Active contours driven by weighted region-scalable fitting energy based on local entropy

Abstract

In this paper, we present a scheme of improvement on the region-scalable fitting (RSF) model proposed by Li et al. (Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing 17(10) (2008) 1940–1949) in terms of robustness to initialization and noise. First, the Gaussian kernel for the RSF energy is replaced with a “mollifying” kernel with compact support. Second, the RSF energy is redefined as a weighted energy integral, where the weight is local entropy deriving from a grey level distribution of image. The total energy functional is then incorporated into a variational level set formulation with two extra internal energy terms. The new RSF model not only handles better intensity inhomogeneity, but also allows for more flexible initialization and more robustness to noise compared to the original RSF model.

Introduction

Image segmentation has always been an essential problem in image processing and computer vision. The goal is to change the representation of an image into something that is more meaningful and easier to analyze. This is a critical intermediate step in many high level tasks such as objects recognition and tracking. So far, a large number of good algorithms and methodologies including active contour models [1] have been proposed for image segmentation. In this study, we focus on implicit active contour models, i.e., active contour models in a level set formulation.

Implicit active contour models have been proved to be a class of efficient techniques for image segmentation [2], [3]. They constitute very interesting applications of the level set method within the active contour framework [1]. The basic idea behind them is that an active contour is implicitly represented as the zero level set of a function in higher dimension (called level set function), and then the level set function is deformed according to an evolution partial differential equation (PDE). A remarkable advantage of implicit models is that the level set evolution allows for automatic topological change of active contour, which is generally impossible in traditional parametric active contour models [4] when direct implementations are performed. Early implicit models [5], [6], [7] first yield an evolution equation of a parameterized contour, and then convert it to the evolution PDE of the level set function whose zero level set represents the parameterized contour. Alternatively, the evolution PDE for level set function can be directly derived by gradient descent from the minimization problem for an energy functional over level set functions [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].

Existing implicit active contour models can be roughly categorized into two basic classes: the edge-based [5], [6], [7], [8], [9] and the region-based [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], depending on image features used for segmentation. Edge-based models utilize typically an edge indicator depending on image gradient to perform contour extraction. They have been successfully used to extract contours defined by gradient, but they have difficulty to detect edges without gradient, for instance very smooth or discontinuous boundaries [10]. Region-based models usually use global and/or local statistics inside regions rather than gradient on edges to find a partition of image domain. They generally have better performances in the presence of weak or discontinuous boundaries. Early popular region-based models tend to rely on intensity homogeneous (roughly constant or smooth) statistics in each of the regions to be segmented. Therefore, they either lack the ability to deal with intensity inhomogeneity like the PC (piecewise constant) model [10] or are excessively expensive in computation like the PS (piecewise smooth) model [12].

Intensity inhomogeneity often occurs in medical images such as X-ray radiography/tomography and magnetic resonance (MR) images [14], [15], [16]. The intensity inhomogeneity usually refers to the slow, nonanatomic intensity variations of the same tissue over the image domain [20]; in MR images, it often appears as an intensity variation across an image, and similar artifacts also in CT images and ultrasound images [16]. Although the presence of intensity inhomogeneity is usually hardly noticeable to a human observer, region-based models such as the PC model [10] are highly sensitive to the spurious variations of image intensities. Thus, segmentation of such medical images usually requires intensity inhomogeneity correction as a preprocessing step [20].

In order to handle directly intensity inhomogeneity within the active contour framework, Li et al. [15] proposed recently a region-scalable fitting (RSF) active contour model (originally termed as local binary fitting (LBF) model [14]) in a variational level set formulation. The RSF model draws upon intensity information in spatially varying local regions depending on a scale parameter, so it is able to deal with intensity inhomogeneity accurately and efficiently. However, the RSF model easily gets stuck in local minimums if the contour is initialized inappropriately. This makes it need user intervention to define the initial contours professionally. The multiple seed initialization used by Vese and Chan [12] also cannot always produce better results than the single seed initialization. Therefore, it is a great challenge to find an efficient way to address the initialization problem of the RSF model.

More recently, Wang et al. [16] proposed a novel local and global intensity fitting (LGIF) model in a variational level set formulation, which combines the advantages of the LBF model [14] and the PC model [10]. They first define an LGIF energy by a linear combination of local intensity fitting (LIF) energy [14] and global intensity fitting (GIF) energy [10] of the form ${\mathcal{E}}^{\mathrm{LGIF}}=(1-\omega ){\mathcal{E}}^{\mathrm{LIF}}+\omega {\mathcal{E}}^{\mathrm{GIF}}$, where the parameter $\omega (0\le \omega \le 1)$ controls the influence of the LIF energy ${\mathcal{E}}^{\mathrm{LIF}}$ and the GIF energy ${\mathcal{E}}^{\mathrm{GIF}}$. The LGIF energy is then incorporated into a variational level set formulation with two extra regularization terms. The resulting evolution of level set function is the gradient flow that minimizes the overall energy functional. Since the PC model [10] is insensitive to initialization as well as noise and the LBF model can handle intensity inhomogeneity, the LGIF model can deal with intensity inhomogeneity by choosing appropriately the value of $\omega$, while robust to initialization and noise. The inconvenience of the LGIF model in applications is mainly that the value of the parameter $\omega$ must be determined and adjusted in reference to initialization and noise level.

In this study, we present an improved version of the RSF model [15] in terms of robustness to initialization and noise. First, the Gaussian kernel for the RSF energy is replaced with a “mollifying” kernel [21] with compact support. Second, the RSF energy is redefined as a weighted energy integral, where the weight function is defined by local entropy that is derived from a grey level distribution of image [22]. The total energy functional is then incorporated into a variational level set formulation with two extra internal energy terms. The new RSF model not only provides desirable segmentation results in the presence of intensity inhomogeneity, but also allows for more flexible localizations of initial contours and more robustness to noise compared to the original RSF model [15].

The remainder of this paper is organized as follows. Section 2 briefly reviews the RSF model. Section 3 introduces the proposed scheme. Section 4 presents experimental results using a set of synthetic and real images, followed by some discussions in Section 5. This paper is summarized in Section 6.