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

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