# Talks

## Sparse Representation Theory

## Sparse Representation Theory

Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk we start by presenting this model and its features, and then turn to describe two special cases of it – the convolutional sparse coding (CSC) and its multi-layered version (ML-CSC). Amazingly, as we will carefully show, ML-CSC provides a solid theoretical foundation to … deep-learning. Alongside this main message of bringing a theoretical backbone to deep-learning, another central message that will accompany us throughout the talk: Generative models for describing data sources enable a systematic way to design algorithms, while also providing a complete mechanism for a theoretical analysis of these algorithms’ performance. This talk is meant for newcomers to this field – no prior knowledge on sparse approximation is assumed.

Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk we start by presenting this model and its features, and then turn to describe two special cases of it – the convolutional sparse coding (CSC) and its multi-layered version (ML-CSC). Amazingly, as we will carefully show, ML-CSC provides a solid theoretical foundation to … deep-learning. Alongside this main message of bringing a theoretical backbone to deep-learning, another central message that will accompany us throughout the talk: Generative models for describing data sources enable a systematic way to design algorithms, while also providing a complete mechanism for a theoretical analysis of these algorithms’ performance. This talk is meant for newcomers to this field – no prior knowledge on sparse approximation is assumed.

Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk we start by presenting this model and its features, and then turn to describe two special cases of it – the convolutional sparse coding (CSC) and its multi-layered version (ML-CSC). Amazingly, as we will carefully show, ML-CSC provides a solid theoretical foundation to … deep-learning. Alongside this main message of bringing a theoretical backbone to deep-learning, another central message that will accompany us throughout the talk: Generative models for describing data sources enable a systematic way to design algorithms, while also providing a complete mechanism for a theoretical analysis of these algorithms’ performance. This talk is meant for newcomers to this field – no prior knowledge on sparse approximation is assumed.

Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk we start by presenting this model, and then turn to describe two special cases of it – the convolutional sparse coding (CSC) and its multi-layered version (ML-CSC). Amazingly, as we will carefully show, ML-CSC provides a solid theoretical foundation to … deep-learning. This talk is meant for newcomers to these fields – no prior knowledge on sparse approximation is assumed.

Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk, we describe a special case of this model— the multi-layered convolutional sparse coding (ML-CSC) construction. As we will carefully show, ML-CSC provides a solid theoretical foundation to the field of deep learning, explaining the used architectures, their performance limits, and prospects for future alternatives.

Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk we start by presenting this model, and then turn to describe two special cases of it — the convolutional sparse coding (CSC) and its multi-layered version (ML-CSC). Amazingly, as we will carefully show, ML-CSC provides a solid theoretical foundation to вЂ¦ deep-learning. This talk is meant for newcomers to these fields – no prior knowledge on sparse approximation is assumed.

Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk we describe two special cases of this model — the convolutional sparse coding (CSC) and its multi-layered version (ML-CSC). We show that the projection of signals (a.k.a. pursuit) to the ML-CSC model leads to various deep convolutional neural network architectures. This connection brings a fresh view to CNN, as we are able to accompany the above by theoretical claims such as uniqueness of the representations throughout the network, and their stable estimation, all guaranteed under simple local sparsity conditions. The ‘take-home-message’ from this talk is this: The ML-CSC model can serve as the theoretical foundation to deep-learning.

Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk we start by presenting this model and its features, and then turn to describe two special cases of it: the convolutional sparse coding (CSC) and its multi-layered version (ML-CSC). Amazingly, as we will carefully show, ML-CSC provides a solid theoretical foundation to the field of deep-learning. Alongside this main message of bringing a theoretical backbone to deep-learning, another central message that will accompany us throughout the talk: Generative models for describing data sources enable a systematic way to design algorithms, while also providing a complete mechanism for a theoretical analysis of these algorithms’ performance. This talk is meant for newcomers to this field – no prior knowledge on sparse approximation is assumed.

