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

TL;DR

This paper introduces higher-order PCA-like rotation-invariant shape descriptors using tensors of moments, enabling more detailed and accurate shape analysis for applications like object recognition and molecular shape comparison.

Contribution

It extends traditional covariance-based shape features to higher-order tensors, providing a method for rotation-invariant shape descriptors with arbitrarily high accuracy.

Findings

Proposes higher-order moment tensors for shape description.

Defines rotation invariants for these higher-order tensors.

Potential applications in object recognition and molecular shape analysis.

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.