Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

Image interpolation using Shearlet based iterative refinement

This paper proposes an image interpolation algorithm exploiting sparse representation for natural images. It involves three main steps: (a) obtaining an initial estimate of the high resolution image using linear methods like FIR filtering, (b) promoting sparsity in a selected dictionary through iterative thresholding, and (c) extracting high frequency information from the approximation to refine the initial estimate. For the sparse modeling, a shearlet dictionary is chosen to yield a multiscale directional representation. The proposed algorithm is compared to several state-of-the-art methods to assess its objective as well as subjective performance. Compared to the cubic spline interpolation method, an average PSNR gain of around 0.8 dB is observed over a dataset of 200 images

Image Super-Resolution Reconstruction Based on L1/2 Sparsity

Based on image sparse representation in the shearlet domain, we proposed a L1/2 sparsity regularized unconvex variation model for image super-resolution. The L1/2 regularizer term constrains the underlying image to have a sparse representation in shearlet domain. The fidelity term restricts the consistency with the measured imaged in terms of the data degradation model. Then, the variable splitting algorithm is used to break down the model into a series of constrained optimization problems which can be solved by alternating direction method of multipliers. Experimental results demonstrate the effectiveness of the proposed method, both in its visual effects and in quantitative terms

Image Deblurring and Super-resolution by Adaptive Sparse Domain Selection and Adaptive Regularization

As a powerful statistical image modeling technique, sparse representation has been successfully used in various image restoration applications. The success of sparse representation owes to the development of l1-norm optimization techniques, and the fact that natural images are intrinsically sparse in some domain. The image restoration quality largely depends on whether the employed sparse domain can represent well the underlying image. Considering that the contents can vary significantly across different images or different patches in a single image, we propose to learn various sets of bases from a pre-collected dataset of example image patches, and then for a given patch to be processed, one set of bases are adaptively selected to characterize the local sparse domain. We further introduce two adaptive regularization terms into the sparse representation framework. First, a set of autoregressive (AR) models are learned from the dataset of example image patches. The best fitted AR models to a given patch are adaptively selected to regularize the image local structures. Second, the image non-local self-similarity is introduced as another regularization term. In addition, the sparsity regularization parameter is adaptively estimated for better image restoration performance. Extensive experiments on image deblurring and super-resolution validate that by using adaptive sparse domain selection and adaptive regularization, the proposed method achieves much better results than many state-of-the-art algorithms in terms of both PSNR and visual perception.Comment: 35 pages. This paper is under review in IEEE TI

Image Super-Resolution Reconstruction Based On L1/2 Sparsity

Based on image sparse representation in the shearlet domain, we proposed a L1/2 sparsity regularized unconvex variation model for image super-resolution. The L1/2 regularizer term constrains the underlying image to have a sparse representation in shearlet domain. The fidelity term restricts the consistency with the measured imaged in terms of the data degradation model. Then, the variable splitting algorithm is used to break down the model into a series of constrained optimization problems which can be solved by alternating direction method of multipliers. Experimental results demonstrate the effectiveness of the proposed method, both in its visual effects and in quantitative terms