Abstract-Sparse and redundant representation modeling of data assumes an ability to describe signals as linear combinations of a few atoms from a pre-specified dictionary. As such, the choice of the dictionary that sparsifies the signals is crucial for the success of this model. In general, the choice of a proper dictionary can be done using one of two ways: (i) building a sparsifying dictionary based on a mathematical model of the data, or (ii) learning a dictionary to perform best on a training set. In this paper we describe the evolution of these two paradigms. As manifestations of the first approach, we cover topics such as wavelets, wavelet packets, contourlets, and curvelets, all aiming to exploit 1-D and 2-D mathematical models for constructing effective dictionaries for signals and images. Dictionary learning takes a different route, attaching the dictionary to a set of examples it is supposed to serve. From the seminal work of Field and Olshausen, through the MOD, the K-SVD, the Generalized PCA and others, this paper surveys the various options such training has to offer, up to the most recent contributions and structures.

The concept of prior probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors assign probability to a signal through various forward measurements of it, while synthesisbased priors seek a reconstruction of the signal as a combination of atom signals. In this paper we describe these two prior classes, focusing on the distinction between them. We show that although when reducing to the complete and under-complete formulations the two become equivalent, in their more interesting overcomplete formulation the two types depart. Focusing on the 1 denoising case, we present several ways of comparing the two types of priors, establishing the existence of an unbridgeable gap between them.

Abstract-An efficient and flexible dictionary structure is proposed for sparse and redundant signal representation. The proposed sparse dictionary is based on a sparsity model of the dictionary atoms over a base dictionary, and takes the form D = ΦA where Φ is a fixed base dictionary and A is sparse. 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 paper 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.

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.

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 paper we concentrate on an alternative, analysis-based model, where an 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 two greedy tailored pursuit algorithms and a penalty function for the dictionary update stage. We demonstrate its effectiveness in several experiments, showing a successful and meaningful recovery of the analysis dictionary.