#### Technical Report USC-IPI-680

“Image Restoration by Spline Functions”

by Mohammad Javad Peyrovian

August 1976

Spline functions, because of their highly desirable interpolating and approximating characteristics, are used as a potential alternative to the conventional pulse approximation method in digital image processing. For uniformly spaced knots, a class of spline functions called B-splines has the useful properties of shift invariance, positiveness, and convolutional and local basis properties. These are exploited in image processing for linear incoherent image systems.

The problem of image degradation in a linear imaging system is described by a superposition integral. For simulation of degradation and restoration by means of a digital computer, the continuous imaging model must be discretized. Thus, a theoretical and experimental study of quadrature formulate, particularly monospline and best quadrature formulae in the sense of Sard, is presented. It is shown that a good choice of degree for a monospline highly depends on the frequency content of the integrand, and in most cases, a cubic monospline generates less error than the pulse approximation method and Newton-Cotes quadrature formulae.

In space-invariant imaging systems, the object and point-spread function are represented by B-splines of degrees m and n. Exploiting the convolutional property, the deterministic part of the blurred image is a spline function of degree m+n+1. A minimum norm principle leading to pseudo-inversion is used for the restoration of space-variant degradations and underdetermined and overdetermined models. Space-variant point-spread functions that describe astigmatism and curvature-of-field are derived and coordinate transformations are applied to reduce the dimensionality. The singular-value-decomposition technique is used for solution of the simplified equations.

For noisy blurred images, a controllable smoothing criteria based on the locally variable statistics and minimization of the second derivative is defined, and the corresponding filter, applicable to both space-variant and space-invariant degradations, is obtained. The parameters of the filter determine the local smoothing window and overall extent of smoothing, and thus the trade-off between resolution and smoothing is controllable in a spatially non-stationary manner. Since the matrices of this filter are banded circular or Toeplitz, efficient algorithms are used for matrix manipulations.

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