A Robust Three-Tier Invariant Representation for 2D Shapes: Enhanced Shape Matching and Analysis Using Manifold Reduction, Eccentricity Transform, and Integral Invariants (Preprint)
Faraz Janan · 2025
BACKGROUND Current methods for analyzing and matching shapes frequently struggle to distinguish subtle structural variations, particularly under conditions involving noise, deformation, or articulations. Existing algorithms often lack robustness and flexibility, relying heavily on local curvature, which may inadequately represent complex structural details essential for precise shape classification and matching. OBJECTIVE To develop a robust and versatile three-tier shape representation pipeline that enhances intra-group similarity and amplifies inter-group differences, thereby providing an invariant representation resilient to noise, articulations, and mechanical deformations. METHODS We propose a novel approach comprising three steps: (1) a manifold-reduction step employing stress minimization to neutralize shape deformations, (2) application of the eccentricity transform (Ecc) to incorporate internal structural information, and (3) integral invariants (II) for robust boundary description. This tripartite framework synergizes differential geometry, topology, and scale-space theory, rigorously evaluated on standard datasets such as the Kimia database. RESULTS Our method significantly outperformed existing shape-matching algorithms, demonstrating notably improved intra-group matching accuracy and effectively enhancing inter-group discrimination. The approach provided substantial resilience against noise, articulations, and bending-induced shape distortions, verified through extensive experimentation and statistical evaluation. CONCLUSIONS The proposed three-tier invariant representation delivers a robust and mathematically sound pipeline suitable for precise shape matching and classification tasks. Its resilience to common shape-analysis challenges makes it highly suitable for practical applications in computational anatomy, biomechanics, medical imaging, and computer-aided geometric design. CLINICALTRIAL Not applicable (omit if required).