Elsevier

Pattern Recognition

Volume 43, Issue 3, March 2010, Pages 603-618
Pattern Recognition

An efficient local Chan–Vese model for image segmentation

https://doi.org/10.1016/j.patcog.2009.08.002Get rights and content

Abstract

In this paper, a new local Chan–Vese (LCV) model is proposed for image segmentation, which is built based on the techniques of curve evolution, local statistical function and level set method. The energy functional for the proposed model consists of three terms, i.e., global term, local term and regularization term. By incorporating the local image information into the proposed model, the images with intensity inhomogeneity can be efficiently segmented. In addition, the time-consuming re-initialization step widely adopted in traditional level set methods can be avoided by introducing a new penalizing energy. To avoid the long iteration process for level set evolution, an efficient termination criterion is presented which is based on the length change of evolving curve. Particularly, we proposed constructing an extended structure tensor (EST) by adding the intensity information into the classical structure tensor for texture image segmentation. It can be found that by combining the EST with our LCV model, the texture image can be efficiently segmented no matter whether it presents intensity inhomogeneity or not. Finally, experiments on some synthetic and real images have demonstrated the efficiency and robustness of our model. Moreover, comparisons with the well-known Chan–Vese (CV) model and recent popular local binary fitting (LBF) model also show that our LCV model can segment images with few iteration times and be less sensitive to the location of initial contour and the selection of governing parameters.

Introduction

Image segmentation has always been a fundamental problem and complex task in the field of image processing and computer vision. Its goal is to change the representation of an image into something that is more meaningful and easier to analyze [1]. In other words, it is used to partition a given image into several parts in each of which the intensity is homogeneous. Up to now, a wide variety of algorithms have been proposed to solve the image segmentation problem. Researchers have also done great efforts to improve the performance of the image segmentation algorithms.

Active contour model, or snakes, proposed by Kass et al. [2], has been proved to be an efficient framework for image segmentation. The fundamental idea of active contour model is to start with a curve around the object to be detected, and the curve moves toward its interior normal and stops on the true boundary of the object based on an energy-minimizing model. The main drawbacks of this method are its sensitivity to initial conditions and the difficulties associated with topological changes like the merging and splitting of the evolving curve. Since the active contour model was proposed, many methods have been proposed to improve it, in which level set method proposed by Osher and Sethian [3] is the most important and successful one.

Level set method is based on active contour model and particularly designed to handle the segmentation of deformable structures. Generally, the classical active contour model uses spline curves to model the boundary of an object. However, the level set method is to use a deformable curve front for approximating the boundary of an object. In the level set framework, the curve is represented by the zero level set of a smooth function, usually called the level set function. Moving the curves can be done by evolving the level set functions instead of directly moving the curves. Therefore, level set methods exhibit interesting elastic behaviors and can efficiently handle the topological changes which is also a main advantage compared with classical active contour model. Formally, the evolution of the curve is driven by a time-dependent partial differential equation (PDE) where the so-called velocity term reflects the image features characterizing the object to be segmented [4].

Generally, a classical level set framework consists of an implicit data representation of a hypersurface, a set of partial differential equations (PDEs) that govern how the curve moves, and the corresponding numerical solution for implementing this method on computers [5].

It should be mentioned that the early edge-based level set methods [6], [7], [8] usually depend on the gradient of the given image for stopping the evolution of the curve. Therefore, these methods can only detect objects with edges defined by the gradient. However, the corresponding discrete gradients are generally bounded and the energy functional will hardly approach zero on the boundaries in practice. So, the evolving curve may pass through the true boundaries, especially for the models in [6], [7], [8].

Recently, region-based level set methods [10], [11], [12], [13] have been proposed and applied to image segmentation filed by incorporating region-based information into the energy functional. Unlike edge-based level set methods using image gradient, region-based methods usually utilize the global region information to stabilize their responses to local variations (such as weak boundaries and noise). Thus, they can obtain a better performance of segmentation than edge-based level set methods, especially for images with weak object boundaries and noise. Among the region-based methods, Chan–Vese model [10] is a representative and popular one.

