Sensing and Control in Symmetric Networks

In engineering applications, one of the major challenges today is to develop reliable and robust control algorithms for complex networked systems. Controllability and observability of such systems play a crucial role in the design process. The underlying network structure may contain symmetries -- caused for example by the coupling of identical building blocks -- and these symmetries lead to repeated eigenvalues in a generic way. This complicates the design of controllers since repeated eigenvalues might decrease the controllability of the system. In this paper, we will analyze the relationship between the controllability and observability of complex networked systems and graph symmetries using results from representation theory. Furthermore, we will propose an algorithm to compute sparse input and output matrices based on projections onto the isotypic components. We will illustrate our results with the aid of two guiding examples, a network with $D_4$ symmetry and the Petersen graph

Algorithms for super-resolution of images based on Sparse Representation and Manifolds

lmage super-resolution is defined as a class of techniques that enhance the spatial resolution of images. Super-resolution methods can be subdivided in single and multi image methods. This thesis focuses on developing algorithms based on mathematical theories for single image super­ resolution problems. lndeed, in arder to estimate an output image, we adopta mixed approach: i.e., we use both a dictionary of patches with sparsity constraints (typical of learning-based methods) and regularization terms (typical of reconstruction-based methods). Although the existing methods already per- form well, they do not take into account the geometry of the data to: regularize the solution, cluster data samples (samples are often clustered using algorithms with the Euclidean distance as a dissimilarity metric), learn dictionaries (they are often learned using PCA or K-SVD). Thus, state-of-the-art methods still suffer from shortcomings. In this work, we proposed three new methods to overcome these deficiencies. First, we developed SE-ASDS (a structure tensor based regularization term) in arder to improve the sharpness of edges. SE-ASDS achieves much better results than many state-of-the- art algorithms. Then, we proposed AGNN and GOC algorithms for determining a local subset of training samples from which a good local model can be computed for recon- structing a given input test sample, where we take into account the underlying geometry of the data. AGNN and GOC methods outperform spectral clustering, soft clustering, and geodesic distance based subset selection in most settings. Next, we proposed aSOB strategy which takes into account the geometry of the data and the dictionary size. The aSOB strategy outperforms both PCA and PGA methods. Finally, we combine all our methods in a unique algorithm, named G2SR. Our proposed G2SR algorithm shows better visual and quantitative results when compared to the results of state-of-the-art methods.Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorTese (Doutorado)Super-resolução de imagens é definido como urna classe de técnicas que melhora a resolução espacial de imagens. Métodos de super-resolução podem ser subdivididos em métodos para urna única imagens e métodos para múltiplas imagens. Esta tese foca no desenvolvimento de algoritmos baseados em teorias matemáticas para problemas de super-resolução de urna única imagem. Com o propósito de estimar urna imagem de saída, nós adotamos urna abordagem mista, ou seja: nós usamos dicionários de patches com restrição de esparsidade (método baseado em aprendizagem) e termos de regularização (método baseado em reconstrução). Embora os métodos existentes sejam eficientes, eles nao levam em consideração a geometria dos dados para: regularizar a solução, clusterizar os dados (dados sao frequentemente clusterizados usando algoritmos com a distancia Euclideana como métrica de dissimilaridade), aprendizado de dicionários (eles sao frequentemente treinados usando PCA ou K-SVD). Portante, os métodos do estado da arte ainda tem algumas deficiencias. Neste trabalho, nós propomos tres métodos originais para superar estas deficiencias. Primeiro, nós desenvolvemos SE-ASDS (um termo de regularização baseado em structure tensor) afim de melhorar a nitidez das bordas das imagens. SE-ASDS alcança resultados muito melhores que os algoritmos do estado da arte. Em seguida, nós propomos os algoritmos AGNN e GOC para determinar um subconjunto de amostras de treinamento a partir das quais um bom modelo local pode ser calculado para reconstruir urna dada amostra de entrada considerando a geometria dos dados. Os métodos AGNN e GOC superamos métodos spectral clustering, soft clustering e os métodos baseados em distancia geodésica na maioria dos casos. Depois, nós propomos o método aSOB que leva em consideração a geometria dos dados e o tamanho do dicionário. O método aSOB supera os métodos PCA e PGA. Finalmente, nós combinamos todos os métodos que propomos em um único algoritmo, a saber, G2SR. Nosso algoritmo G2SR mostra resultados melhores que os métodos do estado da arte em termos de PSRN, SSIM, FSIM e qualidade visual

Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

Fast Orthonormal Sparsifying Transforms Based on Householder Reflectors

Dictionary learning is the task of determining a data-dependent transform that yields a sparse representation of some observed data. The dictionary learning problem is non-convex, and usually solved via computationally complex iterative algorithms. Furthermore, the resulting transforms obtained generally lack structure that permits their fast application to data. To address this issue, this paper develops a framework for learning orthonormal dictionaries which are built from products of a few Householder reflectors. Two algorithms are proposed to learn the reflector coefficients: one that considers a sequential update of the reflectors and one with a simultaneous update of all reflectors that imposes an additional internal orthogonal constraint. The proposed methods have low computational complexity and are shown to converge to local minimum points which can be described in terms of the spectral properties of the matrices involved. The resulting dictionaries balance between the computational complexity and the quality of the sparse representations by controlling the number of Householder reflectors in their product. Simulations of the proposed algorithms are shown in the image processing setting where well-known fast transforms are available for comparisons. The proposed algorithms have favorable reconstruction error and the advantage of a fast implementation relative to the classical, unstructured, dictionaries

ℓ1\ell^1-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?

This paper investigates the problem of signal estimation from undersampled noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, we derive novel recovery guarantees for the $\ell^{1}$-analysis basis pursuit, enabling highly accurate predictions of its sample complexity. The corresponding bounds on the number of required measurements do explicitly depend on the Gram matrix of the analysis operator and therefore particularly account for its mutual coherence structure. Our findings defy conventional wisdom which promotes the sparsity of analysis coefficients as the crucial quantity to study. In fact, this common paradigm breaks down completely in many situations of practical interest, for instance, when applying a redundant (multilevel) frame as analysis prior. By extensive numerical experiments, we demonstrate that, in contrast, our theoretical sampling-rate bounds reliably capture the recovery capability of various examples, such as redundant Haar wavelets systems, total variation, or random frames. The proofs of our main results build upon recent achievements in the convex geometry of data mining problems. More precisely, we establish a sophisticated upper bound on the conic Gaussian mean width that is associated with the underlying $\ell^{1}$-analysis polytope. Due to a novel localization argument, it turns out that the presented framework naturally extends to stable recovery, allowing us to incorporate compressible coefficient sequences as well