Working Locally Thinking Globally: Theoretical Guarantees for Convolutional Sparse Coding

Papyan, Vardan; Sulam, Jeremias; Elad, Michael

doi:10.1109/tsp.2017.2733447

Cited by 104 publications

(106 citation statements)

References 50 publications

(117 reference statements)

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“…The P 0,∞ problem. A new prior for convolutional sparse coding was recently proposed in [33]. Rather than minimizing the 0 "norm" (the total number of nonzero coefficients in the convolutional representation), this model minimizes the 0,∞ group "norm".…”

Section: Of D Computed Efficiently Using Convolutions With Columns Of Dmentioning

confidence: 99%

“…The work in [33] provides guarantees for the success of the standard OMP and 1 relaxation in the ideal and noisy regimes. It proposes optimization methods for solving an approximation of (5) using the Alternating Direction Method of Multipliers (ADMM) [6] to minimize the 1 norm of the representation with additional penalties, which bias the solution towards a small 0,∞ "norm".…”

Section: Of D Computed Efficiently Using Convolutions With Columns Of Dmentioning

confidence: 99%

“…Convolutional sparse coding with the 0,∞ prior has been recently proposed for natural images. However, the algorithms that have so far been introduced for solving it are relaxation based and only tackle the 0,∞ "norm" indirectly, relying on 1 minimization and unconstrained optimization [32], [33].…”

Section: Contributionmentioning

confidence: 99%

“…Conclusions. The greedy algorithms that we have proposed in this work solve the constrained P 0,∞ problem and offer an alternative to the approaches in [32] and [33], which minimize an unconstrained penalized Lagrangian with a convex relaxation to the 1 norm. (c) denoising using the patch-based method from [11]; (d) denoising using the convolutional sparse coding method from [44] (e) denoising using a dictionary trained on the clean training set, 100 atoms and (f ) 256 atoms; (g) denoising using a dictionary trained on the noisy test set, 51 atoms and (h) 100 atoms.…”

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confidence: 99%

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A Greedy Approach to $\ell_{0,\infty}$-Based Convolutional Sparse Coding

Plaut¹,

Giryes²

2019

SIAM J. Imaging Sci.

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Sparse coding techniques for image processing traditionally rely on a processing of small overlapping patches separately followed by averaging. This has the disadvantage that the reconstructed image no longer obeys the sparsity prior used in the processing. For this purpose convolutional sparse coding has been introduced, where a shift-invariant dictionary is used and the sparsity of the recovered image is maintained. Most such strategies target the 0 "norm" or the 1 norm of the whole image, which may create an imbalanced sparsity across various regions in the image. In order to face this challenge, the 0,∞ "norm" has been proposed as an alternative that "operates locally while thinking globally". The approaches taken for tackling the non-convexity of these optimization problems have been either using a convex relaxation or local pursuit algorithms. In this paper, we present an efficient greedy method for sparse coding and dictionary learning, which is specifically tailored to 0,∞, and is based on matching pursuit. We demonstrate the usage of our approach in salt-and-pepper noise removal and image inpainting.1. Introduction. Sparse coding can be described as solving the following minimization problem, known as the P 0 problem [9]:where α ∈ R p is a sparse representation of a signal x ∈ R N in the dictionary D ∈ R N ×p . The columns of D, which are referred to as atoms, are a full and overcomplete set, and we will assume without loss of generality that they are normalized to unit 2 norm. The 0 "norm" 1 returns the number of nonzero elements in a vector, also called the sparsity.When modeling natural images, we allow some deviation from the model rather than seeking a perfect reconstruction:An alternative form, in which the sparsity k is known, is:As the P 0 and P k 0 problems are NP-hard, several approximation techniques have been proposed. Matching Pursuit (MP) [28] is a greedy algorithm that in each iteration updates * Accepted for publication in SIAM Journal on Imaging Sciences (SIIMS).

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Section: Of D Computed Efficiently Using Convolutions With Columns Of Dmentioning

confidence: 99%

Section: Of D Computed Efficiently Using Convolutions With Columns Of Dmentioning

confidence: 99%

Section: Contributionmentioning

confidence: 99%

mentioning

confidence: 99%

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A Greedy Approach to $\ell_{0,\infty}$-Based Convolutional Sparse Coding

Plaut¹,

Giryes²

2019

SIAM J. Imaging Sci.

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“…The convolutional sparse representation model [27], [28], [29] where the dictionary is a concatenation of circulant matrices has been extensively studied in the past. Furthermore, recent work [13] The author is with the Istituto Italiano di Tecnologia (IIT), Genova, Italy. Contact e-mail address: cristian.rusu@iit.it.…”

Section: Introductionmentioning

confidence: 99%

On learning with shift-invariant structures

Rusu

2020

Digital Signal Processing

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In this paper, we describe new results and algorithms, based on circulant matrices, for the task of learning shift-invariant components from training data. We deal with the shift-invariant dictionary learning problem which we formulate using circulant and convolutional matrices (including unions of such matrices), define optimization problems that describe our goals and propose efficient ways to solve them. Based on these findings, we also show how to learn a wavelet-like dictionary from training data. We connect our work with various previous results from the literature and we show the effectiveness of our proposed algorithms using synthetic as well as real ECG signals and images.

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Data‐driven methods for quantitative imaging

Dong,

Flaschel,

Hintermüller

et al. 2024

GAMM-Mitteilungen

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In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative magnetic resonance imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill‐posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically‐oriented overview on how data‐driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.

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Working Locally Thinking Globally: Theoretical Guarantees for Convolutional Sparse Coding

Cited by 104 publications

References 50 publications

A Greedy Approach to $\ell_{0,\infty}$-Based Convolutional Sparse Coding

A Greedy Approach to $\ell_{0,\infty}$-Based Convolutional Sparse Coding

On learning with shift-invariant structures

Data‐driven methods for quantitative imaging

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