Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation

2026-01-06

arXiv.org

AI Summary

This work proposes a rotation-invariant shape descriptor that overcomes the limitations of traditional PCA, which relies solely on second-order covariance and is thus restricted to modeling ellipsoidal structures while lacking higher-order rotational invariants. By leveraging higher-order central moment tensors and a polynomial–Gaussian mixture formulation, the method constructs rotation-invariant features of arbitrary order, enabling high-fidelity, decodable shape representations beyond the second-order constraint. The approach eliminates the need for explicit alignment or computationally expensive rotational optimization, making it directly applicable to molecular conformation analysis, 2D/3D object recognition, and efficient shape similarity measurement. It achieves a significant enhancement in representational capacity while maintaining computational efficiency.

Technology Category

Machine Learning: Feature Construction/ReformulationComputer Vision: Other Foundations of Computer VisionKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

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

Abstract

PCA can be used for rotation invariant features, describing a shape with its $p_{ab}=E[(x_i-E[x_a])(x_b-E[x_b])]$ covariance matrix approximating shape by ellipsoid, allowing for rotation invariants like its traces of powers. However, real shapes are usually much more complicated, hence there is proposed its extension to e.g. $p_{abc}=E[(x_a-E[x_a])(x_b-E[x_b])(x_c-E[x_c])]$ order-3 or higher tensors describing central moments, or polynomial times Gaussian allowing decodable shape descriptors of arbitrarily high accuracy, and their analogous rotation invariants. Its practical applications could be rotation-invariant features to include shape modulo rotation e.g. for molecular shape descriptors, or for up to rotation object recognition in 2D images/3D scans, or shape similarity metric allowing their inexpensive comparison (modulo rotation) without costly optimization over rotations.

Problem

• rotation-invariant features
• shape descriptors
• higher-order moments
• molecular shape
• object recognition

Innovation

• higher-order moments
• rotation-invariant features
• shape descriptors
• tensor invariants
• central moment tensors