Deep neural networks on diffeomorphism groups for optimal shape reparameterization

TL;DR

This paper introduces a neural network-based algorithm for approximating orientation-preserving diffeomorphisms to improve shape alignment, demonstrating universal approximation capabilities and Lipschitz bounds, applicable to curves and surfaces.

Contribution

It presents a novel PyTorch implementation for constructing diffeomorphisms via composition, with proven approximation properties and bounds, advancing shape analysis techniques.

Findings

Algorithm effectively approximates diffeomorphisms for shape alignment.

Universal approximation properties are established for the proposed architecture.

Lipschitz bounds are derived for the constructed diffeomorphisms.

Abstract

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.