Page 306 - Introduction to Information Optics

5.10. Geometric Transform '^9.1

because only in the regions of vicinity of the saddle points can significant
contributions to the value of the integral be established. From Eqs. (5.41) and
(5.48) we see that the required mapping from Eq. (5.45) is satisfied at the saddle
points.
In general, there is more than one saddle point. Therefore, the x -y plane is
subdivided into subareas, each containing only one saddle point. The output
intensity distribution only is taken into consideration; the geometric trans-
formed image is

This technique can be also understood as the phase mask,

exp( jnh(x, y; u, V)/A),

acts as a set of local gratings or local prisms which diffract /(x, y) to G(u, v) at
a set of subareas in the x-y plane where the phase is stationary.

Example 5.4
Find the phase mask amplitude transmittance for logarithmic transfor-
mation.

Solution
The mappings of logarithmic transformation are

u — (p^x) — In x

ln
v = (p 2(y) = >'•

As mentioned in Eq. (5.45), the phase of the mask 0(x, y) bears the relation

x, y) _ 2n
— — in x
dx /.f
d(b(x, y) 2n
~~ ir
_ -5 , — _
Sy A/