Wavelet footprints and frames for signal processing and communication

Linear transforms and expansions are fundamental mathematical tools of signal processing. In particular, the wavelet transform has played an important role in several signal processing tasks, compression being a prime example. A signal can be represented in a basis or a frame. Frames, which are an extension of bases to overcomplete sets of vectors, are sometimes preferred to bases because of their greater design freedom and recently they have had an important impact in signal processing. This thesis focuses on the design of new bases and frames for signal processing and communications. The first contribution of the thesis is the exploration of the use of oversampled filter banks, which represent a possible way to implement frames, to robust communications. Normally, transforms are used as a pure source coding method and a reliable communication is obtained with a channel coder which follows the source coder. In this case, we use frames to achieve efficient source compression and robustness to transmission errors at the same time. In the context of pure source transform coding, we investigate the performance of the wavelet transform and study the dependency of the wavelet coefficients across scales. This analysis leads to the design of a new expansion which provides an efficient representation of piecewise smooth signals. We call footprints the elements of this expansion. The main property of footprints is that they efficiently characterize the singular structures of the signal, which usually carry the main information. We show that algorithms based on footprints outperform wavelet methods in different applications such as denoising, compression and deconvolution. Finally, we study a particular source coding technique called multiple description coding. This technique is used for data transmission over unreliable networks. In multiple description coding, the coder generates several different descriptions of the same signal and the decoder can produce a useful reconstruction of the source with any received subset of these descriptions. We study the problem of multiple description coding of stationary Gaussian sources with memory. First, we compute the multiple description rate distortion region for these sources. Then, we develop an algorithm for the design of optimal critically sampled filter banks for multiple description coding of Gaussian sources.