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The cross correlation or correlation plane, of a 2D signal with is
This can be re-expressed in Fourier space as
where the capital letters denote the Fourier transform of what the lower case letter denotes. So the correlation can then be calculated by inverse Fourier transforming the result.
According to Fresnel Diffraction theory a convex lens of focal length will produce the exact Fourier transform at a distance behind the lens of an object placed distance in front of the lens. So that complex amplitudes are multiplied, the light source must be coherent and is typically from a laser. The input signal and filter are commonly written onto a spatial light modulator (SLM).
A typical arrangement is the 4f correlator. The input signal is written to an SLM which is illuminated with a laser. This is Fourier transformed with a lens and this is then modulated with a second SLM containing the filter. The resultant is again Fourier transformed with a second lens and the correlation result is captured on a camera.
Many filters have been designed to be used with an optical correlator. Some have been proposed to address hardware limitations, others were developed to optimize a merit function or to be invariant under a certain transformation.
The matched filter maximizes the signal-to-noise ratio and is simply obtained by using as a filter the Fourier transform of the reference signal .
The phase-only filter is easier to implement due to limitation of many SLMs and has been shown to be more discriminant than the matched filter.
- A. VanderLugt, Signal detection by complex spatial filtering, IEEE Transactions on Information Theory, vol. 10, 1964, pp. 139-145.
- J. L. Horner and P. D. Gianino, Phase-only matched filtering, Appl. Opt. 23, 1984, 812-816