12:30pm
EQuad E415
Title: Nonlinear Multiresolution Signal Analysis: From Morphological Pyramids to Wavelets.
Abstract: Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the so-called pyramid decomposition scheme. We propose a general axiomatic pyramid decomposition scheme for signal analysis and synthesis. This scheme comprises the following ingredients:
In its original form, the wavelet transform is a linear tool. However, it has been increasingly recognized that nonlinear extensions are possible. A major impulse to the development of nonlinear wavelet transforms has been given by the introduction of the lifting scheme by Sweldens. We present an axiomatic framework, encompassing most existing linear and nonlinear wavelet decompositions, which introduces some, thus far unknown, wavelets based on mathematical morphology, such as the morphological Haar wavelet, both in one and two dimensions. A general and flexible approach for the construction of nonlinear (morphological) wavelets is provided by the lifting scheme. We discuss one example in considerable detail, the max-lifting scheme which has the intriguing property that it preserves local maxima in a signal over a range of scales, depending on how local or global these maxima are.