Within the wide field of sparse approximation, convolutional sparse coding (CSC) has gained increasing attention in recent years. This model assumes a structured-dictionary built as a union of banded Circulant matrices. Most of the attention has been devoted to the practical side of CSC, proposing efficient algorithms for the pursuit problem, and identifying applications that benefit from this model. Interestingly, a systematic theoretical understanding of CSC seems to have been left aside, with the assumption that the existing classical results are sufficient. In this talk we start by presenting a novel analysis of the CSC model and its as- sociated pursuit. Our study is based on the observation that while being global, this model can be characterized and analyzed locally.

We show that uniqueness of the representation, its stability with respect to noise, and successful greedy or convex recovery are all guaranteed assuming that the underlying representation is locally sparse. These new results are much stronger and informative, compared to those obtained by deploying the classical sparse theory. Armed with these new insights, we proceed by proposing a multi-layer extension of this model, ML-CSC, in which signals are assumed to emerge from a cascade of CSC layers. This, in turn, is shown to be tightly connected to Convolutional Neural Networks (CNN), so much so that the forward-pass of the CNN is in fact the Thresholding pursuit serving the ML-CSC model. This connection brings a fresh view to CNN, as we are able to attribute to this architecture theoretical claims such as uniqueness of the representations throughout the network, and their stable estimation, all guaranteed under simple local sparsity conditions. Lastly, identifying the weaknesses in the above scheme, we propose an alternative to the forward-pass algorithm, which is both tightly connected to deconvolutional and recurrent neural networks, and has better theoretical guarantees.

Within the wide field of sparse approximation, convolutional sparse coding (CSC) has gained increasing attention in recent years. This model assumes a structured-dictionary built as a union of banded Circulant matrices. Most of the attention has been devoted to the practical side of CSC, proposing efficient algorithms for the pursuit problem, and identifying applications that benefit from this model. Interestingly, a systematic theoretical understanding of CSC seems to have been left aside, with the assumption that the existing classical results are sufficient. In this talk we start by presenting a novel analysis of the CSC model and its as- sociated pursuit. Our study is based on the observation that while being global, this model can be characterized and analyzed locally. We show that uniqueness of the representation, its stability with respect to noise, and successful greedy or convex recovery are all guaranteed assuming that the underlying representation is locally sparse.

These new results are much stronger and informative, compared to those obtained by deploying the classical sparse theory. Armed with these new insights, we proceed by proposing a multi-layer extension of this model, ML-CSC, in which signals are assumed to emerge from a cascade of CSC layers. This, in turn, is shown to be tightly connected to Convolutional Neural Networks (CNN), so much so that the forward-pass of the CNN is in fact the Thresholding pursuit serving the ML-CSC model. This connection brings a fresh view to CNN, as we are able to attribute to this architecture theoretical claims such as uniqueness of the representations throughout the network, and their stable estimation, all guaranteed under simple local sparsity conditions. Lastly, identifying the weaknesses in the above scheme, we propose an alternative to the forward-pass algorithm, which is both tightly connected to deconvolutional and recurrent neural networks, and has better theoretical guarantees.

Recent work in image processing repeatedly shows highly efficient reconstruction algorithms that lean on modeling of small overlapping patches. Such methods impose a local model in order to regularize a global inverse problem. Why does this work so well? Does this leave room for improvements? What does a local model imply globally on the unknown signal? In this talk we will start from algorithmic attempts that aim to understand this dichotomy in order to narrow the global-local gap. Gradually, we will turn the discussion to a theoretical point of view that provides a deeper understanding of such local models, and their global implications.

In this survey talk I will walk you through a decade of fascinating research activity on “sparse and redundant representations”. We will start with a classic image processing task of noise removal and use it as a platform for the introduction of data models in general, and sparsity and redundancy as specific forces in such models. The emerging model will be shown to lead to a series of key theoretical and numerical questions, which we will handle next. A key problem with the use of sparse and redundant representation modeling is the need for a sparsifying dictionary – we will discuss ways to obtain such a dictionary by learning from examples, and introduce the K-SVD algorithm. Then we will show how all these merge into a coherent theory that can be deployed successfully to various image processing applications.

In this talk we describe the co-sparse analysis model, with emphasis on pursuit algorithms and dictionary learning for it. We present two of our recent activities on this subject: (i) A theoretical study of the Analysis-Thresholding algorithm, exposing measures of goodness for the dictionary that govern the pursuit performance; and (ii) The development of an analysis K-SVD algorithm that trains a dictionary from signal examples and its use for image denoising.

In this talk we describe the co-sparse analysis model, with emphasis on pursuit algorithms and dictionary learning for it. We present two of our recent activities on this subject: (i) A theoretical study of the Analysis-Thresholding algorithm, exposing measures of goodness for the dictionary that govern the pursuit performance; and (ii) The development of an analysis K-SVD algorithm that trains a dictionary from signal examples and its use for image denoising.

In this survey talk I will walk you through a decade of fascinating research activity on “sparse and redundant representations”. We will start with a classic image processing task of noise removal and use it as a platform for the introduction of data models in general, and sparsity and redundancy as specific forces in such models. The emerging model will be shown to lead to a series of key theoretical and numerical questions, which we will handle next. A key problem with the use of sparse and redundant representation modeling is the need for a sparsifying dictionary — we will discuss ways to obtain such a dictionary by learning from examples, and introduce the K-SVD algorithm. Then we will show how all these merge into a coherent theory that can be deployed successfully to various image processing applications.

The synthesis-based sparse representation model for signals has drawn a considerable interest in the past decade. Such a model assumes that the signal of interest can be decomposed as a linear combination of a few atoms from a given dictionary. In this talk we concentrate on an alternative, analysis-based model, where an analysis operator — hereafter referred to as the “Analysis Dictionary” – multiplies the signal, leading to a sparse outcome. While the two alternative models seem to be very close and similar, they are in fact very different. In this talk we define clearly the analysis model and describe how to generate signals from it. We discuss the pursuit denoising problem that seeks the zeros of the signal with respect to the analysis dictionary given noisy measurements. Finally, we explore ideas for learning the analysis dictionary from a set of signal examples. We demonstrate this model’s effectiveness in several experiments, treating synthetic data and real images, showing a successful and meaningful recovery of the analysis dictionary.

In this survey talk I will walk you through a decade of fascinating research activity on “sparse and redundant representations”. We will start with a classic image processing task of noise removal and use it as a platform for the introduction of data models in general, and sparsity and redundancy as specific forces in such models. The emerging model will be shown to lead to a series of key theoretical and numerical questions, which we will handle next. A key problem with the use of sparse and redundant representation modeling is the need for a sparsifying dictionary — we will discuss ways to obtain such a dictionary by learning from examples, and introduce the K-SVD algorithm. Then we will show how all these merge into a coherent theory that can be deployed successfully to various image processing applications.

The synthesis-based sparse representation model for signals has drawn a considerable interest in the past decade. Such a model assumes that the signal of interest can be decomposed as a linear combination of a few atoms from a given dictionary. In this talk we concentrate on an alternative, analysis-based model, where an analysis operator — hereafter referred to as the “Analysis Dictionary” – multiplies the signal, leading to a sparse outcome. While the two alternative models seem to be very close and similar, they are in fact very different. In this talk we define clearly the analysis model and describe how to generate signals from it. We discuss the pursuit denoising problem that seeks the zeros of the signal with respect to the analysis dictionary given noisy measurements. Finally, we explore ideas for learning the analysis dictionary from a set of signal examples. We demonstrate this model’s effectiveness in several experiments, treating synthetic data and real images, showing a successful and meaningful recovery of the analysis dictionary.

