A Fast Karhunen-Loeve Transform for Digital Restoration of Images Degraded by White and Colored Noise
Issue No. 06 - June (1977 vol. 26)
A.K. Jain , Department of Electrical Engineering, State University of New York
The Karhunen-Loeve (KL) transform is known to have certain properties which make it "optimal" for many "mean-square" signal processing applications -. 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 -. 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"  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 . 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.
Image processing, image restoration, Karhunen-Loeve transform, recursive filtering, Wiener filtering.
A.K. Jain, "A Fast Karhunen-Loeve Transform for Digital Restoration of Images Degraded by White and Colored Noise", IEEE Transactions on Computers, vol. 26, no. , pp. 560-571, June 1977, doi:10.1109/TC.1977.1674881