Based on the Mumford–Shah functional [9] for segmentation, Chan and Vese [10] proposed an easily handled model, or called Chan–Vese (CV) model, to detect objects whose boundaries are not necessarily detected by the gradient. Mumford–Shah model was firstly proposed as a general image segmentation model by Mumford and Shah in [9]. Using this model, the image is decomposed into some regions. Inside each region, the original image is approximated by a smooth function. The optimal partition of the image can be found by minimizing the Mumford–Shah functional. Chan and Vese successfully solved the minimization problem by using level set functions, which utilized the global image statistics inside and outside the evolving curve rather than the gradients on the boundaries.

CV model has achieved good performance in image segmentation task due to its ability of obtaining a larger convergence range and handling topological changes naturally. However, it still has some intrinsic limitations. First, CV model generally works for images with intensity homogeneity since it assumes that the intensities in each region always maintain constant. Thus, it often leads to poor segmentation results for images with intensity inhomogeneity due to wrong movement of evolving curves guided by global image information. Second, the segmentation of CV model is usually dependent on the placement of the initial contour, especially for the complicated images. Sometimes, the different results will be obtained on the same image by using different initial contours. Thus, the placement of initial contour is still an important issue for CV model to get successful segmentation in complicated images. Third, the CV model may become time-consuming if the periodical re-initialization step is adopted, which is a technique for periodically re-initializing the level set function to a signed distance function during the evolution. It has been regarded as a numerical remedy for maintaining stable curve evolution and ensuring precise results, which also leads to time-consumption as the side effect.

To solve the limitations of CV model, many efficient implementation schemes have been proposed [14], [15], [16], [17], [18], [19], [20]. For example, in [14], Vese and Chan extended their original model in [10] by using a multiphase level set formulation. However, the involved computation in this model is very expensive, which also limits its applications in practice. In addition, to reduce the computational cost, this method usually requires that the initial contour should be near to the object boundaries. In [15], an initialization scheme for the CV model was introduced, in which the initial curve is found by searching among the externals of the fidelity term in [10]. However, this one-dimensional search method is also time-consuming and fails to work when the gray difference between object and background is small. Xia et al. [16] proposed another initialization method which generates initial closed curves by iteratively connecting edge points obtained by canny detector and morphological filter. This method can efficiently work for some simple images. To reduce the computational load of curve evolution for CV model, the implementation schemes without solving the PDEs were proposed [17], [18]. However, they are still sensitive to the selection of initial curves and sometime sensitive to noise. Li et al. [19] proposed a so-called penalizing energy which acts as a metric to characterize how close the level set function to a signed distance function. This metric can also be adopted by CV model to avoid the re-initialization step. In [20], the penalizing energy proposed in [19] and a discrimination function based on color information was combined into the CV model for segmenting the color images.

To efficiently perform the segmentation of images with intensity inhomogeneity, a new class of models has been proposed which not only utilize region-based techniques but also incorporate the benefits of local information. There have been several literatures [12], [21], [22], [23], [24], [25], [26], [27] which are relevant to the existing works. Paragios et al. [12] presented a method in which edge-based energies and region-based energies were explicitly summed to create a joint energy which was then minimized. In [21], Sum et al. took a similar approach and minimized the sum of a global region-based energy and a local energy based on image contrast. Brox et al. [22] proposed the idea of incorporating localized statistics into a variational framework which shows that segmentation with local means is a first order approximation of the piecewise smooth simplification of the Mumford–Shah functional. Piovano et al. [23] employed convolutions to quickly compute localized statistics and yielded results similar to piecewise-smooth segmentation in a much more efficient manner. The work of An et al. [24] also noted the efficiency of localized approaches versus full piecewise smooth estimation and introduced a way in which localizations at two different scales can be combined to allow sensitivity to both coarse and fine image features. In [25], the authors proposed a similar flow based on computing geodesic curves in the space of localized means rather than approximating a piecewise-smooth model. In [26], a novel localization framework was proposed which allows the region-based energy to be localized in a fully variational way so that objects with heterogeneous statistics can be successfully segmented with the localized energies. Recently, Li et al. proposed an efficient region-based level set method by introducing a local binary fitting (LBF) energy with a kernel function [27]. The LBF model enables the extraction of accurate local image and can be used to segment the images with intensity inhomogeneity. It has attracted extensive attentions for its good segmentation performance in limited iterations. However, the LBF model usually needs to perform four convolution operations at each iteration, which greatly increases the computational complexity. In addition, it is also sensitive to the selection of governing parameters and the location of initial curve.

