Physical consideration of an image in image restoration using Bayes’ formula

We consider image restoration by Bayes’ formula and investigate the relationship between an image and a prior probability from the following two viewpoints: hyperparameter estimation and the accuracy of a restored image. The Q-Ising model is adopted as a prior probability in Bayes’ formula. Not the Q-Ising energy, but the Potts energy plays an important role in the hyperparameter estimation. From the viewpoint of the hyperparameter estimation, the relationship between a natural image and a prior probability is characterized through the Potts energy and magnetization of an image. The Potts energy and magnetization of an image are defined by a set of pixels’ state of an image. The closer to the average Potts energy and magnetization over a prior probability the Potts energy and magnetization of a natural image is, the closer to the true value of a hyperparameter the estimated value of a hyperparameter from a degraded image is. For the accuracy of a restored image, the image which has a smaller Q-Ising energy is better restored by Bayes’ formula composed of the Q-Ising prior. The consideration for the relationship between an image and a prior probability is expected to be valid for a more complicated prior probability.