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A Fast Karhunen-Loeve Transform for Digital Restoration of Images Degraded by White and Colored Noise
June 1977 (vol. 26 no. 6)
pp. 560-571
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 [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.
Index Terms:
Image processing, image restoration, Karhunen-Loeve transform, recursive filtering, Wiener filtering.
Citation:
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. 6, pp. 560-571, June 1977, doi:10.1109/TC.1977.1674881
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