An Iterative Linear Expansion of Thresholds for $\ell_{1}$-Based Image Restoration

Abstract

Abstract-This paper proposes a novel algorithmic framework to solve image restoration problems under sparsity assumptions. As usual, the reconstructed image is the minimum of an objective functional that consists of a data fidelity term and an 1 regularization. However, instead of estimating the reconstructed image that minimizes the objective functional directly, we focus on the restoration process that maps the degraded measurements to the reconstruction. Our idea amounts to parameterizing the process as a linear combination of few elementary thresholding functions (LET) and solve for the linear weighting coefficients by minimizing the objective functional. It is then possible to update the thresholding functions and to iterate this process (i-LET). The key advantage of such a linear parametrization is that the problem size reduces dramatically-each time we only need to solve an optimization problem over the dimension of the linear coefficients (typically less than 10) instead of the whole image dimension. With the elementary thresholding functions satisfying certain constraints, global convergence of the iterated LET algorithm is guaranteed. Experiments on several test images over a wide range of noise levels and different types of convolution kernels clearly indicate that the proposed framework usually outperform state-of-the-art algorithms in terms of both CPU time and number of iterations.Index Terms-Image restoration, sparsity, majorization minimization (MM), iterative reweighted least square (IRLS), thresholding, linear expansion of thresholds (LET).

Deblocking Artifact of Satellite Image Based on Adaptive Soft-Threshold Anisotropic Filter Using Wavelet

“…Blocking artifact removal methods in [22]- [26] and overcomplete wavelet representation in [27], [28] are not suitable for various undistorted satellite images which have texture area and flat area, either. Discrete and continuoustime soft-threshold [29] and a linear expansion of thresholds (LET) [30] can remove blocking artifact and treat the drawbacks of [22]- [28].…”

“…The authors of [29], [30] proposed soft-threshold for removing blocking artifacts. In [29], the iterative softthreshold algorithm (ISTA) was presented for performing a discrete gradient step followed by a soft-threshold operation at each iteration.…”

“…The method of [29] would be difficult to analyze the noise properties in distorted satellite image. In [30], a linear expansion of thresholds (LET) is an excellent estimate of the noiseless image by decomposing the denoising process and optimizing the coefficients of this representation using an estimate of the Stein's unbiased risk estimate (SURE). Unfortunately, they showed unstable and slow convergence performance, in particular for various satellite images.…”

“…Thus, we propose the accelerated gradient of convergence in fusion step for treating these drawbacks of [29] and [30]. The accelerated gradient of convergence is a function value of a kernel which improves convergence performance in estimation, particularly in the fusion of deblocking artifact.…”

“…b. Linear expansion of thresholds An excellent estimate of deblocking Unstable and slow convergence performance, (LET) [30] artifact. in particular for various satellite images.…”