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 H2-metrics with constant coefficients and scale-invariant H2-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.
Accepté le :
DOI : 10.1051/cocv/2018053
Mots-clés : Shape analysis, curve matching, intrinsic metrics, varifolds
@article{COCV_2019__25__A72_0,
author = {Bauer, Martin and Bruveris, Martins and Charon, Nicolas and M{\o}ller-Andersen, Jakob},
title = {A relaxed approach for curve matching with elastic metrics},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018053},
mrnumber = {4036660},
zbl = {07194611},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018053/}
}
TY - JOUR AU - Bauer, Martin AU - Bruveris, Martins AU - Charon, Nicolas AU - Møller-Andersen, Jakob TI - A relaxed approach for curve matching with elastic metrics JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018053/ DO - 10.1051/cocv/2018053 LA - en ID - COCV_2019__25__A72_0 ER -
%0 Journal Article %A Bauer, Martin %A Bruveris, Martins %A Charon, Nicolas %A Møller-Andersen, Jakob %T A relaxed approach for curve matching with elastic metrics %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018053/ %R 10.1051/cocv/2018053 %G en %F COCV_2019__25__A72_0
Bauer, Martin; Bruveris, Martins; Charon, Nicolas; Møller-Andersen, Jakob. A relaxed approach for curve matching with elastic metrics. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 72. doi : 10.1051/cocv/2018053. http://www.numdam.org/articles/10.1051/cocv/2018053/
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