Primal-Dual convex optimization in large deformation diffeomorphic registration with robust regularizers

Abstract

Non-rigid image registration is a highly ill-posed problem. This means that a number of qualitatively different transformations can achieve the same image similarity after registration. This justifies the vast literature on image registration methods with differences on transformation characterization, regularizers, image similarity metrics, optimization methods, and additional constraints [3]. In the last decade, diffeomorphic registration has arisen as a powerful paradigm for non-rigid image registration, with application to Computational Anatomy. Large Deformation Diffeomorphic MetricMapping (LDDMM) [1] and Diffeomorphic Demons [4] are among the most widespread methods for diffeomorphic registration. In both methods, transformations are characterized to belong to an infinite dimensional Riemannian manifold of diffeomorphisms, parameterized by flows of smooth vector fields in the tangent space. The invertibility of the transformations is numerically guaranteed by the use of sufficiently smooth regularizers. Customarily, these methods regularize the problem with the L-norm of some physically meaningful differential expression of the vector fields, or with the Gaussian smoothing of the vector fields. Simultaneously to the development of the diffeomorphic registration paradigm, the computer vision community has shown a growing interest in robust regularizers based on the Total Variation (TV) norm. The popularity of these regularizers has increased thanks to the availability of optimization methods for solving this challenging problem. The ability of TV based regularizers to preserve discontinuities has led these methods to occupy top positions in optical flow benchmark studies and non-rigid image registration evaluations. The purpose of this article is to propose a method for primal-dual optimization of convexified LDDMM problems, formulated with robust regularizers and image similarity norms related to the TV norm. The method is based on Chambolle and Pock algorithm with diagonal preconditioning [2]. Let Di f f (Ω) be the manifold of diffeomorphisms. Let V be the corresponding tangent space at the identity. Let L= Id − γ∆ be the autoadjoint Laplacian operator associated to the scalar product in V , providing the Riemannian metric in Di f f (Ω). The LDDMM variational problem is given by the minimization of the energy functional