MULTIRESOLUTION REPRESENTATIONS USING THE AUTOCORRELATION FUNCTIONS OF COMPACTLY SUPPORTED WAVELETS

We propose a shift-invariant multiresolution representation of signals or images using dilations and translations of the autocorrelation functions of compactly supported wavelets. Although these functions do not form an orthonormal basis, their properties make them useful for signal and image analysis. Unlike wavelet-based orthonormal representations, our representation has 1) symmetric analyzing functions, 2) shift-invariance, 3) associated iterative interpolation schemes, and 4) a simple algorithm for finding the locations of the multiscale edges as zero-crossings. We also develop a noniterative method for reconstructing signals from their zero-crossings (and slopes at these zero-crossings) in our representation. This method reduces the reconstruction problem to that of solving a system of linear algebraic equations.