[Paper Review] Simplifying transforms for general elastic metrics on the space of plane curves

Sebastian Kurtek, Tom Needham|arXiv (Cornell University)|Mar 29, 2018

Morphological variations and asymmetry32 references11 citations

TL;DR

This paper introduces a generalized isometric transform, $F_{a,b}$, that simplifies any elastic metric on the space of plane curves to the flat $L^2$ metric, enabling efficient computation of geodesics and statistical analysis. The method extends existing transforms like the SRVF to a broader family of metrics and demonstrates improved performance in shape classification using variable stretching and bending weights.

ABSTRACT

In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes and averaging of a collection of shapes. The performance of these algorithms depends heavily on the choice of the Riemannian metric. In the setting of plane curve shapes, attention has largely been focused on a two-parameter family of first order Sobolev metrics, referred to as elastic metrics. They are particularly useful due to the existence of simplifying coordinate transformations for particular parameter values, such as the well-known square-root velocity transform. In this paper, we extend the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat $L^2$ metric. We also extend the transforms to treat piecewise linear curves and demonstrate the existence of optimal matchings over the diffeomorphism group in this setting. We conclude the paper with multiple examples of shape geodesics for open and closed curves. We also show the benefits of our approach in a simple classification experiment.

Motivation & Objective

• To develop a unified framework for simplifying general elastic metrics on the space of plane curves to the flat $L^2$ metric via isometric transforms.
• To extend existing simplifying transforms, such as the SRVF, to a broader two-parameter family of elastic metrics defined by stretching and bending weights $a$ and $b$.
• To establish the existence of optimal matchings over the diffeomorphism group for piecewise linear curves under these generalized metrics.
• To demonstrate the utility of the framework through geodesic computation and classification experiments on real shape data.
• To enable advanced statistical analysis (e.g., Karcher means, PCA, regression) by leveraging the flat geometry of the transformed space.

Proposed method

• Propose a new isometric transform $F_{a,b}$ that maps any elastic metric $g^{a,b}$ on the space of plane curves to the standard $L^2$ metric, preserving geodesic structure.
• Define the transform using a weighted combination of curve velocity and curvature, generalizing the square-root velocity transform (SRVF) to arbitrary $a$ and $b$.
• Extend the framework to piecewise linear curves by proving the existence of optimal reparameterizations (diffeomorphic matchings) under the generalized metric.
• Use the transformed space to compute geodesics between shapes via straight-line paths in the $L^2$-equivalent space, significantly simplifying computation.
• Apply the method to real data, including signature classification, using geodesic distances under varying $a/b$ ratios to assess performance.
• Leverage the isometry to transfer statistical tools (e.g., averaging, PCA) from the flat $L^2$ space back to the original shape space.

Experimental results

Research questions

• RQ1 Can a single isometric transform be constructed to simplify all elastic metrics on the space of plane curves to the $L^2$ metric?
• RQ2 How do different parameter choices for stretching and bending ($a$ and $b$) affect the resulting geodesic paths and shape comparisons?
• RQ3 Does the generalized transform preserve optimal matching properties for piecewise linear curves?
• RQ4 Can the framework improve shape classification performance compared to standard $L^2$ or SRVF-based methods?
• RQ5 How can statistical tools like averaging and PCA be efficiently computed in the context of general elastic metrics?

Key findings

• The proposed $F_{a,b}$ transform provides an isometry that maps any elastic metric $g^{a,b}$ to the flat $L^2$ metric, enabling exact geodesic computation via straight-line paths in the transformed space.
• For the signature classification task, the method achieved a classification rate of 97.50% with 19 perfect matches when $\frac{a}{2b} = 1$ or $2$, outperforming baseline $L^2$ and other parameter choices.
• The choice of $\frac{a}{2b}$ significantly affects performance per signature class, with $\frac{a}{2b} = 1$ and $2$ showing complementary strengths across different classes.
• The transform extends to piecewise linear curves, and the existence of optimal diffeomorphic matchings is established under the generalized metric framework.
• The method enables efficient statistical analysis (e.g., geodesic paths, means) by reducing complex Riemannian geometry to simple $L^2$ operations in the transformed space.
• The framework supports future development of data-driven parameter selection and hierarchical Bayesian models for elastic shape analysis.