Diffeomorphic registration of discrete geometric distributions

TL;DR

This paper introduces a novel diffeomorphic registration framework for discrete geometric distributions, enabling more flexible and efficient matching of complex shapes and multi-modal images using direct deformation models.

Contribution

It develops direct deformation models and algorithms for discrete varifolds, extending shape registration to multi-directional and multi-modal geometric data.

Findings

Provides a new numerical setting for curve and surface matching.

Efficiently handles multi-directional and multi-modal shape structures.

Extends diffeomorphic registration to a broader class of geometric objects.

Abstract

This paper proposes a new framework and algorithms to address the problem of diffeomorphic registration on a general class of geometric objects that can be described as discrete distributions of local direction vectors. It builds on both the large deformation diffeomorphic metric mapping (LDDMM) model and the concept of oriented varifolds introduced in previous works like [Kaltenmark 2017]. Unlike previous approaches in which varifold representations are only used as surrogates to define and evaluate fidelity terms, the specificity of this paper is to derive direct deformation models and corresponding matching algorithms for discrete varifolds. We show that it gives on the one hand an alternative numerical setting for curve and surface matching but that it can also handle efficiently more general shape structures, including multi-directional objects or multi-modal images represented as distributions of unit gradient vectors.