Fast Sparse Subband Decomposition Using FIRSP

This paper presents a new fast algorithm for generating sparse signal approximations within an overcomplete subband representation. While the current paper concentrates on sparsifying the Modulated Complex Lapped Transform the theory is applicable to representations composed of a general union of orthonormal bases. We illustrate our method on an audio signal and demonstrate the coding gain of such representations

Nonlinear approximation with redundant multi-component dictionaries

The problem of efficiently representing and approximating digital data is an open challenge and it is of paramount importance for many applications. This dissertation focuses on the approximation of natural signals as an organized combination of mutually connected elements, preserving and at the same time benefiting from their inherent structure. This is done by decomposing a signal onto a multi-component, redundant collection of functions (dictionary), built by the union of several subdictionaries, each of which is designed to capture a specific behavior of the signal. In this way, instead of representing signals as a superposition of sinusoids or wavelets many alternatives are available. In addition, since dictionaries we are interested in are overcomplete, the decomposition is non-unique. This gives us the possibility of adaptation, choosing among many possible representations the one which best fits our purposes. On the other hand, it also requires more complex approximation techniques whose theoretical decomposition capacity and computational load have to be carefully studied. In general, we aim at representing a signal with few and meaningful components. If we are able to represent a piece of information by using only few elements, it means that such elements can capture its main characteristics, allowing to compact the energy carried by a signal into the smallest number of terms. In such a framework, this work also proposes analysis methods which deal with the goal of considering the a priori information available when decomposing a structured signal. Indeed, a natural signal is not only an array of numbers, but an expression of a physical event about which we usually have a deep knowledge. Therefore, we claim that it is worth exploiting its structure, since it can be advantageous not only in helping the analysis process, but also in making the representation of such information more accessible and meaningful. The study of an adaptive image representation inspired and gave birth to this work. We often refer to images and visual information throughout the course of the dissertation. However, the proposed approximation setting extends to many different kinds of structured data and examples are given involving videos and electrocardiogram signals. An important part of this work is constituted by practical applications: first of all we provide very interesting results for image and video compression. Then, we also face the problem of signal denoising and, finally, promising achievements in the field of source separation are presented

Toward sparse and geometry adapted video approximations

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 information) as well as spatial geometry. Clearly, most of past and present strategies used to represent video signals do not exploit properly its spatial geometry. Similarly to the case of images, a very interesting approach seems to be the decomposition of video using large over-complete libraries of basis functions able to represent salient geometric features of the signal. In the framework of video, these features should model 2D geometric video components as well as their temporal evolution, forming spatio-temporal 3D geometric primitives. Through this PhD dissertation, different aspects on the use of adaptivity in video representation are studied looking toward exploiting both aspects of video: its piecewise nature and the geometry. The first part of this work studies the use of localized temporal adaptivity in subband video coding. This is done considering two transformation schemes used for video coding: 3D wavelet representations and motion compensated temporal filtering. A theoretical R-D analysis as well as empirical results demonstrate how temporal adaptivity improves coding performance of moving edges in 3D transform (without motion compensation) based video coding. Adaptivity allows, at the same time, to equally exploit redundancy in non-moving video areas. The analogy between motion compensated video and 1D piecewise-smooth signals is studied as well. This motivates the introduction of local length adaptivity within frame-adaptive motion compensated lifted wavelet decompositions. This allows an optimal rate-distortion performance when video motion trajectories are shorter than the transformation "Group Of Pictures", or when efficient motion compensation can not be ensured. After studying temporal adaptivity, the second part of this thesis is dedicated to understand the fundamentals of how can temporal and spatial geometry be jointly exploited. This work builds on some previous results that considered the representation of spatial geometry in video (but not temporal, i.e, without motion). In order to obtain flexible and efficient (sparse) signal representations, using redundant dictionaries, the use of highly non-linear decomposition algorithms, like Matching Pursuit, is required. General signal representation using these techniques is still quite unexplored. For this reason, previous to the study of video representation, some aspects of non-linear decomposition algorithms and the efficient decomposition of images using Matching Pursuits and a geometric dictionary are investigated. A part of this investigation concerns the study on the influence of using a priori models within approximation non-linear algorithms. Dictionaries with a high internal coherence have some problems to obtain optimally sparse signal representations when used with Matching Pursuits. It is proved, theoretically and empirically, that inserting in this algorithm a priori models allows to improve the capacity to obtain sparse signal approximations, mainly when coherent dictionaries are used. Another point discussed in this preliminary study, on the use of Matching Pursuits, concerns the approach used in this work for the decompositions of video frames and images. The technique proposed in this thesis improves a previous work, where authors had to recur to sub-optimal Matching Pursuit strategies (using Genetic Algorithms), given the size of the functions library. In this work the use of full search strategies is made possible, at the same time that approximation efficiency is significantly improved and computational complexity is reduced. Finally, a priori based Matching Pursuit geometric decompositions are investigated for geometric video representations. Regularity constraints are taken into account to recover the temporal evolution of spatial geometric signal components. The results obtained for coding and multi-modal (audio-visual) signal analysis, clarify many unknowns and show to be promising, encouraging to prosecute research on the subject