Elastic metrics on spaces of Euclidean curves: theory and algorithms. (English) Zbl 07846489
Summary: A main goal in the field of statistical shape analysis is to define computable and informative metrics on spaces of immersed manifolds, such as the space of curves in a Euclidean space. The approach taken in the elastic shape analysis framework is to define such a metric by starting with a reparametrization-invariant Riemannian metric on the space of parametrized shapes and inducing a metric on the quotient by the group of diffeomorphisms. This quotient metric is computed, in practice, by finding a registration of two shapes over the diffeomorphism group. For spaces of Euclidean curves, the initial Riemannian metric is frequently chosen from a two-parameter family of Sobolev metrics, called elastic metrics. Elastic metrics are especially convenient because, for several parameter choices, they are known to be locally isometric to Riemannian metrics for which one is able to solve the geodesic boundary problem explicitly – well-known examples of these local isometries include the complex square root transform of L. Younes et al. [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 19, No. 1, 25–57 (2008; Zbl 1142.58013)] and square root velocity (SRV) transform of A. Srivastava et al. [IEEE transactions on pattern analysis and machine intelligence 33, No. 7, 1415–1428 (2010)]. In this paper, we show that the SRV transform extends to elastic metrics for all choices of parameters, for curves in any dimension, thereby fully generalizing the work of many authors over the past two decades. We give a unified treatment of the elastic metrics: we extend results of A. Trouvé and L. Younes [SIAM J. Control Optim. 39, No. 4, 1112–1135 (2000; Zbl 0983.49009)], M. Bruveris [SIAM J. Math. Anal. 48, No. 6, 4335–4354 (2016; Zbl 1357.58005)] as well as S. Lahiri et al. [Geom. Imaging Comput. 2, No. 3, 133–186 (2015; Zbl 1403.94020)] on the existence of solutions to the registration problem, we develop algorithms for computing distances and geodesics, and we apply these algorithms to metric learning problems, where we learn optimal elastic metric parameters for statistical shape analysis tasks.
MSC:
| 53A04 | Curves in Euclidean and related spaces |
| 53Z50 | Applications of differential geometry to data and computer science |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 68T05 | Learning and adaptive systems in artificial intelligence |
Software:
GitHubReferences:
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