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An optical correlator is an optical computer for comparing two signals by utilising the Fourier transforming properties of a lens. [1] It is commonly used in optics for target tracking and identification.
The correlator has an input signal which is multiplied by some filter in the Fourier domain. An example filter is the matched filter which uses the cross correlation of the two signals.
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 [2] is easier to implement due to limitation of many SLMs and has been shown to be more discriminant than the matched filter.
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Mean-field particle methods are a broad class of interacting type Monte Carlo algorithms for simulating from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability measures can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depends on the distributions of the current random states. A natural way to simulate these sophisticated nonlinear Markov processes is to sample a large number of copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methods these mean-field particle techniques rely on sequential interacting samples. The terminology mean-field reflects the fact that each of the samples interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes. In other words, starting with a chaotic configuration based on independent copies of initial state of the nonlinear Markov chain model, the chaos propagates at any time horizon as the size the system tends to infinity; that is, finite blocks of particles reduces to independent copies of the nonlinear Markov process. This result is called the propagation of chaos property. The terminology "propagation of chaos" originated with the work of Mark Kac in 1976 on a colliding mean-field kinetic gas model.
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