a linear shift invariant operator h(x) , which, together with an 
additive noise term n(x), operates on an input image (or signal) f(x) 
to produce à degraded output image g(x):XFig.3.0. This may be ex- 
pressed as 
gi): m hot) 4 n(x) (3-1) 
The operator h is commonly referred to as the point-spread-function 
(ps£) of the process. This function, however, is for many imaging 
systems, a product of several psf components, where each component is 
assumed to correspond to a linear shift invariant system. Such 
components can be caused by for example diffraction, abberation, 
defocusing, motion blur and atmospheric turbulence (Castleman, 1979). 
The noise term n(x) is assumed to be à random variable. Images 
orginally recorded on photographic film are subject to degradation due 
to film grain noise. In addition to this, the digitizing or sampling 
process generates noise due to quantization, sampling rate and signal 
variation. In equation (3-1) the noise is assumed to be signal in- 
dependent. Unfortunately, in reality this is seldom the case. Much 
of the noise in the process is signal dependent, film grain noise for 
example (Andrews and Hunt 1979, Sondhi 1972, Pratt 1978). However, to 
simplify the description the independent model for the noise will be 
used, 
3.3 Determination of Degradation Parameters 
The aim of image restoration is to estimate the original image f hy 
using the recorded image g and knowledge concerning the degradation 
process. Such estimation, however, requires some form of knowledge 
concerning the degradation function h. An example of a priori de- 
termination is an experiment by McGlamery (1967) where determination of 
the turbulence psf was made. A posteriori determination of the psf may 
be performed by measuring the density of a sharp point in the degraded 
image. It is then assumed that the point in the original image is an 
approximation of the impulse function (Fig 3.2). In the same way the 
psf may be determined a posteriori from lines and edges (Andrews and 
Hunt 1977, Rosen£eld and Kak 1976, Pratt 1978). In the presence of 
noise it is also desirable to have some knowledge about the statistical 
properties of the noise. The noise is usually assumed to be white i.e. 
the expectation of the power spectrum of the noise iS constant. This 
assumption is convenient but somewhat inaccurate (Rosenfeld 
Fig 3.2 Determination of the 
psf. The original sharp point 
to the right is deblurred by 
the psf (left image). 
   
and Kak, 1976). Different restoration techniques require different 
amounts of a priori information about noise. Wiener filtering requires 
knowledge about the noise power spectrum. In constrained restoration, 
however, knowledge of the variance of the noise is nesessary. 
-202pt