A theory for multiresolution signal decomposition: the wavelet representation

Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2^j+1 and 2^j (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L²(Rⁿ), the vector space of measurable, square-integrable n-dimensional functions. In L²(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function ψ(x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed