Time-Frequency Brown Bag Seminar

Wednesday, May 12, 1999

12:30pm

EQuad E415

Speaker: John Goutsias, Center for Imaging Science and Department of Electrical and Computer Engineering, The Johns Hopkins University

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:
  1. The pyramid consists of a (finite or infinite) number of levels such that the information content decreases towards higher levels;
  2. Each step towards a higher level is constituted by an (information-reducing) analysis operator, whereas each step towards a lower level is modeled by an (information-preserving) synthesis operator.
One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. Several examples are described of nonlinear (e.g., morphological) pyramid decomposition schemes.

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.


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