343 research outputs found

    Algorithmic aspects of sparse approximations

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    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 This nice property is no longer valid if the used base is redundant.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

    Sparse Modeling for Image and Vision Processing

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    In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio

    EEG-based brain-computer interfaces using motor-imagery: techniques and challenges.

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    Electroencephalography (EEG)-based brain-computer interfaces (BCIs), particularly those using motor-imagery (MI) data, have the potential to become groundbreaking technologies in both clinical and entertainment settings. MI data is generated when a subject imagines the movement of a limb. This paper reviews state-of-the-art signal processing techniques for MI EEG-based BCIs, with a particular focus on the feature extraction, feature selection and classification techniques used. It also summarizes the main applications of EEG-based BCIs, particularly those based on MI data, and finally presents a detailed discussion of the most prevalent challenges impeding the development and commercialization of EEG-based BCIs

    3D exemplar-based image inpainting in electron microscopy

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    In electron microscopy (EM) a common problem is the non-availability of data, which causes artefacts in reconstructions. In this thesis the goal is to generate artificial data where missing in EM by using exemplar-based inpainting (EBI). We implement an accelerated 3D version tailored to applications in EM, which reduces reconstruction times from days to minutes. We develop intelligent sampling strategies to find optimal data as input for reconstruction methods. Further, we investigate approaches to reduce electron dose and acquisition time. Sparse sampling followed by inpainting is the most promising approach. As common evaluation measures may lead to misinterpretation of results in EM and falsify a subsequent analysis, we propose to use application driven metrics and demonstrate this in a segmentation task. A further application of our technique is the artificial generation of projections in tiltbased EM. EBI is used to generate missing projections, such that the full angular range is covered. Subsequent reconstructions are significantly enhanced in terms of resolution, which facilitates further analysis of samples. In conclusion, EBI proves promising when used as an additional data generation step to tackle the non-availability of data in EM, which is evaluated in selected applications. Enhancing adaptive sampling methods and refining EBI, especially considering the mutual influence, promotes higher throughput in EM using less electron dose while not lessening quality.Ein hĂ€ufig vorkommendes Problem in der Elektronenmikroskopie (EM) ist die NichtverfĂŒgbarkeit von Daten, was zu Artefakten in Rekonstruktionen fĂŒhrt. In dieser Arbeit ist es das Ziel fehlende Daten in der EM kĂŒnstlich zu erzeugen, was durch Exemplar-basiertes Inpainting (EBI) realisiert wird. Wir implementieren eine auf EM zugeschnittene beschleunigte 3D Version, welche es ermöglicht, Rekonstruktionszeiten von Tagen auf Minuten zu reduzieren. Wir entwickeln intelligente Abtaststrategien, um optimale Datenpunkte fĂŒr die Rekonstruktion zu erhalten. AnsĂ€tze zur Reduzierung von Elektronendosis und Aufnahmezeit werden untersucht. Unterabtastung gefolgt von Inpainting fĂŒhrt zu den besten Resultaten. Evaluationsmaße zur Beurteilung der RekonstruktionsqualitĂ€t helfen in der EM oft nicht und können zu falschen SchlĂŒssen fĂŒhren, weswegen anwendungsbasierte Metriken die bessere Wahl darstellen. Dies demonstrieren wir anhand eines Beispiels. Die kĂŒnstliche Erzeugung von Projektionen in der neigungsbasierten Elektronentomographie ist eine weitere Anwendung. EBI wird verwendet um fehlende Projektionen zu generieren. Daraus resultierende Rekonstruktionen weisen eine deutlich erhöhte Auflösung auf. EBI ist ein vielversprechender Ansatz, um nicht verfĂŒgbare Daten in der EM zu generieren. Dies wird auf Basis verschiedener Anwendungen gezeigt und evaluiert. Adaptive Aufnahmestrategien und EBI können also zu einem höheren Durchsatz in der EM fĂŒhren, ohne die BildqualitĂ€t merklich zu verschlechtern

    Dictionary Learning for Sparse Representations With Applications to Blind Source Separation.

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    During the past decade, sparse representation has attracted much attention in the signal processing community. It aims to represent a signal as a linear combination of a small number of elementary signals called atoms. These atoms constitute a dictionary so that a signal can be expressed by the multiplication of the dictionary and a sparse coefficients vector. This leads to two main challenges that are studied in the literature, i.e. sparse coding (find the coding coefficients based on a given dictionary) and dictionary design (find an appropriate dictionary to fit the data). Dictionary design is the focus of this thesis. Traditionally, the signals can be decomposed by the predefined mathematical transform, such as discrete cosine transform (DCT), which forms the so-called analytical approach. In recent years, learning-based methods have been introduced to adapt the dictionary from a set of training data, leading to the technique of dictionary learning. Although this may involve a higher computational complexity, learned dictionaries have the potential to offer improved performance as compared with predefined dictionaries. Dictionary learning algorithm is often achieved by iteratively executing two operations: sparse approximation and dictionary update. We focus on the dictionary update step, where the dictionary is optimized with a given sparsity pattern. A novel framework is proposed to generalize benchmark mechanisms such as the method of optimal directions (MOD) and K-SVD where an arbitrary set of codewords and the corresponding sparse coefficients are simultaneously updated, hence the term simultaneous codeword optimization (SimCO). Moreover, its extended formulation ‘regularized SimCO’ mitigates the major bottleneck of dictionary update caused by the singular points. First and second order optimization procedures are designed to solve the primitive and regularized SimCO. In addition, a tree-structured multi-level representation of dictionary based on clustering is used to speed up the optimization process in the sparse coding stage. This novel dictionary learning algorithm is also applied for solving the underdetermined blind speech separation problem, leading to a multi-stage method, where the separation problem is reformulated as a sparse coding problem, with the dictionary being learned by an adaptive algorithm. Using mutual coherence and sparsity index, the performance of a variety of dictionaries for underdetermined speech separation is compared and analyzed, such as the dictionaries learned from speech mixtures and ground truth speech sources, as well as those predefined by mathematical transforms. Finally, we propose a new method for joint dictionary learning and source separation. Different from the multistage method, the proposed method can simultaneously estimate the mixing matrix, the dictionary and the sources in an alternating and blind manner. The advantages of all the proposed methods are demonstrated over the state-of-the-art methods using extensive numerical tests

