A continuous scale space of diffeomorphisms

TL;DR

This paper introduces a multiscale framework for diffeomorphisms using nested reproducing kernel Hilbert spaces, enabling efficient scale-dependent transformations and solutions to the multiscale LDDMM problem with practical numerical experiments.

Contribution

It develops a novel multiscale diffeomorphic registration method based on a family of reproducing kernel Hilbert spaces, with theoretical guarantees and numerical implementation.

Findings

Successful landmark matching with multiscale LDDMM

Efficient kernel evaluations via numerical approximations

Theoretical proof of existence for multiscale LDDMM solutions

Abstract

In this paper, we define and study a nested family of reproducing kernel Hilbert spaces of vector fields that is indexed by a range of scales, from which we construct a reproducing kernel Hilbert space of scale-dependent vector fields. We provide a characterization of the reproducing kernel of that space, with numerical approximations ensuring quick evaluations when this kernel does not have a closed form. We then introduce a multiscale version of the large deformation diffeomorphic metric mapping (LDDMM) problem and prove the existence of solutions. Finally, we provide numerical experiments performing landmark matching using multiscale LDDMM.