Over-complete transforms have recently become the focus of a wide wealth of research in signal processing, machine learning, statistics and related fields. Their great modelling flexibility allows to find sparse representations and approximations of data that in turn prove to be very efficient in a wide range of applications. Sparse models express signals as linear combinations of a few basis functions called atoms taken from a so-called dictionary. Finding the optimal dictionary from a set of training signals of a given class is the objective of dictionary learning and the main focus of this thesis. The experimental evidence presented here focuses on the processing of audio signals, and the role of sparse algorithms in audio applications is accordingly highlighted. The first main contribution of this thesis is the development of a pitch-synchronous transform where the frame-by-frame analysis of audio data ...

Sparse representations are intensively used in signal processing applications, like image coding, denoising, echo channels modeling, compression, classification and many others. Recent research has shown encouraging results when the sparse signals are created through the use of a learned dictionary. The current study focuses on finding new methods and algorithms, that have a parallel form where possible, for obtaining sparse representations of signals with improved dictionaries that lead to better performance in both representation error and execution time. We attack the general dictionary learning problem by first investigating and proposing new solutions for sparse representation stage and then moving on to the dictionary update stage where we propose a new parallel update strategy. Lastly, we study the effect of the representation algorithms on the dictionary update method. We also researched dictionary learning solutions where the dictionary has a specific form. ...

This thesis deals with developing improved modeling methods for speech and audio processing based on the recent developments in sparse signal representation. In particular, this work is motivated by the need to address some of the limitations of the well-known linear prediction (LP) based all-pole models currently applied in many modern speech and audio processing systems. In the first part of this thesis, we introduce \emph{Sparse Linear Prediction}, a set of speech processing tools created by introducing sparsity constraints into the LP framework. This approach defines predictors that look for a sparse residual rather than a minimum variance one, with direct applications to coding but also consistent with the speech production model of voiced speech, where the excitation of the all-pole filter is model as an impulse train. Introducing sparsity in the LP framework, will also bring to develop the ...

During the last decades, information is being gathered and processed at an explosive rate. This fact gives rise to a very important issue, that is, how to effectively and precisely describe the information content of a given source signal or an ensemble of source signals, such that it can be stored, processed or transmitted by taking into consideration the limitations and capabilities of the several digital devices. One of the fundamental principles of signal processing for decades is the Nyquist-Shannon sampling theorem, which states that the minimum number of samples needed to reconstruct a signal without error is dictated by its bandwidth. However, there are many cases in our everyday life in which sampling at the Nyquist rate results in too many data and thus, demanding an increased processing power, as well as storage requirements. A mathematical theory that emerged ...

Video signals are sequences of natural images, where images are often modeled as piecewise-smooth signals. Hence, video can be seen as a 3D piecewise-smooth signal made of piecewise-smooth regions that move through time. Based on the piecewise-smooth model and on related theoretical work on rate-distortion performance of wavelet and oracle based coding schemes, one can better analyze the appropriate coding strategies that adaptive video codecs need to implement in order to be efficient. Efficient video representations for coding purposes require the use of adaptive signal decompositions able to capture appropriately the structure and redundancy appearing in video signals. Adaptivity needs to be such that it allows for proper modeling of signals in order to represent these with the lowest possible coding cost. Video is a very structured signal with high geometric content. This includes temporal geometry (normally represented by motion ...

This thesis presents novel exemplar-based texture synthesis methods for image prediction (i.e., predictive coding) and image inpainting problems. The main contributions of this study can also be seen as extensions to simple template matching, however the texture synthesis problem here is well-formulated in an optimization framework with different constraints. The image prediction problem has first been put into sparse representations framework by approximating the template with a sparsity constraint. The proposed sparse prediction method with locally and adaptive dictionaries has been shown to give better performance when compared to static waveform (such as DCT) dictionaries, and also to the template matching method. The image prediction problem has later been placed into an online dictionary learning framework by adapting conventional dictionary learning approaches for image prediction. The experimental observations show a better performance when compared to H.264/AVC intra and sparse prediction. ...

This thesis is based on nine papers, all concerned with parameter estimation. The thesis aims at solving problems related to real-world applications such as spectroscopy, DNA sequencing, and audio processing, using sparse modeling heuristics. For the problems considered in this thesis, one is not only concerned with finding the parameters in the signal model, but also to determine the number of signal components present in the measurements. In recent years, developments in sparse modeling have allowed for methods that jointly estimate the parameters in the model and the model order. Based on these achievements, the approach often taken in this thesis is as follows. First, a parametric model of the considered signal is derived, containing different parameters that capture the important characteristics of the signal. When the signal model has been determined, an optimization problem is formed aimed at finding ...

Over the last decade there has been an increasing interest in solutions for the continuous monitoring of health status with wireless, and in particular, wearable devices that provide remote analysis of physiological data. The use of wireless technologies have introduced new problems such as the transmission of a huge amount of data within the constraint of limited battery life devices. The design of an accurate and energy efficient telemonitoring system can be achieved by reducing the amount of data that should be transmitted, which is still a challenging task on devices with both computational and energy constraints. Furthermore, it is not sufficient merely to collect and transmit data, and algorithms that provide real-time analysis are needed. In this thesis, we address the problems of compression and analysis of physiological data using the emerging frameworks of Compressive Sensing (CS) and sparse ...

