It is widely known that the family of projection filters includes the generalized inverse filter, and that the family of parametric projection filters includes parametric generalized projection filters. However, relations between the family of parametric projection filters and constrained least squares filters are not sufficiently clarified. In this paper, we consider relations between parameter estimation and image restoration by these families. As a result, we show that the restored image by the family of parametric projection filters is a maximum penalized likelihood estimator, and that it agrees with the restored image by constrained least squares filter under some suitable conditions.

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.