Learning Co-Sparse Analysis Operators With Separable Structures

Abstract

Abstract-In the co-sparse analysis model a set of filters is applied to a signal out of the signal class of interest yielding sparse filter 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 learn an analysis operator with separable structures, which includes a novel and efficient step size selection rule. Numerical experiments are provided that link the sample complexity to the convergence speed of the SGD algorithm.

A Sparse Analysis-Based Single Image Super-Resolution

“…We observed that the bicubic and GR [19] methods tended to generate blocking and ringing artifacts. Although the SCSR [12], SISU [13], ANR [19], NE + LS [19], GOAL [29], AMSRR [20], and SDS [21] methods improved the visual details and edges, it can be seen that the proposed SASR algorithm had more acceptable performance when reconstructing sharp edges, and showed more detail compared to the other algorithms.…”

“…Although the objective quality of a few HR images obtained by the proposed method had a slight advantage over the other methods, the average quantitative evaluations of the proposed method are competitive to those of the others. [12], SISU [13], GR [19], ANR [19], NE + LS [19], GOAL [29], AMSRR [20], SDS [21], and our proposed SASR. [12], SISU [13], GR [19], ANR [19], NE + LS [19], GOAL [29], AMSRR [20], SDS [21], and our proposed SASR.…”

“…We quantitatively and qualitatively evaluated the performance of the proposed method in comparison with bicubic interpolation, the SCSR [12], SISU [13], ANR [19], GR [19], NE + LS [19], GOAL [29], AMSRR [20], and SDS [21] methods. The sparse representation-based methods (SCSR [12] SISU [13]) and regression-based methods (ANR [19], GR [19], and NE + LS [19]) were used as for baseline methods.…”

“…The sparse representation-based methods (SCSR [12] SISU [13]) and regression-based methods (ANR [19], GR [19], and NE + LS [19]) were used as for baseline methods. Three methods (AMSRR [20], SDS [21], and GOAL [29]) were used as for the state-of-the-art methods. Tables 2-4 show the quantitative comparison for the PSNR, RMSE, and SSIM measures.…”

“…Recently, a different model, sparse analysis, has been proposed [26] as a powerful alternative model for signal processing problems. Extensive research has focused on the high competency of the sparse analysis model [27][28][29]. Ning et al [30] proposed an SR algorithm using iterative refinement and the analysis sparsity model in the Lab color space.…”