A Fast Karhunen-Loeve Transform for Digital Restoration of Images Degraded by White and Colored Noise

The Karhunen-Loeve (KL) transform is known to have certain properties which make it "optimal" for many "mean-square" signal processing applications [1]-[4]. Recently, it has been shown that a class of digital images may be represented by a set of boundary value stochastic difference equations in two dimensions [5]-[7]. If the boundary conditions of this class of images are fixed, then these equations lead to a fast KL transform algorithm. Here this fast KL transform is used for Wiener filtering of images degraded by white or colored noise. Comparisons with "Generalized Wiener Filtering" [1] and conventional Fourier domain filtering are made. It is shown that the two-dimensional Wiener filter is nonseparable so that two-dimensional generalized Wiener filtering is more elaborate than reported in [1]. It is also shown that certain fast KL filters give better signal-to-noise ratio than the conventional Fourier domain Wiener filter and enable determination of an easily computable performance bound. Recursive filtering equations for implementing the fast KL filter on two-dimensional images including both white and colored noise cases are given. These results show that recursive filtering algorithms for images are faster than the transform-domain algorithms and the one-step interpolator algorithm performs very close to the smoothing filter and can be implemented online by introducing a one-step delay.