A Bayesian approach adapted to practical detection and location tasks of a target in additive Gaussian noise with unknown spectral density is developed and studied. The relevance of this theory is first discussed in comparison with a maximum a posteriori solution that has been recently developed [J. Opt. Soc. Am. A 15, 61 (1998)]. The analysis is first performed without considering a particular prior for the spectral density of the noise. General results of the Bayesian approach are thus provided as well as properties of its first-order development, which corresponds to the so-called nonlinear joint-transform correlation frequently used in optical correlators. It is demonstrated that the kernel of the nonlinear filtering is an increasing function of the sum of the spectral density of the reference object and of the input image. Furthermore, it is shown that a power-law mathematical form of the nonlinear filtering is directly related to assumptions on the asymptotic behavior of the prior density probabilities of the unknown spectral density of the noise. These properties constitute new theoretical results in the context of statistical theory concerning the use of nonlinearities in optical correlators.
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