Denoising images corrupted by impulsive noise using projections onto the epigraph set of the total variation function (PES-TV)

In this article, a novel algorithm for denoising images corrupted by impulsive noise is presented. Impulsive noise generates pixels whose gray level values are not consistent with the neighboring pixels. 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. We use Projections onto the Epigraph Set of the TV function (PES-TV) to solve this problem. Unlike other approaches in the literature, the PES-TV method does not require any prior information about the noise variance. It is only capable of utilizing local relations among pixels and 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 those 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. © 2015, Springer-Verlag London

Image restoration and reconstruction using projections onto epigraph set of convex cost fuchtions

Cataloged from PDF version of article.This thesis focuses on image restoration and reconstruction problems. These inverse problems are solved using a convex optimization algorithm based on orthogonal Projections onto the Epigraph Set of a Convex Cost functions (PESC). In order to solve the convex minimization problem, the dimension of the problem is lifted by one and then using the epigraph concept the feasibility sets corresponding to the cost function are defined. Since the cost function is a convex function in R N , the corresponding epigraph set is also a convex set in R N+1. The convex optimization algorithm starts with an arbitrary initial estimate in R N+1 and at each step of the iterative algorithm, an orthogonal projection is performed onto one of the constraint sets associated with the cost function in a sequential manner. The PESC algorithm provides globally optimal solutions for different functions such as total variation, `1-norm, `2-norm, and entropic cost functions. Denoising, deconvolution and compressive sensing are among the applications of PESC algorithm. The Projection onto Epigraph Set of Total Variation function (PES-TV) is used in 2-D applications and for 1-D applications Projection onto Epigraph Set of `1-norm cost function (PES-`1) is utilized. In PES-`1 algorithm, first the observation signal is decomposed using wavelet or pyramidal decomposition. 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 domain such as Fourier, DCT, and/or wavelet domain and transform domain coefficients of the noisy signal are soft-thresholded to reduce noise. Here, the relationship between the standard soft-thresholding based denoising methods and sparsity-based wavelet denoising methods is described. A deterministic soft-threshold estimation method using the epigraph set of `1-norm cost function is presented. It is demonstrated that the size of the `1-ball can be determined using linear algebra. The size of the `1-ball in turn determines the soft-threshold. The PESC, PES-TV and PES-`1 algorithms, are described in detail in this thesis. Extensive simulation results are presented. PESC based inverse restoration and reconstruction algorithm is compared to the state of the art methods in the literature.Tofighi, MohammadM.S

Projections onto the epigraph set of the filtered variation function based deconvolution algorithm

A new deconvolution algorithm based on orthogonal projections onto the hyperplanes and the epigraph set of a convex cost function is presented. In this algorithm, the convex sets corresponding to the cost function are defined by increasing the dimension of the minimization problem by one. The Filtered Variation (FV) function is used as the convex cost function in this algorithm. Since the FV cost function is a convex function in RN, then the corresponding epigraph set is also a convex set in the lifted set in RN+1. At each step of the iterative deconvolution algorithm, starting with an arbitrary initial estimate in RN+1, first the projections onto the hyperplanes are performed to obtain the first deconvolution estimate. Then an orthogonal projection is performed onto the epigraph set of the FV cost function, in order to regularize and denoise the deconvolution estimate, in a sequential manner. The algorithm converges to the deblurred image. © 2016 IEEE