[Paper Review] Manifolds of absolutely continuous curves and the square root velocity framework

Alexander Schmeding|arXiv (Cornell University)|Dec 8, 2016

Morphological variations and asymmetry14 references3 citations

TL;DR

This paper establishes a manifold structure on spaces of absolutely continuous curves with values in strong Riemannian manifolds, generalizing classical results to infinite-dimensional settings. It extends the square root velocity transform (SRVT) framework to manifold-valued curves, proving the SRVT decomposes into smooth (derivative and translation) and continuous (rescaling) components, with full invertibility achieved for Lie group-valued curves via infinite-dimensional Lie theory.

ABSTRACT

A classical result in Riemannian geometry states that the absolutely continuous curves into a (finite-dimensional) Riemannian manifold form an infinite-dimensional manifold. In the present paper this construction and related results are generalised to absolutely continuous curves with values in a strong Riemannian manifolds. As an application we consider extensions of the square root velocity transform (SRVT) framework for shape analysis. Computations in this framework frequently lead to curves which leave the shape space (of smooth curves), and are only contained in a completion. In the vector valued case, this extends the SRVT to a space of absolutely continuous curves. We investigate the situation for shape spaces of manifold valued (absolutely continuous) curves.

Motivation & Objective

• To construct a Banach manifold structure on the space of absolutely continuous curves with values in a strong Riemannian manifold.
• To generalize classical results on manifolds of absolutely continuous curves to infinite-dimensional settings.
• To extend the square root velocity transform (SRVT) framework to shape spaces of manifold-valued absolutely continuous curves.
• To analyze the decomposition of the SRVT into smooth and continuous components in the manifold setting.
• To establish invertibility of the SRVT for Lie group-valued absolutely continuous curves using infinite-dimensional Lie theory.

Proposed method

• Constructs the manifold $AC^p(I,M)$ of absolutely continuous curves with values in a strong Riemannian manifold $M$, modelled on spaces of vector-valued $L^p$-functions.
• Uses a strong Riemannian metric $G$ on $M$ to ensure the topology on tangent spaces coincides with the natural topology, enabling the construction of a Banach manifold structure.
• Introduces $L^p$-bundles over $AC^p(I,M)$ as essential tools for differential geometry in the infinite-dimensional setting.
• Applies local addition and chart techniques to prove the manifold structure, relying on uniform Lipschitz continuity and norm equivalence in local coordinates.
• Analyzes the SRVT as a composition of derivation, translation, and rescaling, proving its decomposition into smooth and continuous parts.
• For Lie group-valued curves, proves the SRVT is a homeomorphism by leveraging recent results in infinite-dimensional Lie theory, enabling pullback of the $L^2$-distance.

Experimental results

Research questions

• RQ1 Can the space of absolutely continuous curves with values in a strong Riemannian manifold be endowed with a Banach manifold structure?
• RQ2 How does the square root velocity transform (SRVT) decompose when extended to manifold-valued absolutely continuous curves?
• RQ3 Is the extended SRVT invertible on the completion of shape spaces for general Riemannian manifolds?
• RQ4 Does the SRVT framework generalize to Lie group-valued curves, and if so, is it a homeomorphism?
• RQ5 What role do strong Riemannian metrics and $L^p$-bundles play in enabling differential geometry on infinite-dimensional manifolds of curves?

Key findings

• The space $AC^p(I,M)$ of absolutely continuous curves with values in a strong Riemannian manifold $M$ forms a Banach manifold modelled on spaces of vector-valued absolutely continuous functions.
• The construction of $AC^p(I,M)$ is independent of the Riemannian metric $G$, relying only on the strong Riemannian structure of $M$.
• The SRVT on manifold-valued absolutely continuous curves splits into a smooth part (derivation and translation) and a continuous part (rescaling), generalizing results from the vector-valued case.
• For Lie group-valued curves, the SRVT is a homeomorphism, allowing the pullback of the $L^2$-distance to define a metric on the completion.
• The $L^p$-bundle over $AC^p(I,M)$ is essential for the differential geometric framework and is constructed using uniform Lipschitz continuity and local chart techniques.
• Invertibility of the extended SRVT on general Riemannian manifolds remains open, though a sketch of a proof is provided in Section 3.7.