In D.O.A. estimation, identification of the signal and the noise subspaces plays an essential role. This identification process was traditionally achieved by the eigenvalue decomposition (EVD) of the spatial correlation matrix of observations or the generalized eigenvalue decomposition (GEVD) of the spatial correlation matrix of observations with respect to that of an observation noise. The framework based on the GEVD is not always an extension of that based on the EVD, since the GEVD is not applicable to the noise-free case which can be resolved by the framework based on the EVD. Moreover, they are not applicable to the case in which the spatial correlation matrix of the noise is singular. Recently, a quotient-singular-value-decomposition-based framework, that can be applied to problems with singular noise correlation matrices, is introduced for noise reduction. However, this framework also can not treat the noise-free case. Thus, we do not have a unified framework of the identification of these subspaces. In this paper, we show that a unified framework of the identification of these subspaces is realized by the concept of proper and improper eigenspaces of the spatial correlation matrix of the noise with respect to that of observations.

A parametric linear filter for a linear observation model usually requires a parameter selection process so that the filter achieves a better filtering performance. Generally, criteria for the parameter selection need not only the filtered solution but also the filter itself with each candidate of the parameter. Obtaining the filter usually costs a large amount of calculations. Thus, an efficient algorithm for the parameter selection is required. In this paper, we propose a fast parameter selection algorithm for linear parametric filters that utilizes a joint diagonalization of two non-negative definite Hermitian matrices.

Practical image restoration filters usually include a parameter that controls regularizability, trade-off between fidelity of a restored image and smoothness of it, and so on. Many criteria for choosing such a parameter have been proposed. However, the relation between these criteria and the squared error of a restored image, which is usually used to evaluate the restoration performance, has not been theoretically substantiated. Sugiyama and Ogawa proposed the subspace information criterion (SIC) for model selection of supervised learning problems and showed that the SIC is an unbiased estimator of the expected squared error between the unknown model function and an estimated one. They also applied it to restoration of images. However, we need an unbiased estimator of the unknown original image to construct the criterion, so it can not be used for general situations. In this paper, we present a modified version of the SIC as a new criterion for choosing a parameter of image restoration filters. Some numerical examples are also shown to verify the efficacy of the proposed criterion.

We consider the situation that plural degraded images are obtained. When no prior knowledge about original images are known, these images are individually restored by an optimum restoration filter, for example, by Wiener Filter or by Projection Filter. If correlations between original images are obtained, some restoration filters based on Wiener Filter or Projection Filter are proposed. In this paper, we deal with the case that some pixels or some parts of original images overlap, and propose a restoration method using a formulae for overlapping. The method is based on Partial Projection Filter. Moreover, we confirm an efficacy of the proposed method by numerical examples.

Optimum filters for an image restoration are formed by a degradation operator, a covariance operator of original images, and one of noise. However, in a practical image restoration problem, the degradation operator and the covariance operators are estimated on the basis of empirical knowledge. Thus, it appears that they differ from the true ones. When we restore a degraded image by an optimum filter belonging to the family of Projection Filters and Parametric Projection Filters, it is shown that small deviations in the degradation operator and the covariance matrix can cause a large deviation in a restored image. In this paper, we propose new optimum filters based on the regularization method called the family of Regularized Projection Filters, and show that they are stable to deviations in operators. Moreover, some numerical examples follow to confirm that our description is valid.

In terms of the formulation of the optimality, image restoration filters can be divided into two streams. One is formulated as an optimization problem in which the fidelity of a restored image is indirectly evaluated, and the other is formulated as an optimization problem based on a direct evaluation. Originally, the formulation of the optimality and the solutions derived from the formulation are identical each other. However in many studies adopting the former stream, an arbitrary choice of a solution without a mathematical ground passes unremarked. In this paper, we discuss the relation between the formulation of the optimality and the solution derived from the formulation from a mathematical point of view, and investigate the relation between a direct style formulation and an indirect one. Through these analyses, we show that the both formulations yield the identical filter in practical situations.

A lot of optimum filters have been proposed for an image restoration problem. Parametric filter, such as Parametric Wiener Filter, Parametric Projection Filter, or Parametric Partial Projection Filter, is often used because it requires to calculate a generalized inverse of one operator. These optimum filters are formed by a degradation operator, a covariance operator of noise, and one of original images. In practice, these operators are estimated based on empirical knowledge. Unfortunately, it happens that such operators differ from the true ones. In this paper, we show the unified formulae of inducing them to clarify their common properties. Moreover, we investigate their properties for perturbation of a degradation operator, a covariance operator of noise, and one of original images. Some numerical examples follow to confirm that our description is valid.