Deep neural networks on diffeomorphism groups for optimal shape reparameterization
This addresses a computationally demanding task in shape analysis for researchers and practitioners, though it appears incremental as it builds on existing optimization methods.
The paper tackles the problem of aligning curves or surfaces in shape analysis by approximating orientation-preserving diffeomorphisms through compositions of elementary ones, implemented in PyTorch, and demonstrates universal approximation properties with bounds on Lipschitz constants.
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.