Complexity

Volume 2019, Article ID 5465289, 17 pages

https://doi.org/10.1155/2019/5465289

## Active Contour Models Based on Block Similarity for Multiple Objects Segmentation

^{1}College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China^{2}Big Data Engineering Laboratory for Teaching Resources & Assessment of Education Quality, Xinxiang 453007, Henan, China^{3}College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

Correspondence should be addressed to Guoqi Liu; moc.361@804080iqouguil

Received 16 March 2019; Revised 21 September 2019; Accepted 14 October 2019; Published 6 November 2019

Academic Editor: Ludovico Minati

Copyright © 2019 Guoqi Liu and Jinjin Wei. 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

For the model of active contours with group similarity (ACGS), a rank constraint for a group of evolving contours is defined to keep the shape similarity. ACGS obtains robust results in extracting a single object with missing or misleading features. However, with one initial contour, it could not extent to multiple objects segmentation because low-rank property will not hold in some image sequences. Besides, ACGS is affected by nontarget objects. In this paper, an active contour model based on block similarity of shapes is proposed to extend the ACGS model to realize multiple objects extraction. For a sequence of image with multiple objects, a model for multiple objects extraction is constructed by combining sparse decomposition and ACGS; second, a block low-rank constraint is proposed to constrain the similarity of these evolving contours in every block; finally, segmentation results are obtained through iterative evolutions. Experimental results show the proposed method could segment images with multiple targets, and it improves the robust segmentation performance of sequence of image when the features of multiobjects are missing or misleading.

#### 1. Introduction

Object segmentation is an important research topic in many fields, such as computer vision and image recognition [1]. Compared with other methods [2, 3], active contour models (ACMs) [4] have many advantages since the resultant contours are closed and quite regular, which are convenient for further applications [5, 6]. The main principle of the ACMs is to define an energy functional and minimize the energy functional to obtain an optimal contour, and the converged result indicates the boundary of the target object to be extracted.

There are broadly two types of ACMs according to the representation of contour, i.e., parametric active contour model [4, 7] and geometric active contour model [8–12]. Geometric active contour model, which is also known as implicit active contour model, has been presented [8] based on level set method (LSM) [8, 10]. LSM offers great flexibility for curve topology. Therefore, topological flexibility is a major advantage of geometric active contour model, which is desirable in detecting multiple objects [11].

Recently, geometric active contour models are widely researched and applied in medical image segmentation. For example, by integrating edge, region [12], or prior information into LSM, geometric active contour models could segment objects in various backgrounds, such as inhomogeneous intensity [13, 14] and object occlusion [15]. By combining edge and region information, distance regularized level set evolution (DRLSE) [16] and completely convex formulation of the Chan–Vese (C-V) [17] image segmentation model (CFCV) [18] have been proposed. Based on the distance regularized term, DRLSE could make the contour evolve robustly and segment multiple objects with proper initialization. With the convex relaxation method, the CFCV model is convex with respective to level set function, which is robust to initialization, and it greatly decreases the iterations and evolution time. In order to deal with inhomogeneous intensity, locally statistical active contour model (LSACM) [19] utilizes the local information to define a map function, which obtains robust segmentation performances.

In prior models based on LSM, shape prior [20, 21] is usually integrated into geometric active contour to segment or track objects. Shape prior has obtained robust results in segmenting objects with complex background [22–25]. However, the shape prior is usually learnt from a large set of annotated data, which is not always accessible in practice. Furthermore, the models with LSM usually have a large computational cost.

In order to avoid training of massive data and effectively deal with the affection caused by the interference information, an active contour with group similarity (ACGS) is proposed for single target extraction in a sequence of image [15]. It can be regarded as an unsupervised prior shape model. ACGS obtains robust results even in some object occlusion and noise. In ACGS, group similarity constraint is measured by low-rank, and low-rank property will not hold if the level set representation is used. Variational methods for image segmentation also have this issue [23–29]. Thus, parametric model is used to represent the contour in ACGS. In the parametric active contour model, contours are usually represented by landmarks. Generally, the parametric model is simple and fast. However, it could not segment multiple objects with one initial contour, since parametric representation cannot describe the topology change of contour. With multiple initial contours, parametric active contour could extract multiple objects. However, the disturbance of nontargets makes the contour farther away from the targets, and it is difficult to determine the locations and sizes of multiple initial contours during initialization.

In this paper, based on ACGS [15] and sparse decomposition method [30, 31], ACGS is extended to realize multiple objects segmentation of image sequence with parametric active contour model. In the proposed method, the sparse decomposition method is integrated into ACGS to realize the multiple objects extraction of sequence of image and decrease the influence of nontarget objects. For the multiple evolving curves of a sequence of image, a block low-rank constraint is used to constrain the shape similarity of these evolving contours in every block.

The rest of this paper is organized as follows. Section 2 introduces some knowledge of ACMs. Problem analysis is stated in Section 3. The proposed model is described in Section 4. Section 5 demonstrates the merits of the proposed method by simulations. Finally, Section 6 concludes the paper with some discussions.

#### 2. Related Work

##### 2.1. Contour Evolution Framework Based on Parametric Active Contour Model

Let denote a gray-level image, where is the image domain and is a real number field. Let define a parametric curve in the 2-D plane and is the number of landmarks, is a landmark on the curve. In parametric ACM, a contour is represented by parametric form, and is a curve parameter. Assume is an artificial time, and represents the evolving curve at time . The initial curve imposed by a force function evolves to converge along its normal direction . The corresponding curve evolution equation is usually written as

For parametric ACM, the Lagrange approach [32] is used to get the above evolution equation. By integrating the smooth constraint into parametric ACM, a curve is deformed to the object boundary by minimizing the following energy functional:where and are the first and second derivatives of with respect to parameter and , are the weighted constants. The external energy is derived from the image , which is usually computed as follows:where is a two dimensional Gaussian function with a standard deviation , , is the convolution operation, and is the gradient operator. Minimizing the above energy functional equation (2) with calculus of variations [32], an Euler equation is obtained as follows:

The first two terms are considered as an internal force, and is viewed as an external force imposing on the deforming curve. Many external force fields in parametric ACMs are proposed, such as gradient vector flow (GVF) vector field [33] and its improvements [34–36].

##### 2.2. Active Contour with Group Similarity

According to the above section, the parametric ACM is usually used to extract a single object. When the object is partially occluded or corrupted by noise, the converged result may be undesired. In order to deal with this problem, ACGS [15] is proposed to robustly extract objects in a sequence of images. In ACGS, the shape conformability term for deforming contours is introduced. That is to say, the energy functional keeping the consistence of evolving contours is integrated into the classical energy functional of ACMs. In ACGS, the nuclear norm , i.e., the sum of singular values of , is introduced. The nuclear norm is a continuous and convex function, and some fast algorithms could be utilized. The energy functional of the ACGS model is defined as follows:

For the image sequence , the set is constituted by a group of evolving contours and is the cardinality of . is the energy functional of a general ACM, such as the parametric geodesic active contours (GAC) [9] and C-V model [17], and . The energy functional of the parametric C-V model in ACGS is defined as follows:where and are the regions inside and outside the contour , and are the mean intensities of and respectively, represents the length of contour , is a parameter, and is usually less sensitive to initialization and has fewer parameters to tune, which makes contour evolve to object boundary.

Through a sequence of evolution , the converged contour is viewed as the final result. It is assumed that contour has points and the size of contour is . Thus, the size of is . From equation (5), the shape conformability is kept and the robustness of evolution is improved since the constraint of nuclear norm is imposed. can keep the elements of similar. With the energy functional and the nuclear norm , contours robustly evolve to the object boundary. Based on the gradient descent method, the evolution equation of in ACGS is iterated as follows:where is the *l*-th iterative solution of , is the time step, and is a parameter. From the above equation (7), is determined by and the gradient .

#### 3. Analysis with ACGS

ACGS is the kind of ACM which utilizes the CV model and combines with the constraint of the matrix’s low-rank property. It mainly solves the segmentation problem of single similar target in a sequence of images when the target feature is missing or misleading. The realization process is expressed as follows: first, some discrete points are used to represent the evolving curves in each image. Then, the shape similarity of these curves is described by the nuclear norm. The similar object in the image group is segmented and restored by the relationship between the size of the matrix’s rank and the similarity degree of target shape.

