The varifold representation of non-oriented shapes for diffeomorphic registration

TL;DR

This paper introduces a varifold-based framework for representing non-oriented shapes in diffeomorphic registration, addressing orientation issues inherent in current-based shape models, and demonstrates its advantages and applications.

Contribution

It develops a novel varifold representation for non-oriented shapes, constructs a Hilbert space structure, and adapts registration algorithms for these shapes within the LDDMM framework.

Findings

Varifold metrics are consistent with shape volume.

Derived formulas for shape variation of the metric.

Successful application to non-oriented shape registration.

Abstract

In this paper, we address the problem of orientation that naturally arises when representing shapes like curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient to model a wide variety of shapes. However, in such approaches, orientation of shapes is a fundamental issue that can lead to several drawbacks in treating certain kind of datasets. More specifically, problems occur with structures like acute pikes because of canceling effects of currents or with data that consists in many disconnected pieces like fiber bundles for which currents require a consistent orientation of all pieces. As a promising alternative to currents, varifolds, introduced in the context of geometric measure theory by F. Almgren, allow the representation of any non-oriented manifold (more generally any non-oriented rectifiable set). In particular, we explain how varifolds can encode numerically non-oriented objects both from the discrete and continuous point of view. We show various ways to build a Hilbert space structure on the set of varifolds based on the theory of reproducing kernels. We show that, unlike the currents' setting, these metrics are consistent with shape volume (theorem 4.1) and we derive a formula for the variation of metric with respect to the shape (theorem 4.2). Finally, we propose a generalization to non-oriented shapes of registration algorithms in the context of Large Deformations Metric Mapping (LDDMM), which we detail with a few examples in the last part of the paper.