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:
- Frame construction via discretization: Starting from a single minimizer, whose particular parameters may however slowly vary over the signal domain, we will determine a discrete set of operators from this continuous phase space. In this way we obtain a function system consisting of a single waveform and its translated/scaled resp. frequency modulated versions. The storage of such data is no problem and should be well balanced, between unnecessary redundancy and an undersampling regime with deterioriated coefficient locality and potential poorness of information with respect to feature extraction. The construction will be extended to application driven phase space definitions.
- M-frames and multiple window systems: Examples show that some applications require mixed representations using more than one such frames simultanously. It will be necessary to find out in which situations two regimes are used in different regions of the phase space, or perhaps even simultaneously, in a well balanced way (like two instruments playing similar tunes at the same time). Thus in addition to the previous aim multi-generator systems require the handling of overlap and transition regions between different expansions.
- Irregular sampling and dictionaries for sparsity: Our aim is thus to explore both the existing knowledge on the irregular sampling problem and recent results in localization theory as well as experimental evidence about merging regions in order to provide well-founded algorithms for signal expansions or signal reconstruction from a mixed set of scalar products.