Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 343159, 11 pages

http://dx.doi.org/10.1155/2015/343159

## Efficient and Enhanced Diffusion of Vector Field for Active Contour Model

^{1}School of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China^{2}Engineering Lab of Intelligence Business and Internet of Things, Henan 453007, China^{3}Engineering Technology Research Center for Computing Intelligence and Data Mining, Henan 453007, China

Received 21 March 2015; Revised 21 June 2015; Accepted 25 June 2015

Academic Editor: Chih-Cheng Hung

Copyright © 2015 Guoqi Liu et al. 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

Gradient vector flow (GVF) is an important external force field for active contour models. Various vector fields based on GVF have been proposed. However, these vector fields are obtained with many iterations and have difficulty in capturing the whole image area. On the other hand, the ability to converge to deep and complex concavity with these vector fields is also needed to improve. In this paper, by analyzing the diffusion equation of GVF, a normalized set is defined and a dynamically normalized constraint of vector fields is used for efficient diffusion, which makes the edge vector diffusing rapidly to the entire image region. In order to improve the ability to converge to concavity, an enhanced diffusion term is integrated into the original energy functional. With the dynamically normalized constraint and enhanced diffusion term, new vector fields of EDGVF (efficient and enhanced diffusion for GVF) and EDNGVF (efficient and enhanced diffusion of NGVF) are obtained. Experimental results demonstrate that vector fields with proposed method capture the entire image and are obtained with less iterations and computational times. In particular, EDNGVF greatly improves the ability to converge to concavity.

#### 1. Introduction

Active contour model (ACM) is one of the most important methods for boundary extraction. It was firstly presented by Kass et al. [1]. There are two foundational types in active contour model, that is, parametric [1, 2] and geometric active contour models [3]. The former represents curves or surfaces explicitly in their parametric forms, while implicit approaches based on level set [4, 5] are adopted in geometric active contour. Compared with the parametric active contour, geometric active contour has advantages in dealing with complex image content. However, geometric active contour always brings much computational cost since curves in geometric active contour are implicitly represented as a level set of higher dimensional functions. In parametric active contour, curve deforms with the help of internal and external forces. Internal force, which is defined by the curve itself, is used to smooth the curve. External force computing from image data pushes the curves toward target boundary. Thus, external force plays a dominating role in converging to target boundary. In this paper, we focus on the external force fields in parametric active contour model.

In parametric active contour, a curve is represented as . The 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 arc length parameter ; the parameters and are the weight constants. External energy is derived from an image, which is usually computed as follows:where is a two-dimensional Gaussian function with standard deviation ; . is the gray intensity of an image. If an image is binary and the background of the image is zero-valued, external energy is calculated as follows:Minimizing the energy functional of (1) with calculus of variations [6], the Euler equation is obtained:The first two terms are considered as the internal force, and is viewed as the external force imposing on the deforming curve. Most of methods are concerned with the external force fields in parametric active contour models, such as gradient vector flow (GVF) vector field [7] and its improvements [8–15].

In this paper, based on the analysis of GVF, a dynamical constraint for the magnitude of vectors is performed by defining a special set. The unit difference of vector magnitude between the nonzero vectors and zero vectors makes the vectors rapidly diffuse to entire image region. On the other hand, by incorporating an enhanced diffusion term, the “equilibrium problems” [16–19] are greatly decreased.

#### 2. Problem Analysis

GVF has a relative large capture range and can extract some U-shaped concavities compared with original vector field. However, GVF always fails to diffuse to entire image [15] and has difficulty in extracting deep concavity. Various methods modify the energy function of GVF to obtain new vector fields, such as GVF in the normal direction (NGVF). Unfortunately, these vector fields based on diffusion equation always need too many iterations (usually cost more than iterations) and have difficulty in enlarging to entire image. For example, we take GVF. In GVF, vector field is calculated by minimizing the following energy functional:where is the edge mapping which is computed as . When is small, the energy is dominated by sum of the squares of the partial derivatives of vector field , yielding a slowly varying field. When is large, the second term dominates the integrand, and minimization is carried out by setting . is a regularization parameter governing the tradeoff between the first term and the second term. Using the calculus of variations, GVF is obtained by solving the following Euler equation:

In homogeneous region (), the Laplacian operator () makes the vectors smoothly diffuse to image region. The linear diffusion process [20] of vector field in homogeneous region is represented as follows:The solution is given as follows:where is the Gaussian function with the standard deviation . According to (8), vectors near object boundaries have strong magnitudes, while the magnitudes of vectors gradually decrease to zero when moving away from the boundaries. The diffusion becomes weak with the increase of the number of iterations; it is difficult to capture entire image. An example for the capture ranges of GVF is shown on the left hand of Figure 1; it cannot capture entire image even with iterations. Thus, the efficiency of generating vector field is to be improved. In order to show the efficiency of generating GVF, the capture ranges with the different iterations are shown on the left hand of Figure 2. The capture range of GVF with 40 iterations does not have obvious difference compared with the effect by 20 iterations. Furthermore, there are not enough strong forces which push contour to the concavity; even the critical points which stop the contour convergence to the concavity appear near the concavity, as shown in the first row of Figure 3.