Bayesian inference to the problem of inverse-halftoning based on statistical mechanics of the Q-Ising model

A Bayesian approach to solve the problem of digital inverse-halftoning [1] is proposed. The inverse-halftoning is a kind of information processing defined as the inverse problem of the halftoning of grayscale images, that is, as the inverse process of representing each grayscale in terms of black and white binary dots by means of the so-called dither method [2]. In our previous study [3], we formulated the inverse process of the halftoning as a combinatorial optimization to find the original grayscale image as the minimum energy state. However, it is worth while for us to look for the alternative or to reconsider the problem from the view point of the Bayesian approach. In the Bayesian approach, we need to specify both the likelihood and the prior to obtain the posterior. As the prior, here we naturally choose P ({z}) ∝ exp[−(J/T )∑n.n.(zx,y − zx′,y′)], where {z} ≡ {zx,y ∈ 0, · · · , Q− 1|x, y = 1, · · · , L} denotes the estimate of the original grayscale image and J, T are the so-called hyper-parameters. The sum ∑ n.n.(· · ·) should be taken for all ingredients of the nearest neighboring pixels. In halftoning process, an original image {ξ} ≡ {ξx,y ∈ 0, · · · , Q − 1|x, y = 1, · · · , L} is converted into the halftone binary dots {τ} ≡ {τx,y ∈ 0, 1|x, y = 1, · · · , L} in terms of the dither method as τx,y = θ(ξx,y − Mx,y) for each pixel, where Mx,y is a threshold at the site (x, y) and θ(· · ·) means a step function. To construct the likelihood, we assume that each pixel of the halftone image τx,y is fluctuated around the successfully converted value: θ(zx,y − Mx,y) and the fluctuation is measured by a Gaussian variable nx,y with mean zero and variance T/h, namely, we assume that one observes τx,y = θ(zx,y − Mx,y) + nx,y instead of θ(zx,y − Mx,y), where nx,y = N (0, T/h). This reads immediately P ({τ}|{z}) ∝ exp[−(h/T )∑(x,y){θ(zx,y −Mx,y) − τx,y}2], and then we have the following posterior distribution: