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

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

Super-resolution reconstruction proposes a fusion of several low-quality images into one higher quality result with better optical resolution. Classic super-resolution techniques strongly rely on the availability of accurate motion estimation for this fusion task. When the motion is estimated inaccurately, as often happens for nonglobal motion fields, annoying artifacts appear in the super-resolved outcome. Encouraged by recent developments on the video denoising problem, where state-of-the-art algorithms are formed with no explicit motion estimation, we seek a super-resolution algorithm of similar nature that will allow processing sequences with general motion patterns. In this paper, we base our solution on the Nonlocal-Means (NLM) algorithm. We show how this denoising method is generalized to become a relatively simple super-resolution algorithm with no explicit motion estimation. Results on several test movies show that the proposed method is very successful in providing super-resolution on general sequences.

Modeling signals by a sparse and redundant representations is drawing a consider-able attention in recent years. Coupled with the ability to train the dictionary using signal examples, these techniques have been shown to lead to state-of-the-art results in a series of recent applications. In this paper we propose a novel structure of such a model for representing image content. The new dictionary is itself a small image, such that every patch in it (in varying location and size) is a possible atom in the representation. We refer to this as the Image-Signature-Dictionary (ISD), and show how it can be trained from image examples. This novel structure enjoys several important features, such as shift and scale flexibilities, and smaller memory and computational requirements, compared to the classical dictionary approach. As a demonstration of these benefits, we present high-quality image denoising results based on this new model.

The K-SVD algorithm is a highly effective method of training overcomplete dictionaries for sparse signal representation. In this report we discuss an efficient implementation of this algorithm, which both accelerates it and reduces its memory consumption. The two basic components of our implementation are the replacement of the exact SVD computation with a much quicker approximation, and the use of the Batch-OMP method for performing the sparse-coding operations. Batch-OMP, which we also present in this report, is an implementation of the Orthogonal Matching Pursuit (OMP) algorithm which is specifically optimized for sparse-coding large sets of signals over the same dictionary. The Batch-OMP implementation is useful for a variety of sparsity-based techniques which involve coding large numbers of signals. In the report, we discuss the Batch-OMP and K-SVD implementations and analyze their complexities. The report is accompanied by Matlab Ⓡ toolboxes which implement these techniques, and can be downloaded at

Abstract—In this paper, we present a patch-based variational Bayesian framework for video processing and demonstrate its potential in denoising, inpainting and deinterlacing. Unlike previous methods based on explicit motion estimation, we propose to embed motion-related information into the relationship among video patches and develop a nonlocal sparsity-based prior for typical video sequences. Specifically, we first extend block matching (Nearest Neighbor search) into patch clustering (k-Nearest-Neighbor search), which represents motion in an implicit and distributed fashion. Then we show how to exploit the sparsity constraint by sorting and packing similar patches, which can be better understood from a manifold perspective. Under the Bayesian framework, we treat both patch clustering result and unobservable data as latent variables and solve the inference problem via variational EM algorithms. A weighted averaging strategy of fusing diverse inference results from overlapped patches is also developed. The effectiveness of patch-based video models is demonstrated by extensive experimental results on a wider range of video materials. Index Terms—video processing, patch-based models, sparsitybased priors, variational Bayesian, variational EM, weighted averaging.

We propose a powerful video denoising algorithm that exploits temporal and spatial redundancy characterizing natural video sequences. The algorithm implements the paradigm of nonlocal grouping and collaborative filtering, where a higher-dimensional transform-domain representation is leveraged to enforce sparsity and thus regularize the data. The proposed algorithm exploits the mutual similarity between 3-D spatiotemporal volumes constructed by tracking blocks along trajectories defined by the motion vectors. Mutually similar volumes are grouped together by stacking them along an additional fourth dimension, thus producing a 4-D structure, termed group, where different types of data correlation exist along the different dimensions: local correlation along the two dimensions of the blocks, temporal correlation along the motion trajectories, and nonlocal spatial correlation (i.e. self-similarity) along the fourth dimension. Collaborative filtering is realized by transforming each group through a decorrelating 4-D separable transform and then by shrinkage and inverse transformation. In this way, collaborative filtering provides estimates for each volume stacked in the group, which are then returned and adaptively aggregated to their original position in the video. Experimental results demonstrate the effectiveness of the proposed procedure which outperforms the state of the art.

An efficient and flexible dictionary structure is proposed for sparse and redundant signal representation. The structure is based on a sparsity model of the dictionary atoms over a base dictionary. The sparse dictionary provides efficient forward and adjoint operators, has a compact representation, and can be effectively trained from given example data. In this, the sparse structure bridges the gap between implicit dictionaries, which have efficient implementations yet lack adaptability, and explicit dictionaries, which are fully adaptable but non-efficient and costly to deploy. In this report we discuss the advantages of sparse dictionaries, and present an efficient algorithm for training them. We demonstrate the advantages of the proposed structure for 3-D image denoising. 1

Abstract—We propose an efficient orthonormal wavelet-domain video denoising algorithm based on an appropriate integration of motion compensation into an adapted version of our recently devised Stein’s unbiased risk estimator-linear expansion of thresholds (SURE-LET) approach. To take full advantage of the strong spatio-temporal correlations of neighboring frames, a global motion compensation followed by a selective blockmatching is first applied to adjacent frames, which increases their temporal correlations without distorting the interframe noise statistics. Then, a multiframe interscale wavelet thresholding is performed to denoise the current central frame. The simulations we made on standard grayscale video sequences for various noise levels demonstrate the efficiency of the proposed solution in reducing additive white Gaussian noise. Obtained at a lighter computational load, our results are even competitive with most state-of-the-art redundant wavelet-based techniques. By using a cycle-spinning strategy, our algorithm is in fact able to outperform these methods. Index Terms—Block-matching, Stein’s unbiased risk estimatorlinear expansion of thresholds (SURE-LET), video denoising, wavelet. I.

Sparse representations of signals, in particular with learned dictionaries, are widely used for state-of-the-art audio, image, and video restoration. In this paper, the problem of denoising and occlusion restoration of 3D range data based on dictionary learning and sparse representations is explored. We consider the 3D surface obtained from a desktop range scanner as an image, where the value of each pixel represents the depth of a point on the 3D surface. Having this image, we apply techniques from dictionary learning and sparse representation to enhance the acquired 3D surface. These techniques use the spare decomposition of the overlapping patches in the image, over an adapted over-complete dictionary, for enhancing the data. We present experimental results of denoising 3D surfaces following this approach. We also propose an algorithm for filling the missing information regions on 3D scans and demonstrate its effectiveness. Our experimental results are on range data obtained from a low-cost structured-light range scanner. Index Terms — Sparse representation, 3D surface denoising, Occlusion restoration. 1.