Abstract-The synthesis-based sparse representation model for signals has drawn 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 paper 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 examples. 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: This includes tailored pursuit algorithms -the Backward Greedy and the Optimized Backward Greedy algorithms, and a penalty function that defines the objective for the dictionary update stage. We demonstrate the effectiveness of the proposed dictionary learning in several experiments, treating synthetic data and real images, and showing a successful and meaningful recovery of the analysis dictionary.

We address single image super-resolution using a statistical prediction model based on sparse representations of low- and high-resolution image patches. The suggested model allows us to avoid any invariance assumption, which is a common practice in sparsity-based approaches treating this task. Prediction of high resolution patches is obtained via MMSE estimation and the resulting scheme has the useful interpretation of a feedforward neural network. To further enhance performance, we suggest data clustering and cascading several levels of the basic algorithm. We suggest a training scheme for the resulting network and demonstrate the capabilities of our algorithm, showing its advantages over existing methods based on a low- and high-resolution dictionary pair, in terms of computational complexity, numerical criteria, and visual appearance. The suggested approach offers a desirable compromise between low computational complexity and reconstruction quality, when comparing it with state-of-the-art methods for single image super-resolution.

Abstract-Signal modeling lies at the core of numerous signal and image processing applications. A recent approach that has drawn considerable attention is sparse representation modeling, in which the signal is assumed to be generated as a combination of a few atoms from a given dictionary. In this work we consider a Bayesian setting and go beyond the classic assumption of independence between the atoms. The main goal of this paper is to introduce a statistical model that takes such dependencies into account and show how this model can be used for sparse signal recovery. We follow the suggestion of two recent works and assume that the sparsity pattern is modeled by a Boltzmann machine, a commonly used graphical model. For general dependency models, exact MAP and MMSE estimation of the sparse representation becomes computationally complex. To simplify the computations, we propose greedy approximations of the MAP and MMSE estimators. We then consider a special case in which exact MAP is feasible, by assuming that the dictionary is unitary and the dependency model corresponds to a certain sparse graph. Exploiting this structure, we develop an efficient message passing algorithm that recovers the underlying signal. When the model parameters defining the underlying graph are unknown, we suggest an algorithm that learns these parameters directly from the data, leading to an iterative scheme for adaptive sparse signal recovery. The effectiveness of our approach is demonstrated on real-life signals -patches of natural images -where we compare the denoising performance to that of previous recovery methods that do not exploit the statistical dependencies.

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The primary challenge in linear inverse problems is to design stable and robust "decoders" to reconstruct high-dimensional vectors from a low-dimensional observation through a linear operator. Sparsity, low-rank, and related assumptions are typically exploited to design decoders which performance is then bounded based on some measure of deviation from the idealized model, typically using a norm.This paper focuses on characterizing the fundamental performance limits that can be expected from an ideal decoder given a general model, i.e., a general subset of "simple" vectors of interest. First, we extend the so-called notion of instance optimality of a decoder to settings where one only wishes to reconstruct some part of the original high dimensional vector from a low-dimensional observation. This covers practical settings such as medical imaging of a region of interest, or audio source separation when one is only interested in estimating the contribution of a specific instrument to a musical recording. We define instance optimality relatively to a model much beyond the traditional framework of sparse recovery, and characterize the existence of an instance optimal decoder in terms of joint properties of the model and the considered linear operator. Noiseless and noise-robust settings are both considered. We show somewhat surprisingly that the existence of noise-aware instance optimal decoders for all noise levels implies the existence of a noise-blind decoder.A consequence of our results is that for models that are rich enough to contain an orthonormal basis, the existence of an 2 / 2 instance optimal decoder is only possible when the linear operator is not substantially dimension-reducing. This covers well-known cases (sparse vectors, low-rank matrices) as well as a number of seemingly new situations (structured sparsity and sparse inverse covariance matrices for instance).We exhibit an operator-dependent norm which, under a model-specific generalization of the Restricted Isometry Property (RIP), always yields a feasible instance optimality property. This norm can be upper bounded by an atomic norm relative to the considered model.