Multiresolution representations using the auto-correlation functions of compactly supported wavelets

A shift-invariant multiresolution representation of signals or images using dilations and translations of the autocorrelation functions of compactly supported wavelets is proposed. Although this set of functions does not form an orthonormal basis, a number of properties of the autocorrelation functions of the compactly supported wavelets make them useful for signal and image analysis. Unlike wavelet-based orthonormal representations, the representation has symmetric analyzing functions, shift invariance, natural and simple iterative interpolation schemes, and a simple algorithm for finding the locations of the multiscale edges as zero crossings. A noniterative method is developed for reconstructing signals from their zero crossings (and slopes at these zero crossings) in the representation. This method reduces the problem to that of solving a system of linear equations