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The matched filter (MF) represents one of the main tools to detect signalsfrom known sources embedded in the noise. In the Gaussian case the noise isassumed to be the realization of a Gaussian random field (GRF). The mostimportant property of the MF, the maximization of the probability of detectionsubject to a constant probability of false detection or false alarm (PFA),makes it one of the most popular techniques. However, the MF technique reliesupon the a priori knowledge of the number and the position of the searchedsignals in the GRF which usually are not available. A typical way out is toassume that the position of a signal coincides with one of the peaks in thematched filtered data. A detection is claimed when the probability that a givenpeak is due only to the noise (i.e. the PFA) is smaller than a prefixedthreshold. In this case the probability density function (PDF) of theamplitudes has to be used for the computation of the PFA, which is differentfrom the Gaussian. Moreover, the probability that a detection is false dependson the number of peaks present in the filtered GRF, the greater the number ofpeaks in a GRF, the higher the probability of peaks due to the noise thatexceed the detection threshold. If not taken into account, the PFA can beseverely underestimated. Many solutions proposed to this problem arenon-parametric hence not able to exploit all the available information. Thislimitation has been overcome by means of two efficient parametric approaches,one based on the PDF of the peak amplitudes of a smooth and isotropic GRFwhereas the other uses the Gumbel distribution (the asymptotic PDF of thecorresponding extreme). Simulations and ALMA maps show that, although the twomethods produce almost identical results, the first is more flexible and allowsus to check the reliability of the detection procedure.

astronomy and astrophysics

Correct estimate of the probability of false detection of the matched  filter in weak-signal detection problems. III (Peak distribution method  versus the Gumbel distribution method)