Statistical Analysis of Gaussian Image Inpainting Problems

Abstract

Bayesian image processing is one of the activity research fields in computer science. Some of the approaches can often be reduced to inference problems on Markov random fields (MRFs). Image inpainting is one such problem. Image inpainting is used to infer intensities in missing regions from an observed image in which the intensities of some pixels are missing. In the inference scheme of inpainting, each observed image is divided into missing regions and nonmissing regions, and it is assumed that it is known which region each pixel belongs to in advance. A deterministic algorithm for the image inpainting problem based on Gaussian MRFs has been proposed. The proposed algorithm returns a unique output image for each observed image. By considering variations among observed images, we can define the statistical performances of the deterministic algorithm. The calculation of statistical performances corresponds to taking the average over all the possible observed images. Statistical performance analysis in probabilistic inference systems is one of the topics of statistical–mechanical informatics, and there are many successful results regarding statistical performance analysis in various research fields of computer sciences. Nishimori and Wong calculated the statistical performance in binary image restorations using MRFs defined on a complete graph. Statistical performance analysis in image restorations using Gaussian MRFs has also been calculated accurately by using the multi-dimensional Gauss integral formulas. Kataoka et al. have proposed approximate methods for evaluating the statistical performances and applied them for binary image restorations using MRFs on square grid graphs. In the present paper, we evaluate statistical performances of image inpaintings by Gaussian MRFs. Our evaluation method is an extension of the idea to evaluate the approximate statistical performance based on the loopy belief propagation in ref. 8, and the calculation of statistical performance is reduced to the problem of solving simultaneous integral equations of distributions of estimated values. We consider an image composed of N pixels on a square grid graph and define a state, corresponding to light intensities, for each pixel as any real number in ð 1;þ1Þ. N pixels are divided into missing regions and non-missing regions. Suppose that x 1⁄4 fxig and y 1⁄4 fyjg are sets of pixels in missing regions and non-missing regions, respectively. Here, i is a label at each pixel belonging to missing regions, whereas j is associated to a pixel in non-missing regions. In this paper, we consider an image in which each pixel is randomly missing with probability p. Let c 1⁄4 fcig; ci 2 f0; 1g; i 1⁄4 1; . . . ; N be a set of state variables that indicate whether pixels are missing or not. ci 1⁄4 0 denotes a pixel i belonging to non-missing regions and ci 1⁄4 1 denotes a pixel i belonging to missing regions. We assume that the posterior probability density function of x, under both c and y are given, is denoted by Pðx j c; yÞ 1⁄4 exp1⁄2 Hðx j c; yÞ =Z, where the energy function Hðx j c; yÞ is defined as Hðx j c; yÞ 1⁄4 1 2 X ði;jÞ2E 2 cicjðxi xjÞ