[Paper Review] Shape analysis on homogeneous spaces

Elena Celledoni|arXiv (Cornell University)|Jan 1, 2017

Advanced Vision and Imaging2 references3 citations

TL;DR

This paper generalizes the Square Root Velocity Transform (SRVT) from vector spaces to Lie groups and homogeneous manifolds, enabling parametrization-invariant shape analysis on curved, symmetric spaces. The key contribution is a unified framework for shape analysis on non-Euclidean geometries, extending SRVT's utility to complex shape spaces like spheres and rotation groups.

ABSTRACT

Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. Shapes are in fact unparametrized curves, evolving on a vector space, on a Lie group or on a manifold. One popular approach to shape analysis is by the use of the Square Root Velocity Transform (SRVT). In this talk we propose a generalisation of the SRVT from vector spaces to Lie groups and to homogeneous manifolds. This is Joint work with S. Eidnes, A. Schmeding.

Motivation & Objective

• To extend the Square Root Velocity Transform (SRVT) beyond vector spaces to non-Euclidean geometries such as Lie groups and homogeneous manifolds.
• To enable parametrization-invariant comparison of shapes on curved spaces, crucial for applications in medical imaging and computer vision.
• To develop a geometric framework that preserves the desirable properties of SRVT—such as isometry and simplification of geodesic computation—on symmetric, non-flat manifolds.
• To provide a theoretical foundation for shape analysis on manifolds where standard Euclidean SRVT fails due to curvature and non-abelian structure.

Proposed method

• Generalize the SRVT by replacing the standard velocity vector with a right-invariant vector field on a Lie group, using the group's Maurer-Cartan form.
• Define a shape space as the quotient of the space of curves on a Lie group by the reparametrization group, analogous to the Euclidean case.
• Construct a metric on the shape space using the right-invariant Riemannian structure of the Lie group, ensuring isometry with the tangent space of the shape space.
• Use the exponential map and logarithmic map on the Lie group to transport shapes and compute geodesics in the shape space.
• Derive the SRVT on homogeneous manifolds by lifting curves to the principal bundle and applying the group-based SRVT.
• Ensure the transformed space inherits the Riemannian structure of the original manifold, enabling efficient computation of shape distances and geodesics.

Experimental results

Research questions

• RQ1 How can the Square Root Velocity Transform be generalized from vector spaces to Lie groups and homogeneous manifolds?
• RQ2 What geometric structures on Lie groups preserve the isometric and simplifying properties of the classical SRVT?
• RQ3 How does the reparametrization-invariant shape space on a Lie group compare to the standard Euclidean shape space?
• RQ4 What is the role of the Maurer-Cartan form in defining the velocity representation on a Lie group?
• RQ5 Can the SRVT framework be extended to non-Euclidean, symmetric spaces such as spheres or rotation groups?

Key findings

• The generalized SRVT preserves the isometric property of the original transform when mapped to the tangent space of the shape space on a Lie group.
• The shape space on a Lie group inherits a Riemannian structure from the right-invariant metric, enabling geodesic computation via the exponential map.
• The transformation decouples the shape from parametrization by embedding curves into the Lie algebra via the logarithmic map.
• The framework allows for consistent shape comparison on curved manifolds such as SO(3) and spheres, where standard SRVT fails.
• The method enables efficient computation of shape distances and interpolation on homogeneous spaces, crucial for applications in computational anatomy.