Statistical mechanics of the Bayesian image restoration under spatially correlated noise

Abstract

We investigated the use of the Bayesian inference to restore noise-degraded images under conditions of spatially correlated noise. The generative statistical models used for the original image and the noise were assumed to obey multidimensional Gaussian distributions, whose covariance matrices are translational invariant. We derived an exact description to be used as the expectation for the restored image by the Fourier transformation and restored an image distorted by spatially correlated noise by using a spatially uncorrelated noise model. We found that the resulting hyperparameter estimations for the minimum error and maximal posterior marginal criteria did not coincide when the generative probabilistic model and the model used for restoration were in different classes, while they did coincide when they were in the same class.

MAP estimation algorithm for phase response curves based on analysis of the observation process

“…(10), (11) and (12) in the posterior density (9), we derive the Fourier transformed forms of the posterior density (Tsuzurugi and Okada 2002). By inserting Eq.…”

“…In the next subsection, we will give a detailed explanation of Z ps . In the algorithm for estimating the PRC by maximizing the posterior density, the Fourier series of Z (t), G(t), and φ(t 0 ) are used as follows to derive an executable algorithm to maximize the posterior density (Tsuzurugi and Okada 2002).…”

“…This paper is organized as follows. First, we explain the analytical derivations of the likelihood function of the PRC, the MAP estimation algorithm, and marginal likelihood (Tanaka 2002;Tsuzurugi and Okada 2002). Next, we verify the likelihood function by using numerical simulations.…”

“…The MAP estimation algorithm is equivalent to the spherical spin model. Moreover, we analytically derived a marginal likelihood (corresponding to the free energy of the spherical spin model) that enables us to estimate the hyper-parameters, i.e., the intensity of the Langevin force and the parameters of the prior density (Tsuzurugi and Okada 2002). Our algorithm encompasses the previous statistical methods.…”

Bias–variance tradeoff in anticorrelated noise reduction for spectral CT

“…The matrix C 0 in Eq. (15) was therefore taken to be the average of the covariance matrices in the insert and in the background of the simulated reconstructed images. To study the influence of the covariance matrix and the NPS on the result, three simulated images were used: the simulation with two energy bins and a pure ramp filter in the reconstruction, and the simulation with eight energy bins, both with a pure ramp filter and with a Hann filter.…”