Comparing Trajectories on the Size and Shape Space

TL;DR

This paper emphasizes the importance of proper shape representation in trajectory analysis and introduces a novel Direct Transport method that improves deformation transport accuracy, especially in affine cases.

Contribution

It proposes a new Direct Transport procedure for better deformation transport in shape trajectory analysis, addressing limitations of existing methods.

Findings

Direct Transport perfectly transports deformation in affine cases.

Existing methods do not preserve deformation in affine cases.

Direct Transport outperforms existing tools in non-affine deformation approximation.

Abstract

In this paper we show that trajectory shape analysis should be performed only after obtaining a proper representation before applying ordination methods. In fact, studying the shape of a trajectory means studying how the deformation changes along each path irrespectively of the actual shape to which these deformations apply. The independence of the deformation from the shape to which it is applied is critical: it implies that any shape variation between individuals at the beginning of each trajectories must be completely filtered out. A Parallel Transport, that can be based on various connection types, is necessary to perform such kind of shape data centering. The Levi Civita connection can also be used to transport a deformation. We demonstrate that this procedure does not preserve deformation even in the ane case. We propose a novel procedure called Direct Transport able to perfectly transport deformation in the ane case and to better approximate non ane deformation in comparison to existing tools.