The synthesis-based sparse representation model for signals has drawn a considerable interest in the past decade. Such a model assumes that the signal of interest can be decomposed as a linear combination of a *few* atoms from a given dictionary. In this talk we concentrate on an alternative, analysis-based model, where an analysis operator — hereafter referred to as the “Analysis Dictionary” – multiplies the signal, leading to a sparse outcome. Our goal is to learn the analysis dictionary from a set of signal examples, and the approach taken is parallel and similar to the one adopted by the K-SVD algorithm that serves the corresponding problem in the synthesis model. We present the development of the algorithm steps, which include a tailored pursuit algorithm termed “Backward Greedy” algorithm and a penalty function for the dictionary update stage. We demonstrate its effectiveness in several experiments, treating synthetic data and real images, showing a successful and meaningful recovery of the analysis dictionary.

In the commonly used sparse representation modeling, the atoms are assumed to be independent of each other when forming the signal. In this talk we shall introduce a statistical model called Boltzman Machine (BM) that enables such dependencies to be taken into account. Adopting a Bayesian point of view, we first treat the pursuit problem – given a signal, and assuming that the model parameters and the dictionary are known, find its sparse representation. We derive the exact MAP estimation, and show that just like in the independent case, this leads to an exponential search problem. We derive two algorithms for its evaluation: a greedy approximation approach for the general case, and an exact estimation that corresponds to a unitary dictionary and banded interaction matrix. We also consider the estimation of the model parameters, learning these parameters directly from training data. We show that given the signals’ representations, this problem can be posed as a convex optimization task by using the Maximum Pseudo-Likelihood (MPL).

This course (4 lectures and one tutorial) brings the core ideas and achievements made in the field of sparse and redundant representation modeling, with emphasis on the impact of this field to image processing applications. The five lectures (given as PPTX and PDF) are organized as follows:

Lecture 1: The core sparse approximation problem and pursuit algorithms that aim to approximate its solution.

Lecture 2: The theory on the uniqueness of the sparsest solution of a linear system, the notion of stability for the noisy case, guarantees for the performance of pursuit algorithms using the mutual coherence and the RIP.

Lecture 3: Signal (and image) models and their importance, the Sparseland model and its use, analysis versus synthesis modeling, a Bayesian estimation point of view, dictionary learning with the MOD and the K-SVD, global and local image denoising, local image inpainting.

Lecture 4: Sparse representations in image processing – image deblurring, global image separation and image inpainting. using dictionary learning for image and video denoising and inpainting, image scale-up using a pair of learned dictionaries, facial image compression with the K-SVD.

This course (5 lectures) brings the core ideas and achievements made in the field of sparse and redundant representation modeling, with emphasis on the impact of this field to image processing applications. The five lectures (given as PPTX and PDF) are organized as follows:

Lecture 1: The core sparse approximation problem and pursuit algorithms that aim to approximate its solution.

Lecture 2: The theory on the uniqueness of the sparsest solution of a linear system, the notion of stability for the noisy case, guarantees for the performance of pursuit algorithms using the mutual coherence and the RIP.

Lecture 3: Signal (and image) models and their importance, the Sparseland model and its use, analysis versus synthesis modeling, a Bayesian estimation point of view.

Lecture 4: First steps in image processing with the Sparseland model – image deblurring, image denoising, image separation, and image inpainting. Global versus local processing of images. Dictionary learning with the MOD and the K-SVD.

Lecture 5: Advanced image processing: Using dictionary learning for image and video denoising and inpainting, image scale-up using a pair of learned dictionaries, Facial image compression with the K-SVD.

This survey talk focuses on the use of sparse and redundant representations and learned dictionaries for image denoising and other related problems. We discuss the the K-SVD algorithm for learning a dictionary that describes the image content efficiently. We then show how to harness this algorithm for image denoising, by working on small patches and forcing sparsity over the trained dictionary. The above is extended to color image denoising and inpainting, video denoising, and facial image compression, leading in all these cases to state of the art results. We conclude with more recent results on the use of several sparse representations for getting better denoising performance. An algorithm to generate such set of representations is developed, and our analysis shows that by this we approximate the minimum-mean-squared-error (MMSE) estimator, thus getting better results.

