Page 119 - Introduction to Information Optics

104 2. Signal Processing with Optics

2.6.1. MELLIN-TRANSFORM PROCESSING

Although complex signal detection is shift invariant, it is sensitive to rotation
and scale variants. In other words, the scale of the input object for which the
complex spatial filter is synthesized must be precisely matched. The miscalmg
problem can be alleviated by using a technique called the Mellin-transform
technique.
The Mellin transformation has been successfully applied in time-vary ing
circuits and in space-variant image restoration, as given by

(lp+i {iq + 1
M(ip, iq) = f(£, w)C >n >d£dn. (2.69)
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The major obstacle to the optical implementation of Mellin transform is the
nonlinear coordinator transformation of the input object. If we replace the
y
x
y
x
space variables £ = e and r\ = e , the Fourier transform of f(e , e ) yields the
Mellin transform of /(£, 77), as given by
00
ff y
M(p, q) = f(e\ e ) exp[-f(px + qyf] dxdy, (2.70)
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where we let M(ip, ig) = M(p, ^) to simplify the notation. An inverse Mellin
transform can also be written by

q)] =/(£, if) = - Af(p, q)^^dpdq. (2.71)

The preceding equation can be made equivalent to the inverse Fourier
transform by replacing the variables £ = exp(x) and r\ — exp(y).
The basic advantage of applying the Mellin transform to optical processing
is its scale-invariant property. In other words, the Mellin transforms of two
different-scale, but otherwise identical, spatial functions are scale invariant; that
is,

+
M 2(p, q) - a«' «>Af ,(/>, q\
where a is an arbitrary factor, M x(p, q) and M 2(p, q) are the Mellin transforms
of /j(x, y) and f 2(x, y), respectively, and / t(x, v) and f 2(x, y) are the identical
but different-scale functions. From the preceding equation we see that the
magnitudes of the Mellin transforms are

\M 2(p,q)\=\Mt(p,q)(, (2.72)