This book introduces sparse and redundant representations with a focus on applications in signal and image processing. It details mathematical modeling for signal sources along with how to use the ...model for tasks such as denoising, restoration and separation.
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.
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.
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.
Sparse representation has shown to be a very powerful model for real world signals, and has enabled the development of applications with notable performance. Combined with the ability to learn a ...dictionary from signal examples, sparsity-inspired algorithms are often achieving state-of-the-art results in a wide variety of tasks. These methods have traditionally been restricted to small dimensions mainly due to the computational constraints that the dictionary learning problem entails. In the context of image processing, this implies handling small image patches. In this work we show how to efficiently handle bigger dimensions and go beyond the small patches in sparsity-based signal and image processing methods. We build our approach based on a new cropped Wavelet decomposition, which enables a multi-scale analysis with virtually no border effects. We then employ this as the base dictionary within a double sparsity model to enable the training of adaptive dictionaries. To cope with the increase of training data, while at the same time improving the training performance, we present an Online Sparse Dictionary Learning (OSDL) algorithm to train this model effectively, enabling it to handle millions of examples. This work shows that dictionary learning can be up-scaled to tackle a new level of signal dimensions, obtaining large adaptable atoms that we call Trainlets.
This paper deals with the single image scale-up problem using sparse-representation modeling. The goal is to recover an original image from its blurred and down-scaled noisy version. Since this ...problem is highly ill-posed, a prior is needed in order to regularize it. The literature offers various ways to address this problem, ranging from simple linear space-invariant interpolation schemes (e.g., bicubic interpolation), to spatially-adaptive and non-linear filters of various sorts. We embark from a recently-proposed successful algorithm by Yang et. al. 1,2, and similarly assume a local Sparse-Land model on image patches, serving as regularization. Several important modifications to the above-mentioned solution are introduced, and are shown to lead to improved results. These modifications include a major simplification of the overall process both in terms of the computational complexity and the algorithm architecture, using a different training approach for the dictionary-pair, and introducing the ability to operate without a training-set by boot-strapping the scale-up task from the given low-resolution image. We demonstrate the results on true images, showing both visual and PSNR improvements.
Single image interpolation is a central and extensively studied problem in image processing. A common approach toward the treatment of this problem in recent years is to divide the given image into ...overlapping patches and process each of them based on a model for natural image patches. Adaptive sparse representation modeling is one such promising image prior, which has been shown to be powerful in filling-in missing pixels in an image. Another force that such algorithms may use is the self-similarity that exists within natural images. Processing groups of related patches together exploits their correspondence, leading often times to improved results. In this paper, we propose a novel image interpolation method, which combines these two forces-nonlocal self-similarities and sparse representation modeling. The proposed method is contrasted with competitive and related algorithms, and demonstrated to achieve state-of-the-art results.
The use of sparse representations in signal and image processing is gradually increasing in the past several years. Obtaining an overcomplete dictionary from a set of signals allows us to represent ...them as a sparse linear combination of dictionary atoms. Pursuit algorithms are then used for signal decomposition. A recent work introduced the K-SVD algorithm, which is a novel method for training overcomplete dictionaries that lead to sparse signal representation. In this work we propose a new method for compressing facial images, based on the K-SVD algorithm. We train K-SVD dictionaries for predefined image patches, and compress each new image according to these dictionaries. The encoding is based on sparse coding of each image patch using the relevant trained dictionary, and the decoding is a simple reconstruction of the patches by linear combination of atoms. An essential pre-process stage for this method is an image alignment procedure, where several facial features are detected and geometrically warped into a canonical spatial location. We present this new method, analyze its results and compare it to several competing compression techniques.
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