Higher-Order Momentum Distributions and Locally Affine LDDMM Registration

TL;DR

This paper introduces higher-order momentum distributions in the LDDMM registration framework, enabling compact, interpretable, and affine-aware deformation representations that improve sparse registration and modeling of complex movements.

Contribution

It presents a novel higher-order momentum approach in LDDMM, allowing local affine transformations and more interpretable, sparse deformation parametrizations.

Findings

Enables registration with fewer parameters.

Provides interpretable local affine deformation descriptions.

Applicable to modeling articulated movement and disease progression.

Abstract

To achieve sparse parametrizations that allows intuitive analysis, we aim to represent deformation with a basis containing interpretable elements, and we wish to use elements that have the description capacity to represent the deformation compactly. To accomplish this, we introduce in this paper higher-order momentum distributions in the LDDMM registration framework. While the zeroth order moments previously used in LDDMM only describe local displacement, the first-order momenta that are proposed here represent a basis that allows local description of affine transformations and subsequent compact description of non-translational movement in a globally non-rigid deformation. The resulting representation contains directly interpretable information from both mathematical and modeling perspectives. We develop the mathematical construction of the registration framework with higher-order momenta, we show the implications for sparse image registration and deformation description, and we provide examples of how the parametrization enables registration with a very low number of parameters. The capacity and interpretability of the parametrization using higher-order momenta lead to natural modeling of articulated movement, and the method promises to be useful for quantifying ventricle expansion and progressing atrophy during Alzheimer's disease.