The bispectrum as a source of phase-sensitive invariants for Fourier descriptors: a group-theoretic approach

Abstract

This paper develops the theory behind the bispectrum, a concept that is well established in statistical signal processing but not, until recently, extended to computer vision as a source of frequency-domain invariants. Recent papers on using the bispectrum in vision show good results when the bispectrum is applied to spherical harmonic models of three-dimensional (3-D) shapes, in particular by improving discrimination over previously-proposed magnitude invariants, and also by allowing detection of neutral pose in human activity detection. The bispectrum has also been formulated for vector spherical harmonics, which have been used in medical imaging for 3-D anatomical modeling. In a paper published in this journal, Smach {\it et al.} use duality theory to establish the completeness of second-order invariants which, as shown here, are the same as the bispectrum. This paper unifies earlier works of various researchers by deriving the bispectrum formula for all compact groups. It also provides a constructive algorithm for recovering functions from their bispectral values on SO(3). The main theoretical result shows that the bispectrum serves as a complete source of invariants for homogeneous spaces of compact groups, including such important domains as the sphere $S^2$.