For CV model using the intensity average only, texture image segmentation is another difficult issue since intensity averages cannot represent the texture information inside and outside the target objects. In many texture images, due to the difference of the intensity averages of neighboring textures, the small textures in objects will be segmented while the desirable whole objects will be not separated. Therefore, other information should be introduced for texture image segmentation. Chan and Vese [10] suggested using texture information or features extracted from the original image, such as the curvature or the orientation of level sets, to overcome the difficulty. However, the proposed texture information in [10] cannot work well in many complicated texture images due to their simple properties. Note that the texture image segmentation greatly relies on the extraction of suitable texture information from the image. Recently, Gabor filters have been efficiently incorporated into level set methods and CV model for the texture image segmentation [12], [28], [29]. Unfortunately, Gabor filters have the fatal drawback that they induce a lot of redundancies and thus lots of feature channels [30]. As another efficient texture representation, the structure tensor [31], [32] is a kind of low dimensional feature computed from the spatial derivatives of the image. It is a common tool for local orientation estimation and image structure analysis which is formed as the outer product of the image gradient with itself. So far, the structure tensor has been applied in many image segmentation algorithms, most notably, in the early geodesic active contours framework [33], [34], [35].

In this paper, we proposed a so-called local Chan–Vese (LCV) model which utilizes both global image information and local image information for image segmentation. The energy functional for the proposed model consists of three parts: global term, local term and regularization term. By using the local image information, the images with intensity inhomogeneity can be efficiently segmented in limited iterations. Moreover, the time-consuming re-initialization step widely adopted in traditional level set methods can be avoided by introducing a new penalizing energy to the regularization term. As a result, the time-consumption is greatly decreased. Specially, the evolving curve in level set evolution process can automatically stop on true boundaries of objects according to a termination criterion which is based on the length change of evolving curve. Finally, we proposed constructing an extended structure tensor (EST) by adding the intensity information into the classical structure tensor for texture image segmentation. By incorporating the EST into the proposed LCV model, the texture image can be easily segmented no matter whether it presents intensity inhomogeneity or not. Moreover, the comparisons with the CV model and recent LBF model show that our LCV model can segment ordinary/texture images with or without intensity inhomogeneity in fewer iterations. Particularly, it can be found that our model is less sensitive to the location of initial contour and the selection of governing parameters.

The rest of this paper is organized as follows: In Section 2, we briefly review the Mumford–Shah model and Chan–Vese (CV) model. Our local Chan–Vese (LCV) model is presented in Section 3. In Section 4, the proposed model is validated by some experiments on synthetic and real images. Finally, some conclusive remarks are included in Section 5.

Section snippets

Mumford–Shah model

The Chan–Vese model is the curve evolution implementation of a piecewise-constant case of the Mumford–Shah model [9]. The Mumford–Shah model is an energy-based method introduced by Mumford and Shah via an energy functional. The basic idea is to find a pair of (u,C) for a given image u0, where u is a nearly piecewise smooth approximation of u0, and C denotes the smooth and closed segmenting curve. The general form for the Mumford–Shah energy functional can be written asEMS(u,C)=Ω|u0(x,y)-u(x,y)|

Local Chan–Vese model

In this section, we shall present and discuss the details of our proposed local Chan–Vese (LCV) model and its numerical implementation. What should be stressed is that our model is defined based on the techniques of curve evolution, local statistical function and level set methods. It is well-known that some traditional level set methods used either the image gradient [6], [7], [8] or the global information [10], [11] to drive the evolving curve(s) towards the true boundaries. However, none of

