In this paper, we present an algorithm for the restoration of images with an unknown, spatially-varying blur. Existing computational methods for image restoration require the assumption that the blur is known and/or spatially-invariant. Our algorithm uses a combination of techniques. First, we section the image, and then treat the sections as a sequence of frames whose unknown PSFs are correlated and approximately spatially-invariant. To estimate the PSFs in each section, phase diversity is used. With the PSF estimates in hand, we then use a technique by Nagy and O’Leary for the restoration of images with a known, spatially-varying blur to restore the image globally. Test results on star cluster data are presented.

In this paper, our focus is on the connections between the methods of (quadratic) regularization for inverse problems and Gaussian Markov random field (GMRF) priors for problems in spatial statistics. We begin with the most standard GMRFs defined on a uniform computational grid, which correspond to the oft-used discrete negative-Laplacian regularization matrix. Next, we present a class of GMRFs that allow for the formation of edges in reconstructed images, and then draw concrete connections between these GMRFs and numerical discretizations of more general diffusion operators. The benefit of the GMRF interpretation of quadratic regularization is that a GMRF is built-up from concrete statistical assumptions about the values of the unknown at each pixel given the values of its neighbors. Thus the regularization term corresponds to a concrete spatial statistical model for the unknown, encapsulated in the prior. Throughout our discussion, strong ties between specific GMRFs, numerical discretizations of diffusion operators, and corresponding regularization matrices, are established. We then show how such GMRF priors can be used for edge-preserving reconstruction of images, in both image deblurring and medical imaging test cases. Moreover, we demonstrate the effectiveness of GMRF priors for data arising from both Gaussian and Poisson noise models.

The noise contained in images collected by a charge coupled device camera is predominantly of Poisson type. This motivates the use of the negative logarithm of the Poisson likelihood in place of the ubiquitous least squares fit-to-data. However if the underlying mathematical model is assumed to have the form z = Au, where A is a linear, compact operator, Poisson likelihood estimation is ill-posed, and hence some form of regularization is required. In Bardsley, J.M. and Vogel, C.R., 2004, A nonnegatively constrained convex programming method for image reconstruction. SIAM Journal on Scientific Computing, 25(4), pp. 1326–1343, a numerical method is presented and analyzed for Tikhonov regularized Poisson likelihood estimation, but no theoretical justification of the approach is given. Our primary objective in this article is to provide such a theoretical justification. We then briefly present the computational method of Bardsley, J.M. and Vogel, C.R., 2004, A nonnegatively constrained convex programming method for image reconstruction. SIAM Journal on Scientific Computing, 25(4), pp. 1326–1343, which is very effective and computationally efficient for this problem. The practical validity of the approach is then demonstrated on a synthetic example from astronomical imaging.

Astronomy & Astrophysics (A&A) is an international journal which publishes papers on all aspects of astronomy and astrophysics

: For a given two-dimensional image, we define the horizontal and vertical increments at a pixel location to be the difference between the intensity values at that pixel and at the neighboring pixels to the right and above, respectively. For a typical image, it makes intuitive sense that the increments will usually be near zero, corresponding to areas of smooth variation in image intensity, but will often have large magnitude, corresponding to edges where sharp intensity changes occur. In this paper, we explore the use of the Laplace increment model, in which the increments are assumed to be independent and identically distributed Laplace random variables – a distribution with heavy tails allowing for large increment values – with zero mean. The prior constructed from the Laplace increment model is very similar to the total variation (TV) prior. We perform a theoretical analysis of its properties, which shows that the Laplace prior yields a regularization scheme with regularized solutions contained in the space of bounded variation, just as for the TV prior. Moreover, numerical experiments indicate that the Laplace prior yields reconstructions that are qualitatively very similar to those obtained using TV.