In the co-sparse analysis model a set of filters is applied to a signal out of the signal class of interest yielding sparse signal responses. As such, it may serve as a prior in inverse problems, or for structural analysis of signals that are known to belong to the signal class. The more the model is adapted to the class, the more reliable it is for these purposes. The task of learning such operators for a given class is therefore a crucial problem. In many applications, it is also required that the filter responses are obtained in a timely manner, which can be achieved by filters with a separable structure. Not only can operators of this sort be efficiently used for computing the filter responses, but they also have the advantage that less training samples are required to obtain a reliable estimate of the operator. The first contribution of this work is to give theoretical evidence for this claim by providing an upper bound for the sample complexity of the learning process. The second is a stochastic gradient descent (SGD) method designed to efficiently learn an analysis operators with separable structures, which incorporates an efficient step size selection. Numerical experiments are provided that link the sample complexity to the convergence speed of the SGD algorithm.
The authors of @cite_10 propose a method specialized on image processing. Instead of a patch-based approach, an image-based model is proposed with the goal of enforcing coherence across overlapping patches. In this framework, which is based on higher-order filter-based Markov Random Field models, all possible patches in the entire image are considered at once during the learning phase. A bi-level optimization scheme is proposed that has at its heart an unconstrained optimization problem w.r.t. the operator, which is solved using a quasi-Newton method.
@cite_10
2169488311 2172174689 1976416062 2198155329
We propose an unsupervised method for learning multi-stage hierarchies of sparse convolutional features. While sparse coding has become an increasingly popular method for learning visual features, it is most often trained at the patch level. Applying the resulting filters convolutionally results in highly redundant codes because overlapping patches are encoded in isolation. By training convolutionally over large image windows, our method reduces the redudancy between feature vectors at neighboring locations and improves the efficiency of the overall representation. In addition to a linear decoder that reconstructs the image from sparse features, our method trains an efficient feed-forward encoder that predicts quasi-sparse features from the input. While patch-based training rarely produces anything but oriented edge detectors, we show that convolutional training produces highly diverse filters, including center-surround filters, corner detectors, cross detectors, and oriented grating detectors. We show that using these filters in multistage convolutional network architecture improves performance on a number of visual recognition and detection tasks. We describe a novel unsupervised method for learning sparse, overcomplete features. The model uses a linear encoder, and a linear decoder preceded by a sparsifying non-linearity that turns a code vector into a quasi-binary sparse code vector. Given an input, the optimal code minimizes the distance between the output of the decoder and the input patch while being as similar as possible to the encoder output. Learning proceeds in a two-phase EM-like fashion: (1) compute the minimum-energy code vector, (2) adjust the parameters of the encoder and decoder so as to decrease the energy. The model produces "stroke detectors" when trained on handwritten numerals, and Gabor-like filters when trained on natural image patches. Inference and learning are very fast, requiring no preprocessing, and no expensive sampling. Using the proposed unsupervised method to initialize the first layer of a convolutional network, we achieved an error rate slightly lower than the best reported result on the MNIST dataset. Finally, an extension of the method is described to learn topographical filter maps. We propose a new high-quality and efficient single-image upscaling technique that extends existing example-based super-resolution frameworks. In our approach we do not rely on an external example database or use the whole input image as a source for example patches. Instead, we follow a local self-similarity assumption on natural images and extract patches from extremely localized regions in the input image. This allows us to reduce considerably the nearest-patch search time without compromising quality in most images. Tests, that we perform and report, show that the local self-similarity assumption holds better for small scaling factors where there are more example patches of greater relevance. We implement these small scalings using dedicated novel nondyadic filter banks, that we derive based on principles that model the upscaling process. Moreover, the new filters are nearly biorthogonal and hence produce high-resolution images that are highly consistent with the input image without solving implicit back-projection equations. The local and explicit nature of our algorithm makes it simple, efficient, and allows a trivial parallel implementation on a GPU. We demonstrate the new method ability to produce high-quality resolution enhancement, its application to video sequences with no algorithmic modification, and its efficiency to perform real-time enhancement of low-resolution video standard into recent high-definition formats. Patch-based low-rank models have shown effective in exploiting spatial redundancy of natural images especially for the application of image denoising. However, two-dimensional low-rank model can not fully exploit the spatio-temporal correlation in larger data sets such as multispectral images and 3D MRIs. In this work, we propose a novel low-rank tensor approximation framework with Laplacian Scale Mixture (LSM) modeling for multi-frame image denoising. First, similar 3D patches are grouped to form a tensor of d-order and high-order Singular Value Decomposition (HOSVD) is applied to the grouped tensor. Then the task of multiframe image denoising is formulated as a Maximum A Posterior (MAP) estimation problem with the LSM prior for tensor coefficients. Both unknown sparse coefficients and hidden LSM parameters can be efficiently estimated by the method of alternating optimization. Specifically, we have derived closed-form solutions for both subproblems. Experimental results on spectral and dynamic MRI images show that the proposed algorithm can better preserve the sharpness of important image structures and outperform several existing state-of-the-art multiframe denoising methods (e.g., BM4D and tensor dictionary learning).