# Optical correlator

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An '''optical correlator''' is an [optical computer](/source/optical_computer) for comparing two signals by utilising the [Fourier transforming properties of a lens](/source/Fourier_optics).<ref>A. VanderLugt, ''[http://deepblue.lib.umich.edu/bitstream/2027.42/8080/5/bad4575.0001.001.pdf Signal detection by complex spatial filtering],'' IEEE Transactions on Information Theory, vol. 10, 1964, pp. 139–145.</ref> It is commonly used in [optics](/source/optics) for target tracking and identification.

==Introduction==

The correlator has an input signal which is multiplied by some filter in the Fourier domain. An example filter is the [matched filter](/source/matched_filter) which uses the [cross correlation](/source/cross_correlation) of the two signals.

The cross correlation or correlation plane, <math>c(x,y)</math> of a 2D signal <math>i(x,y)</math> with <math>h(x,y)</math> is

: <math>c(x,y)=i(x,y) \otimes h^{*}(-x,-y)</math>

This can be re-expressed in Fourier space as

: <math> C(\xi,\eta)=I(\xi,\eta) H^{*}(-\xi,-\eta) </math>

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.

==Implementation==

According to [Fresnel Diffraction](/source/Fresnel_Diffraction) theory a [convex lens](/source/convex_lens) of [focal length](/source/focal_length) <math>f</math> will produce the exact Fourier transform at a distance <math>f</math> behind the lens of an object placed <math>f</math> distance in front of the lens. So that [complex amplitudes](/source/complex_amplitudes) are multiplied, the light source must be [coherent](/source/Coherence_(physics)) and is typically from a [laser](/source/laser). The input signal and filter are commonly written onto a [spatial light modulator](/source/spatial_light_modulator) (SLM).

A typical arrangement is the [4f correlator](/source/Fourier_optics). 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.

==Filter design==

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.

=== Matched filter===

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 <math>r(x,y)</math>.

: <math> H(\xi,\eta) = R(\xi,\eta)</math>

===Phase-only filter===

The phase-only filter<ref>J. L. Horner and P. D. Gianino, ''Phase-only matched filtering'', Appl. Opt. 23, 1984, 812–816</ref> is easier to implement due to limitation of many SLMs and has been shown to be more discriminant than the matched filter.

: <math> H(\xi,\eta) = \frac{R(\xi,\eta)}{ \left\vert R(\xi,\eta)  \right\vert}</math>

==References==
{{Reflist}}

{{DEFAULTSORT:Optical Correlator}}
Category:Optical instruments
Category:Fourier analysis

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Adapted from the Wikipedia article [Optical correlator](https://en.wikipedia.org/wiki/Optical_correlator) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Optical_correlator?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.