Image compression with learnt tree-structured dictionaries

In the present paper we propose a new framework for the construction of meaningful dictionaries for sparse representation of signals. The dictionary approach to coding and compression proves very attractive since decomposing a signal over a redundant set of basis functions allows a parsimonious representation of information. This interest is witnessed by numerous research efforts that have been done in the last years to develop efficient algorithm for the decomposition of signals over redundant sets of functions. However, the effectiveness of such methods strongly depends on the dictionary and on its structure. In this work, we develop a method to learn overcomplete sets of functions from real-world signals. This technique allows the design of dictionaries that can be adapted to a specific class of signals. The found functions are stored in a tree structure. This data structure is used by a Tree-Based Pursuit algorithm to generate sparse approximations of natural signals. Finally, the proposed method is considered in the context of image compression. Results show that the learning Tree-Based approach outperforms state-of-the-art coding technique

Research on structure adaptive multi-atoms matching pursuit algorithm of image sparse representation

图像内容的有效表示是图像处理领域的基本问题。图像的稀疏表示是指用相对较少的数据来表示出目标图像的主要信息。稀疏表示能够更有效地对图像建模，已成为带动压缩感知与图像处理、信号处理、通信等领域发展的核心技术之一，是当前图像处理领域的研究热点与难点，受到国内外学者的广泛关注。本文主要围绕图像稀疏表示理论中过完备字典设计和快速稀疏分解算法两个方面进行了详细和深入的研究，取得的主要研究成果及创新点如下： 1)根据图像的几何结构特性，参考哺乳类动物的视觉系统感知特性，选取二维Gabor函数作为过完备字典的生成函数，建立了可以匹配多种图像结构的Gabor多成分过完备字典。该字典包含平滑、边缘轮廓与纹理三种...Efficient representation of image is the basic problem in digital image processing. Image sparse representation can capture significant information of the original image with relatively less data. Because sparse representation model can effectively represent the image, it becomes one of the core technologies which drive the development of many subjects, such as Compressed Sensing, Signal Processin...学位：工程硕士院系专业：信息科学与技术学院计算机科学系_计算机技术学号：2302009115270

Hybrid Video Coding based on Bidimensional Matching Pursuit

Hybrid video coding combines together two stages: first, motion estimation and compensation predict each frame from the neighboring frames, then the prediction error is coded, reducing the correlation in the spatial domain. In this work, we focus on the latter stage, presenting a scheme that profits from some of the features introduced by the standard H.264/AVC for motion estimation and replaces the transform in the spatial domain. The prediction error is so coded using the matching pursuit algorithm which decomposes the signal over an appositely designed bidimensional, anisotropic, redundant dictionary. Comparisons are made among the proposed technique, H.264, and a DCT-based coding scheme. Moreover, we introduce fast techniques for atom selection, which exploit the spatial localization of the atoms. An adaptive coding scheme aimed at optimizing the resource allocation is also presented, together with a rate-distortion study for the matching pursuit algorithm. Results show that the proposed scheme outperforms the standard DCT, especially at very low bit rates

A Geometrical Study of Matching Pursuit Parametrization

This paper studies the effect of discretizing the parametrization of a dictionary used for Matching Pursuit decompositions of signals. Our approach relies on viewing the continuously parametrized dictionary as an embedded manifold in the signal space on which the tools of differential (Riemannian) geometry can be applied. The main contribution of this paper is twofold. First, we prove that if a discrete dictionary reaches a minimal density criterion, then the corresponding discrete MP (dMP) is equivalent in terms of convergence to a weakened hypothetical continuous MP. Interestingly, the corresponding weakness factor depends on a density measure of the discrete dictionary. Second, we show that the insertion of a simple geometric gradient ascent optimization on the atom dMP selection maintains the previous comparison but with a weakness factor at least two times closer to unity than without optimization. Finally, we present numerical experiments confirming our theoretical predictions for decomposition of signals and images on regular discretizations of dictionary parametrizations.Comment: 26 pages, 8 figure

Algorithmic aspects of sparse approximations

Typical tasks in signal processing may be done in simpler ways or more efficiently if the signals to analyze are represented in a proper way. This thesis deals with some algorithmic problems related to signal approximation, more precisely, in the novel field of sparse approximation using redundant dictionaries of functions. Orthogonal bases permit to approximate signals by just taking the N waveforms whose associated projections have maximal amplitudes. This nice property is no longer valid if the used base is redundant. In fact, finding the best decomposition becomes a NP Hard problem in the general case. Thus, suboptimal heuristics have been developed; the best known ones are Matching Pursuit and Basis Pursuit. Both remain highly complex which prevent them from being used in practice in many situations. The first part of the thesis is concerned with this computational bottleneck. We propose to create a tree structure endowing the dictionary and grouping similar atoms in the same branches. An approximation algorithm, called Tree-Based Pursuit, exploiting this structure is presented. It considerably lowers the cost of finding good approximations with redundant dictionaries. The quality of the representation does not only depend on the approximation algorithm but also on the dictionary used. One of the main advantages of these techniques is that the atoms can be tailored to match the features present in the signal. It might happen that some knowledge about the class of signals to approximate directly leads to the dictionary. For most natural signals, however, the underlying structures are not clearly known and may be obfuscated. Learning dictionaries based on examples is an alternative to manual design and is gaining in interest. Most natural signals exhibit behaviors invariant to translations in space or in time. Thus, we propose an algorithm to learn redundant dictionaries under the translation invariance constraint. In the case of images, the proposed solution is able to recover atoms similar to Gabor functions, line edge detectors and curved edge detectors. The two first categories were already observed and the third one completes the range of natural features and is a major contribution of this algorithm. Sparsity is used to define the efficiency of approximation algorithms as well as to characterize good dictionaries. It directly comes from the fact that these techniques aim at approximating signals with few significant terms. This property was successfully exploited as a dimension reduction method for different signal processing tasks as analysis, de-noising or compression. In the last chapter, we tackle the problem of finding the nearest neighbor to a query signal in a set of signals that have a sparse representation. We take advantage of sparsity to approximate quickly the distance between the query and all elements of the database. In this way, we are able to prune recursively all elements that do not match the query, while providing bounds on the true distance. Validation of this technique on synthetic and real data sets confirms that it could be very well suited to process queries over large databases of compressed signals, avoiding most of the burden of decoding