Page 99 - Introduction to Information Optics

84 2. Signal Processing with Optics

Third term,
convolution

Fourth term,
cross-correlation

Fig. 2.14. Sketch of output diffraction from a FDP.

see that the first and second terms represent the zero-order diffraction, which
appears at the origin of the output plane, and the third and fourth terms are
the convolution and cross-correlation terms, which are diffracted in the
neighborhood of a — — a 0, and a = a 0, respectively, as sketched in Fig. 2.14.
To further show that the processing operation is shift invariant, we let the input
object function translate to a new location; that is, /(x — x 0, y — v 0). The
correlation term of the preceding equation can then be written as

f(x - x 0, y - y 0)f*(x - a - a 0, y - ft) dxdy = R { t(a - a 0 - x 0, ft - >- 0),

(2.38)

where R n represents an autocorrelation function. In view of this result, we see
that the target correlation peak intensity has been translated by the same
amount, for which the operation is indeed shift invariant. A sketch of the
output correlation peak as referenced to the object translation is illustrated in
Fig. 2.15.