    Detection and classification of non-stationary signals using sparse representations in adaptive dictionaries

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    Automatic classification of non-stationary radio frequency (RF) signals is of particular interest in persistent surveillance and remote sensing applications. Such signals are often acquired in noisy, cluttered environments, and may be characterized by complex or unknown analytical models, making feature extraction and classification difficult. This thesis proposes an adaptive classification approach for poorly characterized targets and backgrounds based on sparse representations in non-analytical dictionaries learned from data. Conventional analytical orthogonal dictionaries, e.g., Short Time Fourier and Wavelet Transforms, can be suboptimal for classification of non-stationary signals, as they provide a rigid tiling of the time-frequency space, and are not specifically designed for a particular signal class. They generally do not lead to sparse decompositions (i.e., with very few non-zero coefficients), and use in classification requires separate feature selection algorithms. Pursuit-type decompositions in analytical overcomplete (non-orthogonal) dictionaries yield sparse representations, by design, and work well for signals that are similar to the dictionary elements. The pursuit search, however, has a high computational cost, and the method can perform poorly in the presence of realistic noise and clutter. One such overcomplete analytical dictionary method is also analyzed in this thesis for comparative purposes. The main thrust of the thesis is learning discriminative RF dictionaries directly from data, without relying on analytical constraints or additional knowledge about the signal characteristics. A pursuit search is used over the learned dictionaries to generate sparse classification features in order to identify time windows that contain a target pulse. Two state-of-the-art dictionary learning methods are compared, the K-SVD algorithm and Hebbian learning, in terms of their classification performance as a function of dictionary training parameters. Additionally, a novel hybrid dictionary algorithm is introduced, demonstrating better performance and higher robustness to noise. The issue of dictionary dimensionality is explored and this thesis demonstrates that undercomplete learned dictionaries are suitable for non-stationary RF classification. Results on simulated data sets with varying background clutter and noise levels are presented. Lastly, unsupervised classification with undercomplete learned dictionaries is also demonstrated in satellite imagery analysis

    Graph-based Methods for Visualization and Clustering

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    The amount of data that we produce and consume is larger than it has been at any point in the history of mankind, and it keeps growing exponentially. All this information, gathered in overwhelming volumes, often comes with two problematic characteristics: it is complex and deprived of semantical context. A common step to address those issues is to embed raw data in lower dimensions, by finding a mapping which preserves the similarity between data points from their original space to a new one. Measuring similarity between large sets of high-dimensional objects is, however, problematic for two main reasons: first, high-dimensional points are subject to the curse of dimensionality and second, the number of pairwise distances between points is quadratic with respect to the amount of data points. Both problems can be addressed by using nearest neighbours graphs to understand the structure in data. As a matter of fact, most dimensionality reduction methods use similarity matrices that can be interpreted as graph adjacency matrices. Yet, despite recent progresses, dimensionality reduction is still very challenging when applied to very large datasets. Indeed, although recent methods specifically address the problem of scaleability, processing datasets of millions of elements remain a very lengthy process. In this thesis, we propose new contributions which address the problem of scaleability using the framework of Graph Signal Processing, which extends traditional signal processing to graphs. We do so motivated by the premise that graphs are well suited to represent the structure of the data. In the first part of this thesis, we look at quantitative measures for the evaluation of dimensionality reduction methods. Using tools from graph theory and Graph Signal Processing, we show that specific characteristics related to quality can be assessed by taking measures on the graph, which indirectly validates the hypothesis relating graph to structure. The second contribution is a new method for a fast eigenspace approximation of the graph Laplacian. Using principles of GSP and random matrices, we show that an approximated eigensubpace can be recovered very efficiently, which be used for fast spectral clustering or visualization. Next, we propose a compressive scheme to accelerate any dimensionality reduction technique. The idea is based on compressive sampling and transductive learning on graphs: after computing the embedding for a small subset of data points, we propagate the information everywhere using transductive inference. The key components of this technique are a good sampling strategy to select the subset and the application of transductive learning on graphs. Finally, we address the problem of over-discriminative feature spaces by proposing a hierarchical clustering structure combined with multi-resolution graphs. Using efficient coarsening and refinement procedures on this structure, we show that dimensionality reduction algorithms can be run on intermediate levels and up-sampled to all points leading to a very fast dimensionality reduction method. For all contributions, we provide extensive experiments on both synthetic and natural datasets, including large-scale problems. This allows us to show the pertinence of our models and the validity of our proposed algorithms. Following reproducible principles, we provide everything needed to repeat the examples and the experiments presented in this work
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