Shift- and rotation-invariant object reconstruction using the bispectrum

Triple correlations and their Fourier transforms, called bispectra, have properties desirable for image-sequence analysis and reconstruction. Specifically, the triple correlation of a two-dimensional sequence is shift invariant, vanishes for symmetric probability-distribution-function processes including Gaussian random processes of unknown covariance, and can be used to recover the original sequence uniquely to within a linear phase shift. Discrete analysis is carried out for a deterministic signal in an additive random-noise field. This approach yields discrete algorithms for implementation and permits explicit treatment of the additive noise. Recursive and least-squares fast-Fourier-transform-based algorithms for reconstructing a two-dimensional discrete Fourier transform from the bispectrum are reviewed in detail. While phase retrieval using the leastsquares algorithm requires phase unwrapping, the recursive method requires a more simple correction. The bispectrum is applied to the estimate of a randomly translating or rotating object from a sequence of noisy images. The technique does not require solution of the correspondence problem, which is the primary advantage. The method works in low signal-to-noise-ratio cases, when conventional solutions to estimating the object correspondence may fail. Experimental results presented include application of the method to a sequence of infrared images.