Rigid Motion Invariant Statistical Shape Modeling based on Discrete Fundamental Forms

TL;DR

This paper introduces a motion-invariant statistical shape modeling framework using Lie groups and Riemannian geometry, enabling efficient analysis of large shape populations and improved classification of biological malformations.

Contribution

It presents a novel, alignment-free shape modeling method based on Lie group analysis that handles large deformations efficiently and outperforms existing classifiers in sparse data scenarios.

Findings

Outperforms state-of-the-art classifiers in shape-based classification tasks.

Efficient and robust processing due to explicit Lie group operations.

Effective in analyzing biological shape variability and flattenings.

Abstract

We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. Additionally, as planar configurations form a submanifold in shape space, our representation allows for effective estimation of quasi-isometric surfaces flattenings. We evaluate the performance of our model w.r.t. shape-based classification of hippocampus and femur malformations due to Alzheimer's disease and osteoarthritis, respectively. In particular, we outperform state-of-the-art classifiers based on geometric deep learning as well as statistical shape modeling especially in presence of sparse training data. We evaluate the performance of our model w.r.t. shape-based classification of pathological malformations of the human knee and show that it outperforms the standard Euclidean as well as a recent nonlinear approach especially in presence of sparse training data. To provide insight into the model's ability of capturing biological shape variability, we carry out an analysis of specificity and generalization ability.