ACGS abandons the way of using level set function to evolve contours. Therefore, the ACGS model has the advantages of small computation and fast convergence. However, there are some problems to be considered:(1)ACGS model introduces the low-rank constraint of the matrix. However, the low-rank property of the evolving contour with lever set method will not be valid. The ACGS model could not handle the change of topology, that is, the contour curve does not automatically split or merge. For multitarget sequence of image, ACGS cannot extract multitarget objects with one initial contour.(2)Although multiple contours can be set to extract multitarget objects, it requires more information for initial contour position of each object. The contour evolution is always influenced by nontarget objects, and evolving contours may remain away from the target objects.(3)ACGS is not robust to noise and nontargets, which greatly affects the smoothness of curve evolution.

#### 4. Active Contour Based on Block Similarity

Aiming at the above problems of the ACGS model, in this paper, an active contour based on block similarity (ACBS) is proposed to extend the ACGS model. Firstly, based on ACGS, a model for multiple object extraction task is constructed, which is changed into some single object extraction tasks with sparse decomposition. Secondly, the block shape similarity for the multiple contours is integrated into the evolution equation. Finally, the converged contour is viewed as the final result.

##### 4.1. The Model of ACBS

Assume that the number of the images in a sequence of image is *m*, i.e., . The number of objects of interest to be extracted from each image is *n*, i.e., . The evolving contours can be expressed as , , and is evolved in the *j*-th image to extract the *i*-th object. From the above definition,where is a block of contours, which includes contours. Every block evolves in the images to extract the corresponding objects which have similar shapes.

According to equation (8), the size of is . With the above definitions, the proposed model is written as follows:where is already defined in equation (5) and is a regularized term of evolving contours, which is used to ensure the shape similarity of these evolving contours in every block .

With an alternating minimization method, the energy functional for the block is computed as follows:

With gradient descent method for equation (10), the following evolution equation with initialization is given as follows:

According the above equation, initial block is evolved to converge to the object :where is the initial contour, which evolves in the *j*-th image to extract the *i*-th object.

According to equation (11), the evolution for every block is the same as the ACGS. The proposed method extends ACGS to the ACBS model for multiobject extraction. However, the difficulty lies in the initialization of every block. For a block, as shown in every column of matric in equation (12), there are initial contours to be set in one image.

##### 4.2. The Evolution with ACBS

From the above ACBS model, initial contours evolve to segment the object targets. The initialization for ACBS is difficult since has contours. It is necessary to set initial contours for each image. For parametric ACM, it is difficult to set initial contours because there are some equilibrium points between objects [36]. By setting one initial contour, multiple objects extraction can be changed into multiple tasks of a single object extraction with sparse representation and decomposition method.

###### 4.2.1. Sparse Decomposition

Contour cannot be split and merged automatically in the parametric ACM, thus the problem of multiobject segmentation cannot be divided. In the literature [37–39], the concept of sparse representation is introduced. Based on a dictionary [38] and sparse decomposition of signals, multiple object segmentation is obtained. The basic idea is the decomposition of signals on the overcomplete dictionary to get a simple representation. Sparse representation model is represented as follows:where *Y* is a signal, *D* is a set of basic functions or a dictionary, and *S* is the coefficient matrix of the dictionary. Solving the above equation with some optimization methods [37] (such as greedy algorithm or convex relaxation method), the recovered signal represents as follows:

In order to explain the sparse decomposition method for object segmentation, the flow of sparse decomposition is described in Figure 1, and the corresponding example of the flow is shown in Figure 2. Based on the principle of sparse decomposition, the regional map (regional binary mask) of objects are firstly computed with a clustering operator. Clustering could group an image into some classes of similar characteristics. A clustering algorithm classifies the image into clusters, and the clusters are pairwise disjoint. The clustering result (cluster number is 3 in the classical fuzzy c-means clustering (FCM) [40]) is shown in the second image of the first row of Figure 2.