Deep neural networks on diffeomorphism groups for optimal shape reparametrization
Peer reviewed, Journal article
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Date
2023Metadata
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Original version
Celledoni, E., Glöckner, H., Riseth, J. N. & Schmeding, A. (2023). Deep neural networks on diffeomorphism groups for optimal shape reparametrization. BIT Numerical Mathematics. 63(4). doi: 10.1007/s10543-023-00989-5Abstract
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.