Denosing Using Wavelets and Projections onto the L1-Ball

Both wavelet denoising and denoising methods using the concept of sparsity are based on soft-thresholding. In sparsity-based denoising methods, it is assumed that the original signal is sparse in some transform domains such as the Fourier, DCT, and/or wavelet domain. The transfer domain coefficients of the noisy signal are projected onto L1-balls to reduce noise. In this paper, we establish the relation between the standard soft-thresholding-based denoising methods and sparsity-based wavelet denoising. We introduce a new deterministic soft-threshold estimation method using the epigraph set of L1-ball cost function. It is shown that the size of the L1-ball determined using linear algebra. The size of the L1-ball in turn determines the soft threshold. The key step is an orthogonal projection onto the epigraph set of the L1-norm cost function. The software and detailed information is available here.

Projections Onto the Epigraph Set Of Total Variation Function (PES-TV)

In this article, a novel algorithm for denoising images that are corrupted by impulsive noise is presented. The proposed denoising algorithm is a two step procedure. In the first step, image denoising is formulated as a convex optimization problem, whose constraints are defined as limitations on local variations between neighboring pixels. Projections onto the Epigraph Set of Total Variation function (PES-TV) are performed in the first step. Unlike similar approaches in the literature, the PES-TV method does not require any prior information about the noise variance. The first step is only capable of utilizing local relations among pixels. It does not fully take advantage of correlations between spatially distant areas of an image with similar appearance. In the second step, a Wiener filtering approach is cascaded to the PES-TV based method to take advantage of global correlations in an image. In this step, the image is first divided into blocks and blocks with similar content are jointly denoised using a 3D Wiener filter. The denoising performance of the proposed two-step method was compared against three state of the art denoising methods under various impulsive noise models. The software and detailed information is available here.

Phase and TV Based Convex Sets for Blind Deconvolution for Microscopic Images

In this article, two closed and convex sets for blind deconvolution problem are proposed. Most blurring functions in microscopy are symmetric with respect to the origin. Therefore, they do not modify the phase of the Fourier transform (FT) of the original image. As a result blurred image and the original image have the same FT phase. Therefore, the set of images with a prescribed FT phase can be used as a constraint set in blind deconvolution problems. Another convex set that can be used during the image reconstruction process is the epigraph set of Total Variation (TV) function. This set does not need a prescribed upper bound on the total variation of the image. The upper bound is automatically adjusted according to the current image of the restoration process. Both of these two closed and convex sets can be used as a part of any blind deconvolution algorithm. Simulation examples are presented. The software and detailed information is available here.