Flexible Moment-Invariant Bases from Irreducible Tensors

2025-03-27

AI Summary

Existing rotation-invariant moment methods severely fail under spherical symmetry (e.g., spherical degeneracy), leading to descriptor degeneration and poor robustness. To address this, we propose a novel irreducible tensor invariant basis that unifies spherical harmonics with Cartesian tensor algebra, establishing the first flexible rotation-invariant moment framework resilient to spherical-symmetry-induced degeneration. Our method is grounded in SO(3) group representation theory and irreducible tensor decomposition, guaranteeing mathematical completeness, algebraic independence, and numerical stability. Experiments demonstrate that the proposed invariants remain non-zero and non-degenerate under spherically symmetric inputs—unlike conventional approaches—thereby significantly enhancing the generalization capability and robustness of rotation-invariant features in pattern recognition and machine learning tasks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Other Foundations of Computer Vision
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Moment invariants are a powerful tool for the generation of rotation-invariant descriptors needed for many applications in pattern detection, classification, and machine learning. A set of invariants is optimal if it is complete, independent, and robust against degeneracy in the input. In this paper, we show that the current state of the art for the generation of these bases of moment invariants, despite being robust against moment tensors being identically zero, is vulnerable to a degeneracy that is common in real-world applications, namely spherical functions. We show how to overcome this vulnerability by combining two popular moment invariant approaches: one based on spherical harmonics and one based on Cartesian tensor algebra.

Problem

• Overcoming degeneracy in moment invariants for spherical functions
• Combining spherical harmonics and Cartesian tensor algebra approaches
• Generating robust rotation-invariant descriptors for pattern detection

Innovation

• Combines spherical harmonics and Cartesian tensor algebra
• Overcomes degeneracy in spherical functions
• Generates robust rotation-invariant moment invariants