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220
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combin ..."
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Cited by 427 (36 self)
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A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easily-verifiable conditions under which optimally-sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Online learning for matrix factorization and sparse coding
, 2010
"... Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the large-scale matrix factorization problem that consists of learning the basis set in order to ad ..."
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Cited by 330 (31 self)
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Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the large-scale matrix factorization problem that consists of learning the basis set in order to adapt it to specific data. Variations of this problem include dictionary learning in signal processing, non-negative matrix factorization and sparse principal component analysis. In this paper, we propose to address these tasks with a new online optimization algorithm, based on stochastic approximations, which scales up gracefully to large data sets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range of learning problems. A proof of convergence is presented, along with experiments with natural images and genomic data demonstrating that it leads to state-of-the-art performance in terms of speed and optimization for both small and large data sets.
Sparse subspace clustering
- In CVPR
, 2009
"... We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all oth ..."
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Cited by 241 (14 self)
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We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all other data points. In general, finding such a SR is NP hard. Our key contribution is to show that, under mild assumptions, the SR can be obtained ’exactly ’ by using ℓ1 optimization. The segmentation of the data is obtained by applying spectral clustering to a similarity matrix built from this SR. Our method can handle noise, outliers as well as missing data. We apply our subspace clustering algorithm to the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that our approach significantly outperforms state-of-the-art methods. 1.
Sparse Representation For Computer Vision and Pattern Recognition
, 2009
"... Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on non-traditional applications where the goal is not just to obtain a compact high-fidelity representation of the observed signal, but also to extract semantic information. The choice of ..."
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Cited by 146 (9 self)
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Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on non-traditional applications where the goal is not just to obtain a compact high-fidelity representation of the observed signal, but also to extract semantic information. The choice of dictionary plays a key role in bridging this gap: unconventional dictionaries consisting of, or learned from, the training samples themselves provide the key to obtaining state-of-theart results and to attaching semantic meaning to sparse signal representations. Understanding the good performance of such unconventional dictionaries in turn demands new algorithmic and analytical techniques. This review paper highlights a few representative examples of how the interaction between sparse signal representation and computer vision can enrich both fields, and raises a number of open questions for further study.
Learning multiscale sparse representations for image and video restoration
, 2007
"... Abstract. This paper presents a framework for learning multiscale sparse representations of color images and video with overcomplete dictionaries. A single-scale K-SVD algorithm was introduced in [1], formulating sparse dictionary learning for grayscale image representation as an optimization proble ..."
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Cited by 103 (21 self)
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Abstract. This paper presents a framework for learning multiscale sparse representations of color images and video with overcomplete dictionaries. A single-scale K-SVD algorithm was introduced in [1], formulating sparse dictionary learning for grayscale image representation as an optimization problem, efficiently solved via Orthogonal Matching Pursuit (OMP) and Singular Value Decomposition (SVD). Following this work, we propose a multiscale learned representation, obtained by using an efficient quadtree decomposition of the learned dictionary, and overlapping image patches. The proposed framework provides an alternative to pre-defined dictionaries such as wavelets, and shown to lead to state-of-the-art results in a number of image and video enhancement and restoration applications. This paper describes the proposed framework, and accompanies it by numerous examples demonstrating its strength. Key words. Image and video processing, sparsity, dictionary, multiscale representation, denoising, inpainting, interpolation, learning. AMS subject classifications. 49M27, 62H35
Non-Parametric Bayesian Dictionary Learning for Sparse Image Representations
"... Non-parametric Bayesian techniques are considered for learning dictionaries for sparse image representations, with applications in denoising, inpainting and compressive sensing (CS). The beta process is employed as a prior for learning the dictionary, and this non-parametric method naturally infers ..."
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Cited by 92 (34 self)
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Non-parametric Bayesian techniques are considered for learning dictionaries for sparse image representations, with applications in denoising, inpainting and compressive sensing (CS). The beta process is employed as a prior for learning the dictionary, and this non-parametric method naturally infers an appropriate dictionary size. The Dirichlet process and a probit stick-breaking process are also considered to exploit structure within an image. The proposed method can learn a sparse dictionary in situ; training images may be exploited if available, but they are not required. Further, the noise variance need not be known, and can be nonstationary. Another virtue of the proposed method is that sequential inference can be readily employed, thereby allowing scaling to large images. Several example results are presented, using both Gibbs and variational Bayesian inference, with comparisons to other state-of-the-art approaches.
