Page 127 - Introduction to Information Optics

112 1. Signal Processing with Optics

2.6.5, SIMULATED ANNEALING ALGORITHM

One of the major drawbacks of SDF filter synthesis is the high dynamic
range requirement that deters its practical implementation on a commercially
available SLM, There is, however, an alternative approach toward the syn-
thesis of a composite filter; namely, the simulated annealing (SA) filter.
The SA algorithm is, in fact, a computational optimization process in which
an energy function should be established based on certain optimization criteria.
In other words, the state variables of a physical system can be adjusted until a
global minimum energy state is established. For example, when a system
variable w t is randomly perturbed by AM, the change of the system energy can
be calculated

old
A£ = £ new - E , (2.95)
old
where E new = E(u t + A« ;) and £ = E(u^. Notice that, if A£ < 0, the pertur-
bation AM,, is unconditionally accepted. Otherwise, the acceptance of Aw, is
based on the Boltzmann probability distribution p(A£); that is,

where T is the temperature of the system used in the simulated annealing
algorithm and k is the Boltzmann constant. The process is then repeated by
randomly perturbing each of the state variables and slowly decreasing the
system temperature T, which can avoid the system being trapped in a local
minimum energy state. In other words, by continually adjusting the state
variables of the system and slowly decreasing the system temperature 7; a
global minimum energy state of the system can be found. This is known as the
simulated annealing process. We now illustrate synthesis of a bipolar composite
filter (BCF) using the SA algorithm.
Let us consider two training images, called the target and antitarget set, as
given by

("target set = {t m(x, y)}
{antitarget set = {a m(x, y)},

where m — 1,2,..., M, M is the number of training images. The purpose of
designing a composite filter is to detect the target objects and to reject the
antitarget objects, which are assumed similar to the target objects. Let us
a
denote W\x, y) and W (x, y) as the desired correlation distributions for the
target and antitarget objects, respectively. The mean-square error between the