We present some preliminary results of constructions of biorthogonal wavelets and associated filterbanks with optimality using a tool of pseudoframes for subspaces (PFFS). PFFS extends the theory of frames in that pseudoframe sequences need not reside within the subspace of interests. The flexibility so introduced proves favorably in the construction of biorthogonal wavelets. While past constructions pioneered by Cohen, Daubechies, and Feauveau can be reproduced precisely, results of additional optimalities are also obtained. Some preliminary examples are reported.
A new parameterization approach for the construction of biorthogonal wavelets has revealed the possibility of deriving optimized biorthogonal filterbanks. Such a design methodology incorporates various optimization criteria and the regularity requirement into a single procedure while maintaining the usual perfect reconstruction principle. Because of the parameterization, the additional design procedure becomes fundamentally an unconstrained optimization problem. The method provides examples of new biorthogonal dual filters of slightly shorter length with comparable performance characteristics to that of traditional biorthogonal dual filters of longer length. Image coding using such new filterbanks becomes more efficient while maintaining similar quality. Preliminary studies in image coding will be reported in the presentation.
We present a non-uniform multi-Gabor expansion (MGE) algorithm in which multiple windows and their corresponding translations and modulations are used for signal analysis. Analysis windows of varying supports, each corresponding to a chosen scale of detail, decompose a signal using different translation and modulation parameters for each scale. Structural analysis of dual synthesis waveforms is provided. We show that synthesis waveforms can still be formed from translations and modulations of a set of basic dual windows at each scale. The use of different translation and modulation parameters at each scale allows for a more refined and concise time-frequency representation of the signal, while retaining the versatility of uniform MGEs in representing signal dynamics at many different scales. Examples of implementation and practical application will be presented.