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In this paper, we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of H 2 -metrics with constant coefficients and scale-invariant H 2 -metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.
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Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation
This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curve modulo shape-preserving transformations. We summarize the general construction and theoretical properties of quotient elastic metrics for Euclidean as well as non-Euclidean curves before considering the special case of the square root velocity metric for which the expression of the resulting distance simplifies through a particular transformation. We then examine the different numerical approaches that have been proposed to estimate such distances in practice and in particular to quotient out curve reparametrization in the resulting minimization problems.
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- PAR ID:
- 10645637
- Publisher / Repository:
- Springer International Publishing
- Date Published:
- Page Range / eLocation ID:
- 1 to 35
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Book Chapter:
- https://doi.org/10.1007/978-3-030-03009-4_87-1
