Approximating the Weil-Petersson Metric Geodesics on the Universal Teichmüller space by Singular Solutions

Abstract: We propose and investigate a numerical shooting method for computing geodesics in the Weil-Petersson ($WP$) metric on the universal Teichmüller space T(1). This space, or rather the coset subspace $\PSL_2(\R)\backslash\Diff(S^1)$, has another realization as the space of smooth, simple closed planar curves modulo translations and scalings. This alternate identification of T(1) is a convenient metrization of the space of shapes and provides an immediate application for our algorithm in computer vision. The geodesic equation on T(1) with the $WP$ metric is EPDiff($S^1$), the Euler-Poincare equation on the group of diffeomorphisms of the circle $S^1$, and admits a class of soliton-like solutions named Teichons. Our method relies on approximating the geodesic with these teichon solutions, which have momenta given by a finite linear combination of delta functions. The geodesic equation for this simpler set of solutions is more tractable from the numerical point of view. With a robust numerical integration of this equation, we formulate a shooting method utilizing a cross-ratio matching term. Several examples of geodesics in the space of shapes are demonstrated.

• The paper demonstrates that representing the momentum as a sum of weighted Dirac deltas (teichons) reduces the infinite-dimensional EPDiff equation to a tractable finite system.
• A shooting method combined with cross-ratio-based matching functionals is used to handle Möbius invariance and accurately evolve geodesics on T(1).
• Empirical results validate the approach on synthetic and real-world shapes, while also highlighting limitations due to numerical crowding in highly singular configurations.

Approximate Weil-Petersson Geodesics on Universal Teichmüller Space via Teichons

Introduction

The paper "Approximating the Weil-Petersson Metric Geodesics on the Universal Teichmüller space by Singular Solutions" (1208.2022) investigates the numerical computation of geodesics in the infinite-dimensional universal Teichmüller space $T(1)$, endowed with the Weil-Petersson (WP) metric. The coset description $PSL_2(\mathbb{R}) \backslash \text{Diff}(S^1)$, representing smooth simple closed planar curves up to translations and scalings, provides both a rich mathematical structure and a foundation for relevant applications in shape analysis, computational anatomy, and computer vision.

The central focus is an ansatz-based reduction of the EPDiff($S^1$) equation—originally an infinite-dimensional PDE for geodesics in this metric—to a finite-dimensional system by introducing singular solutions termed "teichons." This reduction enables tractable, robust, and efficient computation of geodesics. The paper systematically develops a numerical shooting method based on this reduction, together with cross-ratio-based matching functionals to overcome the invariance under Möbius group action.

Mathematical Foundations

The WP metric, with negative sectional curvature and strong existence-uniqueness properties for geodesics, provides a powerful tool for shape comparison and statistical analysis. On the Lie algebra level, the WP norm is associated with a non-local, non-diagonal (integro-differential) operator with a null space corresponding to infinitesimal Möbius transformations. The metric thus descends naturally to the coset space $PSL_2(\mathbb{R}) \backslash \text{Diff}(S^1)$, aligning the geometry of $T(1)$ with the structure of complex shapes up to translation and scaling.

Crucially, the geodesic equation on $T(1)$ under the WP metric is an EPDiff equation, analogous to the Camassa-Holm and KdV equations but not known to be completely integrable. However, in the spirit of soliton dynamics, the EPDiff admits "teichon" solutions: momenta supported on Dirac distributions yield strongly localized, finite-dimensional geodesic flows. The resulting finite system preserves the action on the “horizontal” space, orthogonal to Möbius directions.

Teichon Reduction and Numerical Strategy

The $N$-teichon ansatz posits the momentum as a sum of $N$ weighted Dirac deltas. The associated velocity field is constructed by convolution with the explicitly characterized Green’s function of the WP operator, projected away from the Möbius directions. Substituting this ansatz into the EPDiff equation yields a non-linear finite-dimensional system of ODEs for teichon positions and momenta, subject to linear constraints to enforce horizontality.

To connect two shapes—or, more accurately, their footprints in $T(1)$—a shooting method is employed: initial teichon parameters are iteratively adjusted so that the evolved configuration, after integrative forward flow, matches the target up to Möbius indeterminacy. The paper constructs a robust matching functional based on cross-ratios of landmark sets, leveraging their Möbius invariance to enforce consistency in the quotient geometry.

