Constructing reparametrization invariant metrics on spaces of plane curves
This work provides theoretical and computational tools for shape analysis, benefiting fields like computer vision and medical imaging by enabling efficient geodesic computation on curve spaces.
The paper studies a family of Sobolev-type Riemannian metrics on spaces of plane curves, providing explicit geodesic distance formulas for open curves and showing vanishing/non-negative sectional curvatures. For closed curves, they develop a numerical algorithm using the RATTLE method to compute geodesics, demonstrating efficiency and robustness.
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.