Among the many ways to model signals, a recent approach that draws considerable attention is sparse representation modeling. In this model, the signal is assumed to be generated as a random linear combination of a few atoms from a pre-specified dictionary. In this talk we describe our recent work that analyzed two Bayesian denoising algorithms — the Maximum-Aposteriori Probability (MAP) and the Minimum-Mean-Squared-Error (MMSE) estimators, under the assumption that the dictionary is unitary. It is well known that both these estimators lead to a scalar shrinkage on the transformed coefficients, albeit with a different response curve. In this talk we start by deriving closed-form expressions for these shrinkage curves and then analyze their performance. Upper bounds on the MAP and the MMSE estimation errors are derived. We tie these to the error obtained by a so-called oracle estimator, where the support is given, establishing a worst-case gain-factor between the MAP/MMSE estimation errors and the oracle’s performance.

Cleaning of noise from signals is a classical and long-studied problem in signal processing. Algorithms for this task necessarily rely on an a-priori knowledge about the signal characteristics, along with information about the noise properties. For signals that admit sparse representations over a known dictionary, a commonly used denoising technique is to seek the sparsest representation that synthesizes a signal close enough to the corrupted one. As this problem is too complex in general, approximation methods, such as greedy pursuit algorithms, are often employed. In this line of reasoning, we are led to believe that detection of the sparsest representation is key in the success of the denoising goal. Does this means that other competitive and slightly inferior sparse representations are meaningless? Suppose we are served with a group of competing sparse representations, each claiming to explain the signal differently. Can those be fused somehow to lead to a better result? Surprisingly, the answer to this question is positive; merging these representations can form a more accurate, yet dense, estimate of the original signal even when the latter is known to be sparse. In this talk we demonstrate this behavior, propose a practical way to generate such a collection of representations by randomizing the Orthogonal Matching Pursuit (OMP) algorithm, and produce a clear analytical justification for the superiority of the associated Randomized OMP (RandOMP) algorithm. We show that while the Maximum a-posterior Probability (MAP) estimator aims to find and use the sparsest representation, the Minimum Mean-Squared-Error (MMSE) estimator leads to a fusion of representations to form its result. Thus, working with an appropriate mixture of candidate representations, we are surpassing the MAP and tending towards the MMSE estimate, and thereby getting a far more accurate estimation, especially at medium and low SNR. Another topic covered in thistalk concerns the case of a unitary dictionary. In such a case it is well-known that the MAP estimators has a closed-form and exact solution, and OMP is accurately computing it. Can a similar result be derived for MMSE? We show that this is indeed possible, obtaining a recursive formula that computes the MMSE simply and exactly.

In this talk we describe applications such as image denoising and beyond using sparse and redundant representations. Our focus is on ways to perform these tasks with trained dictionaries using the K-SVD algorithm. As trained dictionaries are limited in handling small image patches, we deploy these within a Bayesian reconstruction procedure by forming an image prior that forces every patch in the resulting image to have a sparse representation.

Cleaning of noise from signals is a classical and long-studied problem in signal processing. Algorithms for this task necessarily rely on an a-priori knowledge about the signal characteristics, along with information about the noise properties. For signals that admit sparse representations over a known dictionary, a commonly used denoising technique is to seek the sparsest representation that synthesizes a signal close enough to the corrupted one. As this problem is too complex in general, approximation methods, such as greedy pursuit algorithms, are often employed. In this line of reasoning, we are led to believe that detection of the sparsest representation is key in the success of the denoising goal. Does this means that other competitive and slightly inferior sparse representations are meaningless? Suppose we are served with a group of competing sparse representations, each claiming to explain the signal differently. Can those be fused somehow to lead to a better result? Surprisingly, the answer to this question is positive; merging these representations can form a more accurate, yet dense, estimate of the original signal even when the latter is known to be sparse. In this talk we demonstrate this behavior, propose a practical way to generate such a collection of representations by randomizing the Orthogonal Matching Pursuit (OMP) algorithm, and produce a clear analytical justification for the superiority of the associated Randomized OMP (RandOMP) algorithm. We show that while the Maximum a-posterior Probability (MAP) estimator aims to nd and use the sparsest representation, the Minimum Mean-Squared-Error (MMSE) estimator leads to a fusion of representations to form its result. Thus, working with an appropriate mixture of candidate representations, we are surpassing the MAP and tending towards the MMSE estimate, and thereby getting a far more accurate estimation, especially at medium and low SNR.

