The USC Andrew and Erna Viterbi School of Engineering USC Signal and Image Processing Institute USC Ming Hsieh Department of Electrical Engineering University of Southern California

Technical Report USC-IPI-650

“Least Squares Image Restoration Using Spline Interpolation”

by Hsieh Sheng Hou

February 1976

This dissertation has presented a theoretical analysis and computational technique of the constrained least squares image restoration using spline interpolation. A realistic continuous-discrete physical image model has been adopted throughout the whole formulation. The optical system is assumed to be incoherent, and the general problem of image restoration with space-variant point-spread function degradation has been particularly studied.

A normal equation has been formulation in a finite dimensional space (spline space) for both one-dimensional and two-dimensional imagery. The magnitude of the smoothing parameter in the normal equation allows us the freedom to control parameter in the normal equation allows us the freedom to control resolution in a trade-off of smoothing the restored object. This effect has been demonstrated with experimental results. Constraints that characterize the physical properties of the restored object have been formulated so that they can be imposed on the solution of the normal equation. These constraints are to require the restored object pixels be positive and the energy of it to be equal to that of the degraded image.

Iterative methods for both unconstrained and constrained solutions of the normal equation have been studied in detail. Among the unconstrained methods the conjugate gradient method has been successfully simulated on a computer and strikingly good results have been obtained. Furthermore, the enlargement of images by spline interpolation have been shown by experimental results to be visually pleasing and to retain more picture details as compared with other interpolation methods.

To maximize degrees of freedom and restore objects from a non-uniformly degraded image, it is hypothesized that one would take fewer samples in the blurred area than in the high resolution area. Therefore a method to select the variable sample positions has been proposed so that the structural degrees of freedom of the image system can be optimized.

We have also shown in this dissertation that the singular value decomposition algorithm is a special case of the general method reported. Generalized spline interpolation is also briefly discussed.

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