Mixture of Kernels and Iterated Semidirect Product of Diffeomorphisms Groups

TL;DR

This paper develops a multi-scale variational framework for diffeomorphisms within LDDMM, generalizes semidirect product representations to multiple scales, and demonstrates their application on images, establishing equivalence with mixture of kernels methods.

Contribution

It introduces a detailed multi-scale variational approach for diffeomorphisms, extending semidirect product representations to multiple scales and linking them with mixture of kernels.

Findings

Multi-scale diffeomorphic decomposition demonstrated on synthetic images.

The approach generalizes existing semidirect product representations.

Equivalence established between multi-scale methods and mixture of kernels.

Abstract

In the framework of large deformation diffeomorphic metric mapping (LDDMM), we develop a multi-scale theory for the diffeomorphism group based on previous works. The purpose of the paper is (1) to develop in details a variational approach for multi-scale analysis of diffeomorphisms, (2) to generalise to several scales the semidirect product representation and (3) to illustrate the resulting diffeomorphic decomposition on synthetic and real images. We also show that the approaches presented in other papers and the mixture of kernels are equivalent.