An under-determined linear system of equations Ax = b with non-negativity constraint is considered. It is shown that for matrices A with a row-span intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required sparsity depends on a coherence property of the matrix A. It is further shown that A undergoes a conditioning stage with some degrees of freedom, which may be used to improve the coherence measure and strengthen the claimed result.

In this talk we consider several inverse problems in image processing, using sparse and redundant representations over trained dictionaries. Using the K-SVD algorithm, we obtain a dictionary that describes the image content effectively. Two training options are considered: using the corrupted image itself, or training on a corpus of high-quality image database. Since the K-SVD is limited in handling small image patches, we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image. We show how such Bayesian treatment leads to a simple and effective denoising algorithm for gray-level images with state-of-the-art denoising performance. We then extend these results to color images, handling their denoising, inpainting, and demosaicing. Following the above ideas, with necessary modifications to avoid color artifacts and over-fitting, we present stat-of-the art results in each of these applications. Another extension considered is video denoising — we demonstrate how the above method can be extended to work with 3D patches, propagate the dictionary from one frame to another, and get both improved denoising performance while also reducing substantially the computational load per pixel.

In signal and image processing, we often use transforms in order to simplify operations or to enable better treatment to the given data. A recent trend in these fields is the use of over complete linear transforms that lead to a sparse description of signals. This new breed of methods is more difficult to use, often requiring more computations. Still, they are much more effective in applications such as signal compression and inverse problems. In fact, much of the success attributed to the wavelet transform in recent years, is directly related to the above-mentioned trend. In this talk we will present a survey of this recent path of research, and its main results. We will discuss both the theoretic and the application sides to this field. No previous knowledge is assumed (… just common sense, and little bit of linear algebra).

A recent trend in signal, image, and data analysis is the use of overcomplete linear transforms that lead to a sparse description of the processed data. This new breed of methods is more difficult to use, but they are much more effective in applications such as data compression and solution of inverse problems. In this talk we present a wide angle view to this recent path of research.

In signal and image processing, we often use transforms in order to simplify operations or to enable better treatment to the given data. A recent trend in these fields is the use of overcomplete linear transforms that lead to a sparse description of signals. This new breed of methods is more difficult to use, often requiring more computations. Still, they are much more effective in applications such as signal compression and inverse problems. In fact, much of the success attributed to the wavelet transform in recent years, is directly related to the above-mentioned trend. In this talk we will present a survey of this recent path of research, its main results, and the involved players and their contributions. We will discuss both the theoretic and the application sides to this field. No previous knowledge is assumed.

Transforming signals is typically done in order to simplify their representations. Among the many ways to do so, the use of linear combinations taken over a redundant dictionary is appealing due to both its simplicity and its diversity. Choosing the sparsest of all solutions aligns well with our desire for a simple signal description, and this also leads to uniqueness. Since the search for the sparsest representation is NP-hard, methods such as the Basis-Pursuit (BP) and the Matching Pursuit (MP) have been proposed in the mid 90’s to approximate the desired sparse solution.

The pioneering work by Donoho and Huo (’99) started a sequence of research efforts, all aiming to theoretically understand the quality of approximations obtained by the pursuit algorithms, and the limits to their success. A careful study established that both BP and MP algorithms are expected to lead to the sparsest of all representations if indeed such solution is sparse enough. Later work generalized these results to the case where error is allowed in the representation. Very recent results addressed the same analysis from a probabilistic point of view, finding bounds on the average performance, and showing a close resemblance to empirical evidence.

