The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise

Abstract

In 1993, Snyder et al investigated the maximum-likelihood (ML) approach to the deconvolution of images acquired by a charge-coupled-device camera and proved that the iterative method proposed by Llacer and Nuñez in 1990 can be derived from the expectation-maximization method of Dempster et al for the solution of ML problems. The utility of the approach was shown on the reconstruction of images of the Hubble space Telescope. This problem deserves further investigation because it can be important in the deconvolution of images of faint objects provided by next-generation ground-based telescopes that will be characterized by large collecting areas and advanced adaptive optics. In this paper, we first prove the existence of solutions of the ML problem by investigating the properties of the negative log of the likelihood function. Next, we show that the iterative method proposed by the above-mentioned authors is a scaled gradient method for the constrained minimization of this function in the closed and convex cone of the non-negative vectors and that, if it is convergent, the limit is a solution of the constrained ML problem. Moreover, by looking for the asymptotic behavior in the regime of high numbers of photons, we find an approximation that, as proved by numerical experiments, works well for any number of photons, thus providing an efficient implementation of the algorithm. In the case of image deconvolution, we also extend the method to take into account boundary effects and multiple images of the same object. The approximation proposed in this paper is tested on a few numerical examples.

Restoration of images degraded by signal-dependent noise based on energy minimization: an empirical study

“…Φðj∇ðu i ÞjÞ; (10) where ∇ stands for image gradient and j · j denotes l 2 norm. The function Φ is referred to as "potential function."…”

Distributed approach for deblurring large images with shift-variant blur

“…However, there are some fast approximations proposed in [1,2]; see [3] for detailed comparison between the different approximations. If we approximate the noise in observed image by a non-stationary white Gaussian noise as considered in [4] (see [5] for more refined noise model), the discrete image formation model can be written as: y = H x + ε. The matrix H ∈ R m×n denotes the bluring operator.…”

“…Motivation: Image deblurring is a well studied topic, and there is a vast literature [4,5,7,8,10] for moderate size (few megapixels) images with shift-invariant blur. The applications in astronomy, satellite imagery and others are able to capture extremely large images (up to gigapixels) of wide fieldof-views suffering from shift-variant blur.…”

An EM Approach for Time-Variant Poisson-Gaussian Model Parameter Estimation

“…Among existing works dealing with Poisson-Gaussian noise, a number of methods have addressed noise identification problems [5][6][7][8][9][10][11], as well as denoising [9,[12][13][14][15][16] and reconstruction [17][18][19][20][21]. The developed algorithms are useful in various areas such as digital photography [8], medicine [22], biology [23] and astronomy [18].…”