Page 110 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 91

(a) (b)

(c) (d)

FIGURE 3.8 (a) The analyzed image, (b) the reference image, (c) the
amplitude of their correlation in the Fourier domain, and (d) the amplitude of
their correlation in the fractional Fourier domain for x = /2 and y = /4.

−1
with U 1 = U = U f ( x , y ) and U 3 = U f (− /2, − /2), which pro-
2
vides the shift-variant pattern recognition. The shift tolerance condi-
tion is usually expressed in the form 4,15 v x,y
x,y cot x,y 1, where
v x,y is the allowed shift of the pattern on the scene with respect to the
reference one used for filter design and
x,y is the pattern width in the
x and y directions, correspondingly.
Thus, if we choose different fractional angles for two orthogonal
coordinates, such that x = /2 and y = /4 and the same filter as
we have used before, then 0 will be recognized only on the middle line
of the number set, as shown in Fig. 3.8d, where the amplitude of the
fractional correlation is displayed. Therefore, the fractional correlation