#### Technical Report USC-IPI-610

“Fast Computational Techniques for Pseudoinverse and Wiener Image Restoration”

by Faramarz Davarian

August 1975

A fast minimum mean-square error technique for restoring images degraded by blur is presented in this dissertation.

The phenomenon of linear shift-invariant degradation in an incoherent optical system can be described by means of a convolution integral. Since digital processing of pictorial data requires discretization of this integral by means of a quadrature technique, a theoretical study of a broad class of quadrature formulae is first presented.

The discrete image degradation phenomenon is modeled by two distinct vector space formulations: dark background objects correspond to a model possessing an overdetermined blur matrix; objects with unknown background, however, result in a system that is underdetermined. It is shown that these models become equivalent if the background of the object is artificially set to zero by processing the observed image. This fact results in introduction of a fast restoration technique in the absence of noise.

The noisy restoration problem is resolved by employing Wiener estimation. It is shown that with proper arrangement of the observed image data, the covariance matrix of the object becomes a circulant matrix. Hence, the Fourier domain properties of circulants gives rise to a computationally efficient Wiener restoration technique results in a suboptimal, but faster, restoration filter. It is shown that the computational saving gained by this approximation is significant, while the increase in the error variance is quite small.

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