Distributed signal or information processing using Internet of Things (IoT), facilitates real-time monitoring of signals, for example, environmental pollutants, health indicators, and electric energy consumption in a smart city. Despite the promising capabilities of IoTs, these distributed deployments often face the challenge of data privacy and communication rate constraints. In traditional machine learning, training data is moved to a data center, which requires massive data movement from distributed IoT devices to a third-party location, thus raising concerns over privacy and inefficient use of communication resources. Moreover, the growing network size, model size, and data volume combined lead to unusual complexity in the design of optimization algorithms beyond the compute capability of a single device. This necessitates novel system architectures to ensure stable and secure operations of such networks. Federated learning (FL) architecture, a novel distributed learning paradigm introduced by McMahan ...

Chapter 2 contains a short introduction to the fundamentals of compressed sensing theory, which is the larger context of this thesis. We start with introducing the key concepts of sparsity and sparse representations of signals. We discuss the central problem of compressed sensing, i.e. how to adequately recover sparse signals from a small number of measurements, as well as the multiple formulations of the reconstruction problem. A large part of the chapter is devoted to some of the most important conditions necessary and/or sufficient to guarantee accurate recovery. The aim is to introduce the reader to the basic results, without the burden of detailed proofs. In addition, we also present a few of the popular reconstruction and optimization algorithms that we use throughout the thesis. Chapter 3 presents an alternative sparsity model known as analysis sparsity, that offers similar recovery ...

This thesis deals with the coding of images with multiple coding schemes and their dynamic selection. In our society of information highways, electronic communication is taking everyday a bigger place in our lives. The number of transmitted images is also increasing everyday. Therefore, research on image compression is still an active area. However, the current trend is to add several functionalities to the compression scheme such as progressiveness for more comfortable browsing of web-sites or databases. Classical image coding schemes have a rigid structure. They usually process an image as a whole and treat the pixels as a simple signal with no particular characteristics. Second generation schemes use the concept of objects in an image, and introduce a model of the human visual system in the design of the coding scheme. Dynamic coding schemes, as their name tells us, make ...

This thesis deals with the modeling of structured signals under different sparsity constraints. Many phenomena exhibit an inherent structure that may be exploited when setting up models, examples include audio waves, radar, sonar, and image objects. These structures allow us to model, identify, and classify the processes, enabling parameter estimation for, e.g., identification, localisation, and tracking. In this work, such structures are exploited, with the goal to achieve efficient localisation and tracking of a structured source signal. Specifically, two scenarios are considered. In papers A and B, the aim is to find a sparse subset of a structured signal such that the signal parameters and source locations may be estimated in an optimal way. For the sparse subset selection, a combinatorial optimization problem is approximately solved by means of convex relaxation, with the results of allowing for different types of ...

Due to significant advances made over the last few decades in the areas of (wireless) networking, communications and microprocessor fabrication, the use of sensor networks to observe physical phenomena is rapidly becoming commonplace. Over this period, many aspects of sensor networks have been explored, yet a thorough understanding of how to analyse and process the vast amounts of sensor data collected remains an open area of research. This work, therefore, aims to provide theoretical, as well as practical, advances this area. In particular, we consider the problem of inferring certain underlying properties of the monitored phenomena, from our sensor measurements. Within mathematics, this is commonly formulated as an inverse problem; whereas in signal processing, it appears as a (multidimensional) sampling and reconstruction problem. Indeed it is well known that inverse problems are notoriously ill-posed and very demanding to solve; meanwhile ...

With the joint development of networking and digital coding technologies multimedia and more particularly video services are clearly becoming one of the major consumers of the new information networks. The rapid growth of the Internet and computer industry however results in a very heterogeneous infrastructure commonly overloaded. Video service providers have nevertheless to oer to their clients the best possible quality according to their respective capabilities and communication channel status. The Quality of Service is not only inuenced by the compression artifacts, but also by unavoidable packet losses. Hence, the packet video stream has clearly to fulll possibly contradictory requirements, that are coding eciency and robustness to data loss. The rst contribution of this thesis is the complete modeling of the video Quality of Service (QoS) in standard and more particularly MPEG-2 applications. The performance of Forward Error Control (FEC) ...

This doctorate thesis focuses on sparse regression, a statistical modeling tool for selecting valuable predictors in underdetermined linear models. By imposing different constraints on the structure of the variable vector in the regression problem, one obtains estimates which have sparse supports, i.e., where only a few of the elements in the response variable have non-zero values. The thesis collects six papers which, to a varying extent, deals with the applications, implementations, modifications, translations, and other analysis of such problems. Sparse regression is often used to approximate additive models with intricate, non-linear, non-smooth or otherwise problematic functions, by creating an underdetermined model consisting of candidate values for these functions, and linear response variables which selects among the candidates. Sparse regression is therefore a widely used tool in applications such as, e.g., image processing, audio processing, seismological and biomedical modeling, but is ...