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This research extends Principal Component Analysis to higher-order tensors to generate rotation-invariant shape descriptors, offering a mathematically rigorous method for object recognition and 3D scene understanding without costly rotational optimization. By utilizing central moments up to arbitrary orders, the approach provides decodable shape features suitable for complex applications like molecular modeling and detailed 3D scan analysis. The work targets computer vision practitioners seeking improved accuracy in shape similarity metrics while maintaining computational efficiency.
Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation
arXiv:2601.03326v2 Announce Type: replace-cross 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 maybe also for 3D scene understanding, or shape similarity metric allowing inexpensive comparison of objects modulo rotation avoiding costly optimization over rotations.
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 maybe also for 3D scene understanding, or shape similarity metric allowing inexpensive comparison of objects modulo rotation avoiding costly optimization over rotations.
Computer Vision and Pattern Recognition (cs.CV); Machine Learning (cs.LG)
arXiv:2601.03326 [cs.CV]
(or arXiv:2601.03326v2 [cs.CV] for this version)
https://doi.org/10.48550/arXiv.2601.03326 Focus to learn more arXiv-issued DOI via DataCite
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