All these results lead to the ability to use the pursuit algorithms with clear understanding of their expected behavior, in what Stanley Osher would have called “emotionally uninvolved” manner. This paves the way for future transforms that will be based on (i) overcomplete (redundant) representations, (ii) linear in constructing signals, and non-linear in their decomposition, and (iii) sparsity as their core force. Furthermore, as signal transforms, signal compression, and inverse problems, are all tangled together, we are now armed with new and effective tools when addressing many problems in signal and image processing.

In this talk we present a survey of this recent path of research, its main results, and the involved players and their contributions. We will discuss both the theoretic and the application sides to this field. No previous knowledge is assumed.

In this talk we present a novel method for separating images into texture and piecewise smooth parts, and show how this formulation can also lead to image inpainting. Our separation and inpainting process exploits both the variational and the sparsity mechanisms, by combining the Basis Pursuit Denoising (BPDN) algorithm and the Total-Variation (TV) regularization scheme.

The basic idea in this work is the use of two appropriate dictionaries, one for the representation of textures, and the other for the natural scene parts, assumed to be piece-wise-smooth. Both dictionaries are chosen such that they lead to sparse representations over one type of image-content (either texture or piecewise smooth). The use of the BPDN with the two augmented dictionaries leads to the desired separation, along with noise removal as a by-product. As the need to choose a proper dictionary for natural scene is very hard, a TV regularization is employed to better direct the separation process.

This concept of separation via sparse and over-complete representation of the image is shown to have a direct and natural extension to image inpainting. When some of the pixels in known locations in the image are missing, the same separation formulation can be changed to fit the problem of decomposing the image while filling in the holes. Thus, as a by-product of the separation we achieve inpainting. This approach should be compared to a recently published inpainting system by Bertalmio, Vese, Sapiro, and Osher. We will present several experimental results that validate the algorithm’s performance.

When over-complete dictionary is used, the search for the sparsest of all representations amounts to the need to solve a highly complicated combinatorial problem, with complexity growing exponenentially with the number of atoms in the dictionary. The Basis Pursuit algorithm (BP) (by Chen, Donoho, and Saundrers, 1995) was shown to be a highly effective tool for approximating the solution for the above problem.

In this talk we describe a fascinating analysis that explains the surprising empirical success of the PB. We start by presenting Donoho and Huo’s results, and then show how these results could be further improved. We first discuss the case of pair of ortho-bases and show tighter bounds on the required sparsity of the signal representation that guarantees BP-optimality. We then turn to extend previous results for arbitrary dictionaries, showing that all previous work falls as a special case of the new theory. Finally, we show how the obtained analysis results can be used to solve the problem of separating a volume of voxels into a sparse set of points, lines, and planes.

Transforming a signal is typically done in order to simplify its representation. Among the many ways to represent signals, the use of a linear combination taken from an over-complete dictionary is appealing due to its ability to cover a diversity of signal behaviors in one transform. However, with this benefit comes the problem of non-unique representation. Choosing the sparsest of all representations aligns well with our desire for simple signal description, but searching such representation becomes a hard to solve non-convex and highly non-smooth optimization problem. The “Basis-Pursuit” (BP) algorithm (Chen, Donoho, and Saunders – 1995) is a stable approximate solver for the above task, replacing the non-convex L_0 minimization with a L_1 norm. A later work by Donoho and Huo (1998) theoretically proved that for specific dictionaries built from pair of ortho-bases and for sparse enough representations, the BP algorithm is indeed optimal.

In this talk we start by describing Donoho and Huo’s work on the BP algorithm, and then show how these results could be further improved. We first discuss the case of pair of ortho-bases and show tighter bounds on the required sparsity of the signal representation that guarantees BP-optimality. We then turn to extend previous results for arbitrary dictionaries, showing that all previous work falls as a special case of the new theory. Finally, we show work in progress that relates the Basis Pursuit algorithm used as a non-linear filtering to the Bayesian approach. We show how TV, Wavelet denoising, general Bayesian priors, and specifically the bilateral filter are all special cases of the Basis Pursuit for difference choices of dictionaries.