Algorithmic Aspects of Pyramidal Tours with Restricted Jump-Backs

Abstract

Typical tasks in signal processing may be done in simpler ways ormore efficiently if the signals to analyze are represented in a properway. This thesis deals with some algorithmic problems related tosignal approximation, more precisely, in the novel field of sparseapproximation using redundant dictionaries of functions.Orthogonal bases permit to approximate signals by just taking theN waveforms whose associated projections have maximal amplitudes.This nice property is no longer valid if the used base is redundant.In fact, finding the best decomposition becomes a NP Hard problemin the general case. Thus, suboptimal heuristics have been developed;the best known ones are Matching Pursuit and Basis Pursuit. Bothremain highly complex which prevent them from being used in practicein many situations. The first part of the thesis is concerned withthis computational bottleneck. We propose to create a tree structureendowing the dictionary and grouping similar atoms in the same branches.An approximation algorithm, called Tree-Based Pursuit, exploitingthis structure is presented. It considerably lowers the cost of findinggood approximations with redundant dictionaries.The quality of the representation does not only depend on the approximationalgorithm but also on the dictionary used. One of the main advantagesof these techniques is that the atoms can be tailored to match thefeatures present in the signal. It might happen that some knowledgeabout the class of signals to approximate directly leads to the dictionary.For most natural signals, however, the underlying structures arenot clearly known and may be obfuscated. Learning dictionaries basedon examples is an alternative to manual design and is gaining ininterest. Most natural signals exhibit behaviors invariant to translationsin space or in time. Thus, we propose an algorithm to learn redundantdictionaries under the translation invariance constraint. In thecase of images, the proposed solution is able to recover atoms similarto Gabor functions, line edge detectors and curved edge detectors.The two first categories were already observed and the third onecompletes the range of natural features and is a major contributionof this algorithm.Sparsity is used to define the efficiency of approximation algorithmsas well as to characterize good dictionaries. It directly comes fromthe fact that these techniques aim at approximating signals withfew significant terms. This property was successfully exploited asa dimension reduction method for different signal processing tasksas analysis, de-noising or compression. In the last chapter, we tacklethe problem of finding the nearest neighbor to a query signal ina set of signals that have a sparse representation. We take advantageof sparsity to approximate quickly the distance between the queryand all elements of the database. In this way, we are able to prunerecursively all elements that do not match the query, while providingbounds on the true distance. Validation of this technique on syntheticand real data sets confirms that it could be very well suited toprocess queries over large databases of compressed signals, avoidingmost of the burden of decoding.