In this paper, we present a blind deconvolution algorithm based on the total variational (TV) minimization method proposed in [11]. The motivation for regularizing with the TV norm is that it is extremely effective for recovering edges of images [11] as well as some blurring functions, e.g. motion blur and out-of-focus blur. An alternating minimization (AM) implicit iterative scheme is devised to recover the image and simultaneously identify the point spread function (PSF). Numerical results indicate that the iterative scheme is quite robust, converges very fast (especially for discontinuous blur) and both the image and the PSF can be recovered under the presence of high noise level. Finally, we remark that PSF's without sharp edges, e.g. Gaussian blur, can also be identified through the TV approach. I. Introduction It is well-known that recovering the image u (resp. the PSF k) with known PSF (resp. image) is a mathematically ill-posed problem. One of the most successful regularizatio...

We present a very simple inpainting algorithm for reconstruction of small missing and damaged portions of images that is two to three orders of magnitude faster than current methods while producing comparable results.

. We consider the use of multigrid methods for solving certain differential-convolution equations which arise in regularized image deconvolution problems. We first point out that the usual smoothing procedures (e.g. relaxation smoothers) do not work well for these types of problems because the high frequency error components are not smoothed out. To overcome this problem, we propose to use optimal fast-transform preconditioned conjugate gradient smoothers. The motivation is to combine the advantages of multigrid (mesh independence) and fast transform based methods (clustering of eigenvalues for the convolution operator). Numerical results for Tikhonov regularization with the identity and the Laplacian operators show that the resulting method is effective. However, preliminary results for total variation regularization show that this case is much more difficult and further analysis is required. 1 Introduction In PDE based image processing, we often need to solve differentialconvolution...

In this dissertation, we analyze three classes of iterative methods which are often used as preconditioners for Krylov subspace methods, for the solution of large and sparse linear systems arising from the discretization of partial differential equations. In addition, we propose algorithms for image processing applications and multiple right-hand side problems. The first class is the incomplete LU factorization preconditioners, an intrinsic sequential algorithm. We develop a parallel implementation of ILU(0) and devise a strategy for a priori memory allocation crucial for ILU(k) parallelization. The second class is the sparse approximate inverse (SPAI) preconditioners. We improve and extend its applicability to elliptic PDEs by using wavelets which converts smoothness, often found in the Green's function...

The relative Newton algorithm, previously proposed for quasi maximum likelihood blind source separation and blind deconvolution of one-dimensional signals is generalized for blind deconvolution of images. Smooth approximation of the absolute value is used in modelling the log probability density function, which is suitable for sparse sources. In addition, we propose a method of sparsification, which allows blind deconvolution of sources with arbitrary distribution, and show how to find optimal sparsifying transformations by training.

In this report, we propose combining the Total Variation denoising method with the high loss wavelet compression for high noise level images. Numerical experiments show that TVdenoising can bring more wavelet coecients closer to zero thereby making the compression more ecient while the salient features (edges) of the images can still be retained.

A family preconditioners for the solution of discrete linear systems arising in regularized ill-posed problems is presented. These preconditioners are based on a two-level splitting of the solution space, and were previously developed by Hanke and Vogel for positive definite regularization operators. The work presented here extends previous results to the case where the regularization operator has a nontrivial null space. Key words: Conjugate Gradient, Preconditioners, Iterative Methods, Image Deblurring 1 Introduction Our goal is to efficiently solve very large symmetric positive definite linear systems of the form Au = b (1a) where A = K K + ffL: (1b) The matrix K is assumed to be highly ill-conditioned and full, the matrix L is sparse and symmetric positive semidefinite, and ff is a small positive Preprint submitted to Elsevier Preprint 24 March parameter. Systems with this structure arise, for example, in image deblurring. Blurred, noisy image data is modeled by d ij = Z Z...