Statistical Mechanics of Inverse Halftoning

For many years, researchers have investigated information science, such as image analysis (Besag, 1974, Winkler, 1995, Cressie, 1993). Especially, image restoration has been studied as a fundamental problem in information science. In a recent development of this field, theoretical physicists have applied statistical mechanics to information based on analogy between statistical mechanics and Bayesian inference via the maximizer of the posterior (MPM) estimate (Nishimori, 2001). In this field, many techniques in statistical mechanics have been applied to various problems. Following the strategy, the present author has applied statistical mechanics to image restoration using the plane rotator model (Saika & Nishimori, 2002) and phase retrieval (Saika & Nishimori, 2005). Recently, statistical mechanical approach for information becomes an established field called as statistical mechanical informatics. Now statistical mechanics has been applied to many problems in various areas, such as information communication and quantum computation. In print technology, many techniques have been proposed to print images with high quality. Especially, a technique called as digital halftoning (Ulichney, 1987) is essential to convert an original image into a halftone image expressed as a set of black and white dots which are visually similar to the original image through human vision system. A lot of techniques have been proposed for this problem, such as the dither method (Bayer, 1973). On the other hand, the inverse of digital halftoning is called as inverse halftoning and then the purpose is to reconstruct the original image from the halftone image (Miceli, C. M. & Parker, K. J., 1992). A lot of techniques have been proposed. From the practical point of view, Wong (Wong, 1995) has proposed statistical smoothing to inverse halftoning for halftone images. Then, Stevenson (Stevenson, 1995) has constructed the MAP estimation for halftone dithered images. In this article, we demonstrate recent development of our researches both on theoretical and practical aspects of inverse halftoning for halftone images obtained by the dither and error diffusion methods (Ulichney, 1987). As shown in Fig. 1, our strategy for this problem is based on the analogy between statistical mechanics and the Bayesian inference via the maximizer of the posterior (MPM) estimate (Fig. 2) and is then to propose the statistical mechanical techniques for this problem. First, we construct a Bayesian probabilistic formulation for inverse halftoning utilizing statistical mechanics of the Q-Ising model (Saika, et al., 2009, Saika & Okamoto, 2010). Then, we clarify the statistical performance of the present method using both the Monte Carlo simulation for a set of the snapshots of the Q-Ising model and the analytical estimate via the infinite-range model.