Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation
For computer vision and molecular shape analysis, this provides a method to obtain detailed rotation-invariant shape descriptors, potentially improving object recognition and shape similarity metrics.
The paper extends PCA-like rotation-invariant features to higher-order tensors (e.g., third-order central moments) and polynomial-times-Gaussian descriptors, enabling arbitrarily accurate shape descriptors modulo rotation. This allows for inexpensive rotation-invariant shape comparison without costly optimization.
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.