Hierarchic recursive image enhancement

The problem that is solved in this paper can be formulated as: given an observation of an image against the background of additive noise and given the statistics of the image and the noise, find an optimal estimate of the image such that the computer-time and storage requirements of the estimator are modest for images of, say points or more. A discrete-time vector-scanning model is derived that describes the statistics of a large class of images. The optimal linear smoother-with regard to the least-squares criterion-is formulated in a recursive manner as a combination of two Kalman filters. It is observed that in the model the covariance matrices are Toeplitz matrices. It is shown that the z transform defines a one-to-one relation between Toeplitz matrices and functions of a complex variable. This reduces the Riccati equation to a scalar equation in the domain. It is further shown that multiplication by a Toeplitz matrix can be performed recursively by two linear dynamical systems. This leads to an algorithm which is not only recursive in the "time" parameter of the state space model but also in the index of the elements of the state vector. This so-called hierarchic recursive method has modest computational requirements.