SCALE INVARIANT OPTICAL CORRELATION
USING MELLIN TRANSFORMS
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VanderLugt Filter (VLF)
The total incident intensity at the level of the film is determined by the interference of
two distributions of amplitude(coming from L2 and from P). The inclined wave plane is
represented by its complex amplitude in the plane P2 .
An optical filter having the remarkable property to permit the efficient control of the
amplitude and of the phase of a transfer function at the same time even if they are only
formed in purely absorbent figures.
The total intensity incident at each point on the recording medium is
determined by the interference of the two mutually coherent amplitude
distributions present. The tilted plane wave incident from the prism produces
a field distribution
where the spatial frequency a = sin Ɵ/ λ
So the total in intensity distribution may be written as
We can write H as an amplitude distribution A and a phase distribution Ψ
As a final step in the synthesis of the frequency-plane mask, the exposed film is
developed to produce a transparency which has an amplitude transmittance that is
proportional to the intensity distribution that was incident during exposure. Thus the
amplitude transmittance of the filter is of the form
Now the field strength transmitted by the mask then obeys the proportionality
Finally we take FT of the desired response impulse by either Modified Mach-
Zehnder interferometer or modified Rayleigh interferometer
we see that the third output term yields a convolution of h and g, centred at coordinates
(0, -ah f ) in the (x3, y3) plane. Similarly, the fourth term may be rewritten as
which is the crosscorrelation of g and h, centered at coordinates (0, ah f ) in the (x3, y3) plane.
The Mellin transform of a function g(t) is defined by
SCALE INVARIANT OPTICAL CORRELATION USING MELLIN TRANSFORMS
If we take s , an imaginary quantity s = j2πf and substitute ξ = exp(-x)
M (j2πf ) = =
which we see is nothing but the Fourier transform of the function g(e-x).
Means it is possible to perform a Mellin transform with an optical Fourier
transforming system provided the input is introduced in a "stretched" coordinate
system, in which the natural space variable is logarithmically stretched (x = - In ξ ).
Such a stretch can be introduced, for example, by driving the deflection voltage of a
cathode ray tube through a logarithmic amplifier and writing onto an SLM with the
resulting stretched signal.
Here we see that the magnitude is independent of scale-size changes in the input.
To prove this we replace M( g (ξ ) ) by M( g (aξ ) ) , and we see the invariance in the
introduction to Fourier optics by Joseph W. Goodman .