Elastic Metrics on Spaces of Euclidean Curves: Theory and Algorithms

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 reparameterization-invariant Riemannian metric on the space of parameterized 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 explictly -- well-known examples of these local isometries include the complex square root transform of Younes, Michor, Mumford and Shah and square root velocity (SRV) transform of Srivastava, Klassen, Joshi and Jermyn. 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 Trouv\'{e} and Younes, Bruveris as well as Lahiri, Robinson and Klassen 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

A relaxed approach for curve matching with elastic metrics

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.Comment: 27 page

The Square Root Velocity Framework for Curves in a Homogeneous Space

In this paper we study the shape space of curves with values in a homogeneous space $M = G/K$, where $G$ is a Lie group and $K$ is a compact Lie subgroup. We generalize the square root velocity framework to obtain a reparametrization invariant metric on the space of curves in $M$. By identifying curves in $M$ with their horizontal lifts in $G$, geodesics then can be computed. We can also mod out by reparametrizations and by rigid motions of $M$. In each of these quotient spaces, we can compute Karcher means, geodesics, and perform principal component analysis. We present numerical examples including the analysis of a set of hurricane paths.Comment: To appear in 3rd International Workshop on Diff-CVML Workshop, CVPR 201

Constructing reparametrization invariant metrics on spaces of plane curves

Metrics on shape space are used to describe deformations that take one shape to another, and to determine a distance between them. We study a family of metrics on the space of curves, that includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space $\text{Imm}(S^1,\mathbb R^2)$ of parametrized plane curves and the quotient space $\text{Imm}(S^1,\mathbb R^2)/\text{Diff}(S^1)$ of unparametrized curves. For the space of open parametrized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parametrized and are non-negative on the space of unparametrized open curves. For the metric, which is induced by the "R-transform", we provide a numerical algorithm that computes geodesics between unparameterised, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests that demonstrate it's efficiency and robustness.Comment: 27 pages, 4 figures. Extended versio