolution masks corresponding to the image orientation in the eight compass directions. Then, the fractional order differentiation image is obtained and applied to the level set method. Secondly, we defined a new energy functional based on local image information and fractional order differentiation image; the proposed model not only can describe the input image more accurately but also can deal with intensity inhomogeneity. Local fitting term can enhance the ability of the model to deal with intensity inhomogeneity. The defined penalty term is used to reduce the occurrence of false boundaries. Finally, in order to eliminate the time-consuming step of reinitialization and ensure stable evolution of level set function, the Gaussian filtering method is used. Experiments on synthetic and real images show that the proposed model is efficient for images with intensity inhomogeneity and flexible to initial contour.

#### 1. Introduction

Image segmentation is a fundamental and challenging task in automatic image processing and computer vision. It aims to divide an image into a number of nonoverlapping regions that are more meaningful and are easy to analyze. So, the quality of segmentation directly affects the results of the following high level tasks. Up till now, numerous techniques for image segmentation have been proposed to solve the problem of image segmentation, such as thresholding, edge detection, clustering, region growth, and active contour method. Among those methods, active contour models (ACMs) have been proven to be one of the successful methods for image segmentation. However, images in natural world are often corrupted by intensity inhomogeneity, which makes image difficult to segment accurately.

Since first introduced by Kass et al. [1], ACMs have received more and more attention. The basic idea is that the dynamic curves evolve controlled by an appropriate energy functional and toward the target boundaries. Over the last two decades, many researchers have done great efforts to improve the performance of it and have proposed many different ACMs, which can be categorized into two categories: parametric ACMs and geometric ACMs. However, the parametric ACMs cannot deal with the topological changes like splitting and merging of the evolving curve. In geometric ACMs, this problem can be handled by using the level set method, which is proposed by Osher and Sethian [2]. Differentiating from the pure partial differential equation models whose evolution equation was directly constructed, the variational level set method [3] first defines an energy functional by level set function, and then the evolution equation is obtained via minimizing the energy functional. When the evolution of level set function stops, we can get the final contour represented by the zero level set. One of the most popular variational level set models is the CV model [4], which is built based on the assumption that the intensities in foreground and background keep constant. The CV model has been widely applied to two-phase image segmentation. However, it usually fails to segment images with intensity inhomogeneity. In order to deal with this problem, Li et al. proposed a famous local region-based model named LBF [5, 6] (local binary fitting). The energy functional is defined by using the local intensity statistical information. Although the LBF model can work well on images with intensity inhomogeneity, it is sensitive to initial contour.

In the image processing, methods based on fractional order differentiation appear to give better performance than traditional ones. In fact, fractional order differentiation, which can date back to three hundred years ago, is a generalization of the ordinary differentiation. Due to more precise derivatives of arbitrary order, fractional order differentiation can provide the best description for many natural phenomena and has been successfully applied to many fields such as signal processing and automatic control [7, 8]. Though fractional order differentiation has been uses in image denoising [9], image enhancement [10], and image segmentation [11], its application in the area of image processing is just an emerging branch to study.

In the present work, we proposed a novel active contour model based on the fractional order differentiation and level set method to segment images with intensity inhomogeneity. Many natural images are always corrupted by noise and low contrast. The traditional methods based on integer order differentiation improve image visual quality while enhancing noise. However, factional order differentiation can better describe image data than integer order differentiation and get better results. So, we construct filter convolution masks based on the fractional order differentiation. In order to effectively estimate the fractional order differential of digital image and ensure the factional mask windows are invariant to rotation, filter masks are defined in eight directions. And then the filtered image is obtained by the convolution between original image and filter masks. In the level set method, on one hand, local image information plays a vital role in successful segmentation of images with intensity inhomogeneity. So the proposed model defined a new local fitting term to cope with intensity inhomogeneity. On the other hand, false boundaries often appear; therefore, a new penalty term is defined to solve this problem. Finally, the Gaussian filtering method is used to eliminate the time-consuming step of reinitialization and ensure stable evolution of level set function. Experiments on synthetic and real images show that the proposed model is efficient for images with intensity inhomogeneity and flexible to initial contour.

In summary, the main contributions of the paper are as follows. (a) We use the filtered image resulting from fractional order differentiation as a guide image to accurately estimate the local information. The increased quality of this guide image improves the performance of the proposed model. (b) Based on the observation of the Heaviside function, we propose a new penalty function which is used to reduce the occurrence of false boundary. (c) Some numerical experiments are presented to analyze the improvement due to using filter convolution masks based on fractional order.

This paper is organized as follows. In Section 2, we briefly review two classic level set models for image segmentation and indicate their limitations. Section 3 shows a new level set model in detail. Section 4 presents experimental results on real and synthetic images using the proposed model. Finally, conclusions are drawn in Section 5.

#### 2. Related Work

##### 2.1. CV Model

Chan and Vese [4] proposed a classic level set model for image segmentation based on the assumption that the input image is piecewise constant. Let be a given image on the image domain . The CV model can be expressed as minimization of the following energy functional: where , , and are fixed nonnegative parameters. is gradient operator. is the intensity at a point in . and are two constants that denote the average intensities in the regions inside and outside the contour , respectively. is the level set function, and is the Heaviside function.

By minimizing the energy functional (1), we obtain the following formulation: where is divergence operator. The two piecewise constants and are given by

In (2), the first term computes the length of contour, which smooths the evolution curves. The second term is the global image fitting force based on the global image information to drive the contour evolution toward the object boundaries. If the intensities inside or outside the contour are inhomogeneity, and may be far different from the real image data. So the CV model usually fails to segment images with intensity inhomogeneity.

##### 2.2. LBF Model

To segment images with intensity inhomogeneity, Li et al. proposed the LBF model [5]. The local statistical information is obtained by introducing a kernel function. They defined the energy functional as follows: where is Gaussian kernel with standard deviation . , , , and are fixed parameters. and locally approximate the intensities inside and outside contour in a widow, respectively.

By minimizing the energy functional (4) with regard to level set function , we can obtain the following equation: where and are functions: with where and .

In (5), the first term is the penalty term to regularize the level set function, which avoids the initialization step. The second term is the length term and the last term is the local image fitting force that controls the contour evolution. The LBF model can effectively segment inhomogeneity images. However, LBF model is sensitive to initial contour. In other words, it has restrictive requirement for the location of the initial contour.

##### 2.3. Grünwald-Letnikov (GL) Definition

The fractional order differentiation is a generalization of the ordinary differentiation and has started to play a very important role in image processing. The definition of fractional order differentiation is studied by many researchers from different views and more than one exist in literatures. In this paper, we use GL definition, which can be expressed as where , and is the Gamma function. The explicit numerical approximation can be expressed as where , which can be considered as the coefficients of the Taylor series expansions of the corresponding “generating” functions, defined as follows [12]:

#### 3. The Proposed Model

##### 3.1. Fractional Order Image

In [13], Tian et al. define four directions of fractional order masks: on - and -axes directions, expressed by , , , and . It is shown in Figure 1, where , , , , and .