Fast Global Minimization of the Active Contour/Snake Model

Abstract

The active contour/snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. The only drawback of this model is the existence of local minima in the active contour energy, which makes the initial guess critical to get satisfactory results. In this paper, we propose to solve this problem by determining a global minimum of the active contour model. Our approach is based on the unification of image segmentation and image denoising tasks into a global minimization framework. More precisely, we propose to unify three well-known image variational models, namely the snake model, the Rudin-Osher-Fatemi denoising model and the Mumford-Shah segmentation model. We will establish theorems with proofs to determine the existence of a global minimum of the active contour model. From a numerical point of view, we propose a new practical way to solve the active contour propagation problem toward object boundaries through a dual formulation of the minimization problem. The dual formulation, easy to implement, allows us a fast global minimization of the snake energy. It avoids the usual drawback in the level set approach that consists of initializing the active contour in a distance function and re-initializing it periodically during the evolution, which is time-consuming. We apply our segmentation algorithms on synthetic and real-world images, such as texture images and medical images, to emphasize the performances of our model compared with other segmentation models.

Variational Sar Image Segmentation Based on the G0 Model and an Augmented Lagrangian Method

“…Nevertheless, directly using of the gradient descent algorithm can be inefficient due to the non-convex and nondifferentiable property of the energy functional. In this paper, those two problems are tackled by convex relaxing and variable splitting techniques [23,24]. As a result, the optimizing problem becomes a constrained one.…”

“…Furthermore, the non-convexity of the minimizing problem also makes the adoption of efficient convex optimizing algorithms impossible. To overcome this problem, Chan et al [22] and Bresson et al [23] proposed a convex relaxing technique to convert the non-convex minimization problems in the form of (17) into convex ones. The admissible set of the solution is relaxed to [0, 1], which is a convex set.…”

“…However, the speed of the standard level set method is slow, which is a main obstacle for real applications [19]. Recently, there is a trend to develop efficient optimizing methods for variational image processing approaches [20][21][22][23]. Among various methods, the class of finding global minimum of variational models has drawn much attention [22,23].…”

“…Recently, there is a trend to develop efficient optimizing methods for variational image processing approaches [20][21][22][23]. Among various methods, the class of finding global minimum of variational models has drawn much attention [22,23]. As the original energy functionals are generally non-convex and non-differentiable, they are hard to be minimized directly.…”

“…As the original energy functionals are generally non-convex and non-differentiable, they are hard to be minimized directly. In [22,23], the original energy functional is firstly converted to be convex and differentiable. After the converting, matured convex optimizing tools can be used to minimize the converted energy functional efficiently.…”