#### Technical Report USC-IPI-520

“Digital Image Restoration Under a Regression Model - The Unconstrained, Linear Equality and Inequality Constrained Approaches”

by Nelson Delfino d'Avila Mascarenhas

January 1974

The problem of restoring images degraded by blur and corrupted by noise is considered in this dissertation.

The Fredholm integral equation of the first kind in a two-dimensional form adequately describes the linear model. A discretization is performed by using quadrature methods. By transforming the two-dimensional array into a vector format a regression model results. The overdetermined and underdetermined cases are considered in detail, with the derivation of the estimators, their covariance matrices, confidence intervals and hypothesis testing involving parametric functions of pixel values. The problem of ill conditioning is examined for atmospheric turbulence and diffraction limited spread functions. The results of the restoration of simulated pictures under separable spread functions are presented.

In order to solve the ill conditioning of the restoration problem, a priori information in the form of deterministic constraints is proposed. A comparison with existing methods like Wiener filter, smoothing and regularizing techniques is made. Linear equality constraints reduce the variance of the estimators, but some bias may be introduced if the constraints are not valid. A combination of estimation and hypothesis testing is proposed to decide if a reduction of the mean square error (taking into account both bias and variance) occurs. Experimental results show that more acceptable restored pictures are obtained in the restoration.

Linear inequality constraints are incorporated by means of a quadratic programming formulation. The natural constraint of nonnegativeness of pixel values is handled in a formal way, as well as other types of restrictions that can be described by linear inequalities. Experimental results indicate a substantial improvement in the restoration even for the ill conditioned situation.

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