This thesis introduces novel wavelet-based semi-parametric centralized and distributed compression methods for a class of piecewise smooth functions. Our proposed compression schemes are based on a non-conventional transform coding structure with simple independent encoders and a complex joint decoder.Current centralized state-of-the-art compression schemes are based on the conventional structure where an encoder is relatively complex and nonlinear. In addition, the setting usually allows the encoder to observe the entire source. Recently, there has been an increasing need for compression schemes where the encoder is lower in complexity and, instead, the decoder has to handle more computationally intensive tasks. Furthermore, the setup may involve multiple encoders, where each one can only partially observe the source. Such scenario is often referred to as distributed source coding.In the first part, we focus on the dual situation of the centralized compression where the encoder is linear and the decoder is nonlinear. Our analysis is centered around a class of 1-D piecewise smooth functions. We show that, by incorporating parametric estimation into the decoding procedure, it is possible to achieve the same distortionrate performance as that of a conventional wavelet-based compression scheme. We also present a new constructive approach to parametric estimation based on the sampling results of signals with finite rate of innovation.The second part of the thesis focuses on the distributed compression scenario, where each independent encoder partially observes the 1-D piecewise smooth function. We propose a new wavelet-based distributed compression scheme that uses parametric estiiii mation to perform joint decoding. Our distortion-rate analysis shows that it is possible for the proposed scheme to achieve that same compression performance as that of a joint encoding scheme.Lastly, we apply the proposed theoretical framework in the context of distributed image and video compression. We start by considering a simplified model of the video signal and show that we can achieve distortion-rate performance close to that of a joint encoding scheme. We then present practical compression schemes for real world signals.Our simulations confirm the improvement in performance over classical schemes, both in terms of the PSNR and the visual quality.iv

This paper proposes a new approach to distributed video coding. Distributed video coding is a new paradigm in video coding, which is based on the concept of decoding with side information at the decoder. Such a coding scheme employs a low-complexity encoder, making it well suited for low-power devices such as mobile video cameras.The uniqueness of our work lies in the combined use of discrete wavelet transform (DWT) and the concept of sampling of signals with …nite rate of innovation (FRI).1 This enables the decoder to retrieve the motion parameters and reconstruct the video sequence from the low-resolution version of each transmitted frame. Unlike the currently existing practical coders, we do not employ traditional channel coding technique. For a simple video sequence with a …xed background, our preliminary results show that the proposed coding scheme can achieve a better PSNR than JPEG2000-intraframe coding at low bit rates.

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