RUI: Development of Frame Extensions and their Applications, II

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A continuing project relating to applications of pseudoframes for subspaces (PFFS) is proposed. Pseudoframes are frame-like systems for the linear analysis and synthesis of a subspace X. They differ from frames in that both analysis and synthesis sequences are not necessarily contained in X. Theme 1 of the project studies constructions of biorthogonal wavelets (via PFFS) that incorporate the regularity requirement and the perfect reconstruction principle with practical optimal design criteria. Theme 2 studies the application of the geometrical properties of PFFS in simultaneous noise removal and signal reconstruction. PFFS generate non-orthogonal projections onto X along any complementary subspace of X. This enables a technique of noise removal by having the null space of the projection to contain the noise subspace. Studies in theme 3 will be on the applications of PFFS to sampling and linear inverse problems in noisy environments. Given a convolution or (non-ideal) sampling operation, one can apply PFFS to specify an available reconstruction subspace X for the best (reconstruction) result. We will study examples of how accessible reconstruction subspaces X should be selected and how reconstruction or deconvolution is carried out via PFFS. Performance of these techniques will also be studied.All signal analysis systems break a signal into components. The principal novelty that we introduce to signal analysis is the idea of PFFS systems. PFFS is a tool to analyze a subspace X while standing outside of X. In doing so, one is given the freedom and additional "room" to perform signal decomposition and analysis in X by using advanced means not available in X. Consequently, PFFS systems are more flexible than most others in that they can be designed to produce the most useful components for denoising and signal reconstruction purposes. This study will develop multiple PFFS systems designed to solve a number of different signal analysis problems in broadband communications, target recognition, and signal/image restoration/de-blurring, etc. In particular, we hope to advance the art of image compression to achieve high compression rate with the least artificial distortions such as check-boarding effect (showing checkerboards on the decompressed image) and ringing effect (showing blurring on the edges). Similarly, we expect to develop tools for improving speech signal compression by reducing leakage from one frequency band to another. More generally, the methods we will develop can be tailored for noise removal and accurate reconstruction of digital signals in a number of different signal processing environments.