A Theory for Multiresolution Signal Decomposition: The Wavelet Representation

Multiresolution representations are very effective for analyzing the information content of images. We study the properties of the operator which approximates a signal at a given resolution. We show that the difference of information between the approximation of a signal at the resolutions 2’ + ’ and 2jcan be extracted by decomposing this signal on a wavelet orthonormal basis of L*(R”). In LL(R ), a wavelet orthonormal basis is a family of functions ( @ w (2’~ n)) ,,,“jEZt, which is built by dilating and translating a unique function t+r (xl. This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror lilters. For images, the wavelet representation differentiates several spatial orientations. We study the application of this representation to data compression in image coding, texture discrimination and fractal analysis.