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.

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In this paper, we discuss the restoration of noise-degraded images through Bayesian inference. Superimposed noise is usually assumed to be uncorrelated between pixels. Here, we discuss spatially correlated noise. The generative statistical models for the original image and the noise are assumed to obey multi-dimensional Gaussian distributions whose covariance matrixes are translational invariant. We can derive an exact description to be used as the expectation for the restored image by means of Fourier transfonnation. We have attempted to restore a distorted image with spatially correlated noise by using a spatially uncorrelated noise model. We found that the resultant values of the hyperparameter estimations for minimum error and maximal posterior marginal criteria do not coincide with each other when the generative probabilistic model and the model used for the restoration process belong to different classes, while they coinside with each other when these two probablistic models belong to same class.

Many associative memory models with synaptic decay such as the forgetting model and the zero-order decay model have been proposed and studied so far. The previous studies showed the relation between the storage capacity C and the synaptic decay coefficient α in each synaptic decay model. However, with the exceptions of a few studies, they did not compare the network retrieval performance between different synaptic decay models. We formulate the associative memory model with the β-th-order synaptic decay as an extension of the zero-order decay model. The parameter β denotes the synaptic decay order or the degree of the synaptic decay term, which enables us to compare the retrieval performance between different synaptic decay models. Using numerical simulations, we investigate the relation between the synaptic decay coefficient α and the storage capacity C of the network by varying the synaptic decay order β. The results show that the properties of the synaptic decay model are constant for a large decay order β. Moreover, we search the minimum β to avoid overloading and the optimal β to maximize the network retrieval performance. The minimum integer value of β to avoid overloading is −1. The optimal integer value of β to maximize the network retrieval performance is 1, i.e., the degree of the forgetting model, and the suboptimal integer β is 0, i.e., that of the zero-order synaptic decay model.