This tutorial explains the processing of the annealing machine in the demonstration application image noise reduction".
This is a problem of estimating the original image from a black and white image with noises.
Generally, assuming that the following conditions hold, expect it to work properly on an annealing machine.
In this problem, since the color of the pixel is binary black or white, we correspond it to the spin
value -1 or +1 of CMOS annealing machine.
From (a) above, here we set the value of the i-th pixel of the input image as $y_i$ and the value of the ith pixel of the noise-reduced image as $s_i$.
To make the pixel values of the input image and output image roughly the same, we use the following cost function.
Here $V$ represents a set of pixels. Since $y_i s_i$ becomes +1 when the i-th pixel value of the input and output image is the same and becomes -1 when the pixel value is different, add the signed $-$ to all pixels so that the cost is reduced when the pixel values are the same.
Additonally considering the condition that "(b) In output images after noise reduction, adjacent pixel values are often the same.", the cost function is as follows.
Here $E$ represents a collection of adjacent pixel pairs. When these are added together, the following final cost function is obtained.