Notably, the cross-ratio-based comparators are derived from the Delaunay triangulation of target landmark sets. This encoding provides a practical, projectively-invariant method to drive optimization in the shooting process. Constraints on the teichon momenta, dictated by the horizontal subspace, are incorporated via appropriate projection of gradients, and the optimization exploits natural gradient and conjugate gradient techniques on Riemannian manifolds for accelerated convergence.

Implementation and Numerical Results

The algorithm is applied to both synthetic and real-world shape datasets. The authors demonstrate the ability to compute WP-geodesics for a range of shapes, including ellipses (with aspect ratios up to six), biological contours (hippocampus slices), and generic planar shapes from the MPEG-7 database. Empirical comparisons to prior methods—specifically energy-minimization-based approaches—indicate near-coincidence in geodesic lengths and angle sums in non-crowded cases, confirming the accuracy of the teichon-based algorithm.

A series of numerical experiments validate:

• Linear scaling of WP distances with ellipse aspect ratios up to the crowding threshold, and precise agreement with previous benchmark algorithms.
• Empirical verification of negative curvature on $T(1)$: triangle angle sums fall strictly below $\pi$, matching theoretical predictions for non-positive curvature.
• Practical success in handling complex biological contours, with landmark flows and geodesic evolutions stably computed as long as the crowding condition is not pathological.

The main limitation is the "crowding" phenomenon: as evolved landmarks or teichons become too closely spaced (order $\sqrt{\epsilon_\text{mach}}$), numerical integration, gradient calculation, and optimization inevitably collapse due to floating-point degeneracy of the Gram matrix from the WP Green’s function. Shapes with sharp, highly localized features or elongated filaments violate these separation bounds, at which point the method ceases to provide meaningful results.

Theoretical and Practical Implications

The approach detailed in the paper provides one of the first reliable, efficient, and theoretically principled mechanisms for computing WP-geodesics in $T(1)$ via a finite-dimensional reduction. The adoption of teichon soliton solutions, combined with Möbius-invariant matching criteria, anchors geodesic shooting in a firm geometrical and analytical framework with immediate practical significance for computational anatomy, latent shape statistics, and computer vision scenarios requiring consistent shape interpolation and comparison.

The uniqueness of geodesics, ensured by negative curvature, allows for well-posed statistical constructs (e.g., the Karcher mean and tangent space analysis). This, in turn, sets the stage for large-scale applications in medical imaging and pattern theory, where quantification and comparison of anatomical variation are central.

The teichon method outperforms historical energy minimization algorithms in terms of convergence and accuracy in non-crowded cases and is competitive with recent alternatives based on other minimization schemes. Its shooting structure guarantees an exact (up to integration error) geodesic, rather than an inexact, energy-minimizing path.

Prospects and Limitations

While robust in standard shape analysis workflows, the algorithm's primary limitation arises from crowding: the inability, due to numerical singularity, to resolve shapes with highly compressed or elongated zones on the unit circle. Overcoming this limitation may require adaptive representation, higher-precision arithmetic, or fundamentally different regularization strategies. The authors suggest that related techniques from advanced conformal mapping theory could be integrated for amelioration, but this is left for future work.

Potential future directions include:

• Exploiting the teichon machinery for large-scale statistical shape analysis and database mining, leveraging uniqueness and tangent-space linearization for constructing means and performing inference on populations of shapes.
• Augmenting the teichon ansatz to incorporate adaptivity or sparsity, possibly via multi-scale or hierarchical construction, to alleviate difficulties posed by crowding and to resolve more singular features.
• Transfer of the methodology to related infinite-dimensional homogeneous spaces, and extension to quotient geometries encountered in higher-genus Teichmüller theory.

Conclusion

The paper delivers an analytically grounded, numerically effective strategy for generating WP-geodesics on the universal Teichmüller space by representing the flow with singular teichon solutions. The marriage of finite-dimensional ODE-based evolution and Möbius-invariant matching establishes a solid computational foundation for high-dimensional shape analysis, within the theoretical constraints posed by numerical crowding. While significant challenges remain for pathological (crowded) cases, the approach substantially advances the state of the art for applications demanding geodesic paths and statistics on shape spaces modeled by $T(1)$ (1208.2022).