Frames and Signal Expansions

Many important function systems used in signal/image processing start from basic atoms, (mother wavelets, Gaussians) and obtain a complete functions system (a bases, a discrete or continuous frame, or a dictionary) by applying a set of elementary modifications, such as shift, scaling, frequency shift, rotation, shears to these basic waveforms. Group theoretical methods provide a solid mathematical basis for the associated transforms, e.g. wavelet or windowed Fourier transforms. Efficient algorithms useful for coding/decoding and for signal analysis (denosing, pattern matching, compression etc.) can be built on this basis. Often the Fourier transform, realized via the FFT is at its core. Signal expansions relying on alternative optimization principles are likely to allow the use of similar principles. Recent advances on localization and the raising theory of fusion frames will allow to obtain a similar situation (from theory to efficient signal expansions) in the situations where optimal parameters are slowly varying, and expansions adapt well to the given data, in order to bring up application dependent features in a very flexible way.

We have the following aims: