Unitary representations of groups and the implications for wavelet analysis.

The purpose of this line of research is to use powerful tools from an abstract area of mathematics (the theory of unitary representations of locally compact groups) to discover novel methods for efficiently analyzing, storing and manipulating multidimensional signals. An audio recording is a good example of a one dimensional signal while an image, such as a photograph or CT scan is a good example of a two dimensional signal. In the last 30 years, there has been a major revolution in signal and image processing with the introduction and development of a family of techniques collectively called "wavelet analysis". The impact has been profound. From digitized music now universally compressed in MP3 or a similar format to functional MRI (magnetic resonance imaging) to digitized finger prints in a searchable database, all of our lives have felt this revolution in signal processing. Many of the algorithms which execute the signal processing are based on properties of unitary representations. For example, to analyze an image one moves a small, easily managed piece of image (the wavelet) around using affine motions. These are combinations of translations, rotations, flips, stretches and shears. All these motions form what is called the affine group. When these motions act on the collective of all possible 2-dimensional signals (images), the result is called a (unitary) representation of the affine group. The abstract theory of unitary representations guides us to select particular smaller collections of affine motions to use for efficient analysis. The recently developed shearlet transform which is effective in detecting curved edges in images arose in exactly this fashion. We have just succeeded in introducing crystal symmetry groups into multi-dimensional signal analysis. Moreover, we have a completely novel method for four dimensional (think 3D in motion) signals. Work now needs to be done to use this basis to develop methods to take advantage of inherent features of multi-dimensional signals for efficient analysis and storage. Our team will play a part in that development, guiding the way through the systematic design of the blueprints (underlying theory).