Most blurring processes can be approximated by convolution integrals, also known as Fredholm integral equations of the first kind [4]. The blurring is characterized by a Point-Spread Function (PSF) or impulse response. The PSF is the output of the imaging system for an input point source. All the blurring processes considered in this thesis are linear and have a spatially invariant PSF.
For discrete image processing, the convolution integral is replaced by a sum. The blurry image x(n,m) is obtained from the original image s(n,m) by this convolution:
The function h(n,m) is the discrete Point Spread Function for the imaging system. Also of interest is the Discrete Fourier Transform (DFT) representation of the point-spread function, given by
for 
and 
.
H(u,v) gives a set of coefficients for plane waves of various frequencies
and orientations. These plane waves, called spatial frequency
components, reconstruct the PSF exactly when multiplied by the
coefficients H(u,v) and summed.
The function H(u,v) is referred to as the transfer function,
or system frequency response. By examining |H(u,v)|, one
can quickly determine which spatial frequency components are
passed or attenuated by the imaging system.
As an example, consider this 3x3 mask which can be used to model small amounts of blurring:
The DFT of this mask is:
Figure 1.1: |H(u,v)| for a 3x3 blurring mask, with N=M=33
Figure 1.1 shows a plot of |H(u,v)|. Near 
,
the transfer function has 
. This indicates that
low-frequency components are passed. Near the perimeter of the plot,
, meaning that high frequency components are
blocked.