Page 108 - Introduction to Information Optics

2.5. Image Processing with Optics 93

2.5.2. IMAGE RESTORATION

One of the most interesting applications in optical processing is image
restoration. As contrasted with the optimum correlation detection that uses the
maximum signal-to-noise ratio criterion, for optimum image restoration one
uses the minimum mean-square error criterion. Since this restoration spatial
filter is restricted by the same physical constraints as Eqs. (2.28) and (2.29), we
will discuss the synthesis of a practical filter that avoids the optimization. In
other words, the example provided in the following is by no means optimum.
We know that a distorted image can be, in principle, restored by an inverse
filter, as given by

(2-55)

where D(p) represents the distorting spectral distribution. In other words, a
distorted image, as described in the Fourier domain, can be written as

G(p) = S(p)D(p),

where G(p) and S(p) are the distorted and undistorted image spectral distribu-
tions, respectively. Thus, we see that an image can be restored by inverse
filtering, such as

S(p) = G(p)H(p). (2.56)

Let us now assume that the transmission function of a linear srneared-point
(blurred) image can be written as

0, otherwise,

where A£ is the smeared length. To restore the image, we seek for an inverse
filter as given by

,-
(1

In view of the preceding equation, we quickly note that the filter is not
physically realizable, since it has an infinite number of poles. This precisely
corresponds to some information loss due to smearing. In other words, to