Centralized and Distributed Semiparametric Compression of Piecewise Smooth Functions

Abstract

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

Sampling at the rate of innovation: theory and applications

“…Functions satisfying Strang-Fix conditions are extensively used in wavelet theory and the above result provides an intriguing connection between sampling of FRI signals and wavelets. This connection allows to combine FRI theory with wavelets to develop efficient centralized and distributed algorithms for the compression of piecewise smooth functions [38,39]. Finally, it is possible to show that any device whose input and output are related by linear differential equations can be turned into an exponential reproducing kernel and can therefore be used to sample FRI signals [9].…”

“…By modeling the quantization error and any model mismatch as additive noise, one can use the CRB to estimate the performance of this compression strategy. The rate-distortion behavior of this FRI-based algorithm is [38,39]:…”

Generalized methods and solvers for noise removal from piecewise constant signals. II. New methods

“…Here, by PWS, we mean a signal that has a finite isolated set of discontinuities (jumps), and everywhere else the function has one or many continuous derivatives. The PWS noise removal problem has attracted considerable attention, in particular from those applying wavelet analysis in the signal and image processing communities (Chaisinthop and Dragotti, 2009;Mallat, 2009). For PWS signals, the level-set model is no longer parsimonious (but see the stack or threshold decomposition representation that is of central importance to morphological signal processing (Arce, 2005)).…”

Approximate Strang-Fix: sampling infinite streams of Diracs with any kernel

“…4,15 FRI sampling theory has also had impact in specific applications such as image super-resolution, 2 for depth sensing, 13 for calcium transient detection 16 and in compression. 6,11 In this paper, we present a new framework for the sampling of FRI signals using arbitrary kernels and also present a local algorithm for the robust reconstruction of infinite streams of Diracs. The paper is organised as follows: in the next section we provide an overview of the theory of sampling signals with FRI, we present the local reconstruction algorithm in Sec.…”