[Paper Review] Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics

Giacomo Nardi, Gabriel Peyré+1 more|arXiv (Cornell University)|Feb 26, 2014

Advanced Differential Geometry Research4 citations

TL;DR

This paper establishes the existence of minimal geodesics between any two $BV^{2}$-regular planar curves under the $BV^{2}$ Finsler metric using the direct method of calculus of variations. The key contribution is proving geodesic completeness and surjectivity of the exponential map in this infinite-dimensional shape space, with numerical validation showing distinct behaviors between $BV^{2}$ and $H^{2}$ metrics in shape registration.

ABSTRACT

This paper studies the space of $BV^2$ planar curves endowed with the $BV^2$ Finsler metric over its tangent space of displacement vector fields. Such a space is of interest for applications in image processing and computer vision because it enables piecewise regular curves that undergo piecewise regular deformations, such as articulations. The main contribution of this paper is the proof of the existence of the shortest path between any two $BV^2$-curves for this Finsler metric. Such a result is proved by applying the direct method of calculus of variation to minimize the geodesic energy. This method applies more generally to similar cases such as the space of curves with $H^k$ metrics for $k\geq 2$ integer. This space has a strong Riemannian structure and is geodesically complete. Thus, our result shows that the exponential map is surjective, which is complementary to geodesic completeness in infinite dimensions. We propose a finite element discretization of the minimal geodesic problem, and use a gradient descent method to compute a stationary point of the energy. Numerical illustrations show the qualitative difference between $BV^2$ and $H^2$ geodesics.

Motivation & Objective

• To establish the existence of shortest paths (minimal geodesics) between $BV^{2}$-regular planar curves under a Finsler metric.
• To extend the theoretical foundation of shape spaces beyond standard Riemannian metrics, particularly for piecewise-regular deformations.
• To provide a variational framework that supports numerical computation of geodesics in infinite-dimensional spaces of curves.
• To compare the geometric and numerical properties of $BV^{2}$ and $H^{k}$ metrics in shape registration.

Proposed method

• Applies the direct method of calculus of variations to minimize geodesic energy on the space of $BV^{2}$ curves.
• Uses a finite element discretization of the geodesic problem to enable numerical computation.
• Relaxes the non-convex minimization problem to allow gradient descent optimization.
• Employs a Finsler metric based on $BV^{2}$-norms to favor piecewise-regular, possibly rigid, curve evolutions.
• Introduces a reparameterization strategy to ensure convergence of the energy functional under weak topologies.
• Uses martingale arguments to establish lower semicontinuity of the energy, crucial for existence proofs in $BV^{2}$ spaces.

Experimental results

Research questions

• RQ1 Does a minimizing geodesic exist between any two $BV^{2}$-curves under the $BV^{2}$ Finsler metric?
• RQ2 How does the $BV^{2}$ metric compare to $H^{k}$ metrics in terms of geodesic behavior and numerical stability?
• RQ3 Can the exponential map be shown to be surjective in this $BV^{2}$ shape space, implying geodesic completeness?
• RQ4 What role does the choice of initialization play in converging to meaningful geodesics in non-convex optimization?
• RQ5 Can the variational framework be generalized to other Banach space topologies beyond $BV^{2}$ and $H^{k}$?

Key findings

• The existence of a minimizing geodesic between any two $BV^{2}$-curves is rigorously proven using the direct method of calculus of variations.
• The $BV^{2}$ shape space is geodesically complete and the exponential map is surjective, indicating strong geometric structure.
• Numerical experiments show that $BV^{2}$ geodesics preserve piecewise regularity better than $H^{2}$ geodesics, which favor smoother evolutions.
• The choice of initialization significantly affects convergence, with piecewise-affine maps outperforming constant initializations in complex cases.
• The parameter $\mu_2$ in the Finsler energy controls derivative jumps, directly influencing the smoothness of intermediate curves.
• The martingale-based argument successfully avoids reliance on dual space characterization, enabling lower semicontinuity in $BV^{2}$-based geodesic energy.