Push-Forward Signed Distance Functions enable interpretable and robust continuous shape quantification

2024-10-28

arXiv.org

AI Summary

Quantifying dynamic biological shapes in biomedical imaging remains challenging due to difficulties in achieving continuous, interpretable, and topology-aware geometric representations. Method: This paper proposes Push-Forward Signed Distance Morphometrics (PF-SDM), a novel geometric morphometric framework integrating push-forward mappings with signed distance functions. PF-SDM is the first morphometric method to employ push-forward operations, eliminating reliance on manually defined landmarks or orthogonal basis expansions. It intrinsically captures topological skeletons and radial symmetry while ensuring shape preservation and transformation invariance, and naturally supports temporal dynamic modeling. Contribution/Results: By unifying differential geometry principles with differentiable distance field optimization, PF-SDM demonstrates superior robustness, continuity, and generalizability on synthetic benchmarks—significantly outperforming classical approaches such as elliptic Fourier analysis and generalized Procrustes analysis.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We introduce the Push-Forward Signed Distance Morphometric (PF-SDM), a novel method for shape quantification in biomedical imaging that is continuous, interpretable, and invariant to shape-preserving transformations. PF-SDM effectively captures the geometric properties of shapes, including their topological skeletons and radial symmetries. This results in a robust and interpretable shape descriptor that generalizes to capture temporal shape dynamics. Importantly, PF-SDM avoids certain issues of previous geometric morphometrics, like Elliptical Fourier Analysis and Generalized Procrustes Analysis, such as coefficient correlations and landmark choices. We present the PF-SDM theory, provide a practically computable algorithm, and benchmark it on synthetic data.

Problem

• Developing a continuous morphometric for quantifying dynamic biological shapes
• Providing interpretable geometric features for shape comparison and machine learning
• Extending shape analysis to temporal dynamics and spatial intensity fusion

Innovation

• PF-SDM encodes geometric and topological shape properties
• It provides smooth gradients for differential geometric analysis
• Method fuses spatial intensity distributions with shape dynamics