Smoothing penalty methods for Euler's elastica image processing and discretization error analysis

Abstract

Euler's elastica model has been widely used in image processing and computer vision. It consists of a regularization term capturing the geometrical feature of the image and a fidelity term enforcing the priori information to be preserved. Since the curvature term in the objective function makes it a challenging nonconvex and nonsmooth optimization model, most existing algorithms do not have convergence theory for it. On the other hand, whether the discrete Euler's elastica model converges to the continuous one is an interesting issue, which has not been stressed in the existing works. This thesis is divided into two parts. In the first part, we propose a penalty relaxation algorithm with mathematical guarantee to find a stationary point of discretized Euler's elastica model. To deal with the nonsmoothness of Euler's elastica model, we first introduce a smoothing relaxation problem, and then propose an exact penalty method to solve it. We present first order optimality conditions and establish the relationships between Euler's elastica model, the smoothing relaxation problem and the penalty problem in theory regarding optimal solutions and stationary points. Moreover, we propose an efficient block coordinate descent algorithm to solve the penalty problem by taking advantages of convexity of its subproblems. We prove global convergence of the algorithm to a stationary point of the penalty problem. Finally, we apply the proposed algorithm to denoise the optical coherence tomography images with real data from an optometry clinic and show the efficiency of the proposed method for image processing using Euler's elastica model. In the second part, we consider a discrete form of Euler's elastica model by using central difference discretization and establish a relationship between optimal values of the continuous Euler's elastica model and the discrete Euler's elastica model. We use extension, injection and projection operators to build the connection between the variables in the continuous space and discrete space. Fundamental properties of operators and minimizers of continuous and discrete models are studied under some assumptions. Based on these properties, we derive that the optimal value of the discrete model is less than the optimal value of the continuous model plus some error.