Page 387 - Introduction to Information Optics

372 7, Pattern Recognition with Optics

phase transformation can be written as

T\J(x, y)] = - 2n. (7.10)

Applied to the JTC, the phase representation of the input can be written as
pf(x + a, y) + pr(x — a, y), where pr(x, y) — e l<pr(x y)
' represents the phase refer-
ence function. The corresponding output cross-correlation peak intensities is

2
C(±2«,0) = pf(x, y)pr*(x, y) dx dy = \A\ 2 2 7.11)

for cp r(x, y) = T[/(x, j;)], and A is a constant proportional to the size of the
reference function.
The phase representation JTC (PJTC) is indeed an optimum correlator,
regardless of nonzero-mean noise; that is,

SNR < El\PF n(p, q)^ dpdq. (7.12)

The equality holds if and only if <f> r(x, y) = T[/(x, >')], where £[] denotes the
ensemble average and F n(p, q) is the noise spectrum. Thus, the PJTC is an
optimal filtering system, independent of the mean value of the additive noise.
A CJTC often loses its pattern discriminability whenever a false target is
similar to the reference function or the object is heavily embedded in a noisy
background. For comparison, let the two images of an M60 tank and a T72
tank be embedded in a noisy background, as shown in Fig. 7.20a, where an
M60 tank is used as the reference target. The corresponding output correlation
distributions are shown in Figs. 7.20b and c, in which we see that CJTC fails
to detect the M60 tank.

7.2.5. ITERATIVE JOINT-TRANSFORM DETECTION

Hybrid JTC is known for its simplicity of operation and real-time pattern
recognition. By exploiting the flexibility of the computer, iterative operation is
capable of improving its performance. In other words, it is rather convenient
to feedback the output data into the input for further improvement of the
operation.
Here we illustrate the iterative feedback to a composite filtering (as an
example) to improve detection accuracy. Let us assume that the input scene to
a JTC contains N reference patterns and a target 0(x, y) to be detected, as