Learning A Discriminative Dictionary for Sparse Coding via Label Consistent K-SVD
"... A label consistent K-SVD (LC-KSVD) algorithm to learn a discriminative dictionary for sparse coding is presented. In addition to using class labels of training data, we also associate label information with each dictionary item (columns of the dictionary matrix) to enforce discriminability in sparse ..."
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Cited by 88 (9 self)
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A label consistent K-SVD (LC-KSVD) algorithm to learn a discriminative dictionary for sparse coding is presented. In addition to using class labels of training data, we also associate label information with each dictionary item (columns of the dictionary matrix) to enforce discriminability in sparse codes during the dictionary learning process. More specifically, we introduce a new label consistent constraint called ‘discriminative sparse-code error ’ and combine it with the reconstruction error and the classification error to form a unified objective function. The optimal solution is efficiently obtained using the K-SVD algorithm. Our algorithm learns a single over-complete dictionary and an optimal linear classifier jointly. It yields dictionaries so that feature points with the same class labels have similar sparse codes. Experimental results demonstrate that our algorithm outperforms many recently proposed sparse coding techniques for face and object category recognition under the same learning conditions. 1.
Task-Driven Dictionary Learning
"... Abstract—Modeling data with linear combinations of a few elements from a learned dictionary has been the focus of much recent research in machine learning, neuroscience, and signal processing. For signals such as natural images that admit such sparse representations, it is now well established that ..."
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Cited by 86 (3 self)
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Abstract—Modeling data with linear combinations of a few elements from a learned dictionary has been the focus of much recent research in machine learning, neuroscience, and signal processing. For signals such as natural images that admit such sparse representations, it is now well established that these models are well suited to restoration tasks. In this context, learning the dictionary amounts to solving a large-scale matrix factorization problem, which can be done efficiently with classical optimization tools. The same approach has also been used for learning features from data for other purposes, e.g., image classification, but tuning the dictionary in a supervised way for these tasks has proven to be more difficult. In this paper, we present a general formulation for supervised dictionary learning adapted to a wide variety of tasks, and present an efficient algorithm for solving the corresponding optimization problem. Experiments on handwritten digit classification, digital art identification, nonlinear inverse image problems, and compressed sensing demonstrate that our approach is effective in large-scale settings, and is well suited to supervised and semi-supervised classification, as well as regression tasks for data that admit sparse representations. Index Terms—Basis pursuit, Lasso, dictionary learning, matrix factorization, semi-supervised learning, compressed sensing. Ç 1
On the Role of Sparse and Redundant Representations in Image Processing
- PROCEEDINGS OF THE IEEE – SPECIAL ISSUE ON APPLICATIONS OF SPARSE REPRESENTATION AND COMPRESSIVE SENSING
, 2009
"... Much of the progress made in image processing in the past decades can be attributed to better modeling of image content, and a wise deployment of these models in relevant applications. This path of models spans from the simple ℓ2-norm smoothness, through robust, thus edge preserving, measures of smo ..."
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Cited by 78 (1 self)
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Much of the progress made in image processing in the past decades can be attributed to better modeling of image content, and a wise deployment of these models in relevant applications. This path of models spans from the simple ℓ2-norm smoothness, through robust, thus edge preserving, measures of smoothness (e.g. total variation), and till the very recent models that employ sparse and redundant representations. In this paper, we review the role of this recent model in image processing, its rationale, and models related to it. As it turns out, the field of image processing is one of the main beneficiaries from the recent progress made in the theory and practice of sparse and redundant representations. We discuss ways to employ these tools for various image processing tasks, and present several applications in which state-of-the-art results are obtained.
Supervised translation-invariant sparse coding
- IN: IEEE CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION
, 2010
"... In this paper, we propose a novel supervised hierarchical sparse coding model based on local image descriptors for classification tasks. The supervised dictionary training is performed via back-projection, by minimizing the training error of classifying the image level features, which are extracted ..."
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Cited by 76 (8 self)
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In this paper, we propose a novel supervised hierarchical sparse coding model based on local image descriptors for classification tasks. The supervised dictionary training is performed via back-projection, by minimizing the training error of classifying the image level features, which are extracted by max pooling over the sparse codes within a spatial pyramid. Such a max pooling procedure across multiple spatial scales offer the model translation invariant properties, similar to the Convolutional Neural Network (CNN). Experiments show that our supervised dictionary improves the performance of the proposed model significantly over the unsupervised dictionary, leading to state-of-the-art performance on diverse image databases. Further more, our supervised model targets learning linear features, implying its great potential in handling large scale datasets in real applications.
