Page 122 - Introduction to Information Optics

2.6. Algorithms for Processing 107

Since the circular harmonic function is determined by different angles ma but
not by a simple a, it is evident that object rotation poses severe problems for
conventional matched filtering. Nevertheless, by using one of the circular
harmonic functions as the reference functions, such as
m(
/ ref(r, 0) = F m(ry \ (2.81)
the central correlation value between the target /(r. 8 + a) and the reference
/(r, 8) can be written as
im
C(a) = A me \
The corresponding intensity is independent from a, such as
2
C(a)| = /I,;, (2.82)
for which we see that it is independent of the object orientation.
Needless to say, the implementation of circular harmonic processing can be
by using either a Fourier domain filter or a spatial domain filter in a JTC. One
of the major advantages of using a JTC is robust environmental factors.
However, JTPS is input scene dependent, so the detection or correlation
efficiency is affected by the input scene; for example, strong background noise,
multi-input objects, and other problems, which cause poor correlation per-
formance. Nevertheless, these disadvantages can be alleviated by using non-
zero-order JTC, as will be described in Sec. 7.2.

2.6.3. HOMOMORPHIC PROCESSING

The optical processing discussed so far relies on linear spatial invariant
operation. There are, however, some nonlinear processing operations that can
be carried out by optics. One such nonlinear processing operation worth
mentioning is homomorphic processing, described in Fig. 2.31. Note that
homomorphic processing has been successfully applied in digital signal pro-
cessing. In this section, we illustrate logarithmic processing for target detection
in a multiplicative noise regime. Let us assume an input object is contaminated
by a multiplicative noise, as written by

n(x,y)f(x,y). (2.83)

Since matched filtering is optimum under an additive white Gaussian noise
assumption, our first task is to convert the multiplicative noise to an additive
noise by means of logarithmic transformation, such as

log[H(x, j;)/(x, >')] = log n(x, y) + log /(x, y). (2.84)