Blind image restoration by anisotropic regularization

This paper presents anisotropic regularization techniques to exploit the piecewise smoothness of the image and the point spread function (PSF) in order to mitigate the severe lack of information encountered in blind restoration of shift-invariantly and shift-variantly blurred images. The new techniques, which are derived from anisotropic diffusion, adapt both the degree and direction of regularization to the spatial activities and orientations of the image and the PSF. This matches the piecewise smoothness of the image and the PSF which may be characterized by sharp transitions in magnitude and by the anisotropic nature of these transitions. For shift-variantly blurred images whose underlying PSFs may differ from one pixel to another, we parameterize the PSF and then apply the anisotropic regularization techniques. This is demonstrated for linear motion blur and out-of-focus blur. Alternating minimization is used to reduce the computational load and algorithmic complexity.

A regularization approach to joint blur identification and image restoration

The primary difficulty with blind image restoration, or joint blur identification and image restoration, is insufficient information. This calls for proper incorporation of a priori knowledge about the image and the point-spread function (PSF). A well-known space-adaptive regularization method for image restoration is extended to address this problem. This new method effectively utilizes, among others, the piecewise smoothness of both the image and the PSF. It attempts to minimize a cost function consisting of a restoration error measure and two regularization terms (one for the image and the other for the blur) subject to other hard constraints. A scale problem inherent to the cost function is identified, which, if not properly treated, may hinder the minimization/blind restoration process. Alternating minimization is proposed to solve this problem so that algorithmic efficiency as well as simplicity is significantly increased. Two implementations of alternating minimization based on steepest descent and conjugate gradient methods are presented. Good performance is observed with numerically and photographically blurred images, even though no stringent assumptions about the structure of the underlying blur operator is made.

Blur identification by residual spectral matching

The estimation of the point spread function (PSF) for blur identification, often a necessary first step in the restoration of real images, method is presented. The PSF estimate is chosen from a collection of candidate PSFs, which may be constructed using a parametric model or from experimental measurements. The PSF estimate is selected to provide the best match between the restoration residual power spectrum and its expected value, derived under the assumption that the candidate PSF is equal to the true PSF. Several distance measures were studied to determine which one provides the best match. The a priori knowledge required is the noise variance and the original image spectrum. The estimation of these statistics is discussed, and the sensitivity of the method to the estimates is examined analytically and by simulations. The method successfully identified blurs in both synthetically and optically blurred images.

Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation

The application of regularization to ill-conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution. The value of the regularization parameter depends on the variance of the noise in the data. The problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined. An error analysis based on an objective mean-square-error (MSE) criterion is used to motivate regularization. Two approaches for choosing the regularization parameter and estimating the noise variance are proposed. The proposed and existing methods are compared and their relationship to linear minimum-mean-square-error filtering is examined. Experiments are presented that verify the theoretical results.

Maximum likelihood parametric blur identification based on a continuous spatial domain model

A formulation for maximum-likelihood (ML) blur identification based on parametric modeling of the blur in the continuous spatial coordinates is proposed. Unlike previous ML blur identification methods based on discrete spatial domain blur models, this formulation makes it possible to find the ML estimate of the extent, as well as other parameters, of arbitrary point spread functions that admit a closed-form parametric description in the continuous coordinates. Experimental results are presented for the cases of 1-D uniform motion blur, 2-D out-of-focus blur, and 2-D truncated Gaussian blur at different signal-to-noise ratios.