Landmark shape spaces with induced metrics

2026-07-30

AI Summary

This work addresses the challenge of landmark-based shape modeling by simultaneously preserving invariance under Kendall’s shape space, preventing landmark collisions, and accommodating a variable number of landmarks. To this end, we integrate Kendall’s shape space with the Riemannian structure induced by right-invariant Sobolev metrics on the diffeomorphism group, and introduce a filtered elastic operator whose null space precisely corresponds to rigid motions. This formulation eliminates global rigid transformations and scale while retaining local rigidity, thereby effectively avoiding landmark collisions. Within a Riemannian quotient space framework, we combine numerical geodesic computation with matching algorithms to construct a novel shape space endowed with a regular metric. The resulting framework supports an arbitrary number of landmarks, maintains shape invariance, and enables stable and efficient geodesic and matching computations.

Abstract

We present a unification of Kendall's landmark shape spaces, where rigid motions are factored out and scale fixed on landmark configurations equipped with Euclidean geometry, with landmark configuration spaces carrying Riemannian metrics descending from right-invariant Sobolev metrics on the diffeomorphism group. The resulting new landmark shape spaces achieve the defining properties of both approaches: The regularity of the descending metric prevents landmarks from colliding, the metric is defined in the ambient space independent of the number of landmarks, local rigid transformations are preserved, global rigid motions are removed, and scale fixed. To achieve this, we define a particular Sobolev-type operator, the screened elasticity operator, whose null-space consists exactly of the rigid motions, we show how this operator descends to achieve the desired geometry, and we present approaches to solving matching problems and computing geodesics numerically. The resulting construction allows the use of landmark configuration spaces with sufficiently regular metrics in applications while retaining the shape invariances that are a hallmark of Kendall's shape spaces.

Problem

• landmark shape spaces
• Riemannian metrics
• rigid motions
• Sobolev metrics
• shape invariance

Innovation

• landmark shape spaces
• Sobolev metrics
• screened elasticity operator
• diffeomorphism group
• geodesic computation