Experimental results

In this Section, we shall present the experimental results of our local Chan–Vese (LCV) model on some synthetic and real images. The proposed model was implemented by Matlab 7 on a computer with Intel Core 2 Duo 2.2GHz CPU, 2G RAM, and Windows XP operating system. The processing time referred later in this section starts after choosing the initial contour. We used the same parameters of the time-step Δt=0.1, the grid spacing h=1, ε=1 (for Hε(z) and δε(z)), the window size of averaging

Conclusions and future works

In this paper, we propose a new local Chan–Vese (LCV) model for image segmentation, which is based on the techniques of curve evolution, local statistical function and level set theory. The energy functional for the proposed model consists of global term, local term and regularization term. By incorporating the local image information into our model, the images with intensity inhomogeneity can be efficiently segmented. To avoid the time-consuming re-initialization step, a new penalizing energy

Acknowledgements

This work was supported by the grants of the National Science Foundation of China, Nos. 60873012, 60805021, 60705007 & 30700161, the grant from the National Basic Research Program of China (973 Program), No.2007CB311002, the grants from the National High Technology Research and Development Program of China (863 Program), Nos. 2007AA01Z167, the grant of the Guide Project of Innovative Base of Chinese Academy of Sciences (CAS), No.KSCX1-YW-R-30, the grant of the Graduate Students’ Scientific

About the Author—XIAO-FENG WANG received the B.Sc. degree in Computer and Science Technology from Anhui University, Hefei, China, in 1999, the M.Sc. degree in Pattern Recognition and Intelligent System from Institute of Intelligent Machines (IIM), Chinese Academy of Sciences, Hefei, China, in 2005. He is now in pursuit for Ph.D. degree in Pattern Recognition and Intelligent System in University of Science and Technology of China (USTC). His research interests include pattern recognition, image

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    About the Author—XIAO-FENG WANG received the B.Sc. degree in Computer and Science Technology from Anhui University, Hefei, China, in 1999, the M.Sc. degree in Pattern Recognition and Intelligent System from Institute of Intelligent Machines (IIM), Chinese Academy of Sciences, Hefei, China, in 2005. He is now in pursuit for Ph.D. degree in Pattern Recognition and Intelligent System in University of Science and Technology of China (USTC). His research interests include pattern recognition, image processing and data mining.

    About the Author—DE-SHUANG HUANG received the B.Sc., M.Sc. and Ph.D. degrees all in Electronic Engineering from Institute of Electronic Engineering, Hefei, China, National Defense University of Science and Technology, Changsha, China, and Xidian University, Xian, China, in 1986, 1989 and 1993, respectively. During the 1993–1997 period he was a postdoctoral student, respectively, in Beijing Institute of Technology and in National Key Laboratory of Pattern Recognition, Chinese Academy of Sciences Beijing, China. In September, 2000, he joined the Institute of Intelligent Machines, Chinese Academy of Sciences as the Recipient of “Hundred Talents Program of CAS”. From September 2000 to March 2001, he worked as Research Associate in Hong Kong Polytechnic University. From April 2002 to June 2003, he worked as Research Fellow in City University of Hong Kong. From August to September 2003, he visited the George Washington University as visiting professor, Washington DC, USA. From October to December 2003, he worked as Research Fellow in Hong Kong Polytechnic University. From July to December 2004, he worked as the University Fellow in Hong Kong Baptist University. Dr. Huang is currently a senior member of the IEEE.

    About the Author—HUAN XU received the B.Sc. degree in Central China Normal University, majored in Computer Software Science and minored in Mathematics, in 2006, the M.Sc. degree in Pattern Recognition and Intelligent System from University of Science and Technology of China (USTC), in 2009. She is now in pursuit for Ph.D. degree in the Electrical and Computer Engineering Department of Louisiana State University. Her research interests include image analysis, visual and geometric computing.

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