Left panel: histogram H(zmax) vs. the theoretical PDF υTh(zmax), mean PDF υMean(zmax) and the maximum likelihood PDF υML(zmax) of the value zmax of the highest peak of a zero-mean unit-variance GRF with autocorrelation given by a circular Gaussian with dispersion set to three pixels. The numerical experiment is based on the simulation of 5 × 103 GRF’s of size 500 × 500 pixels. υTh(zmax) has been computed using Eqs. (18) and (8) with Np obtained as explained in Vio17, whereas υmean(zmax) has been computed with Np given by the mean number of peaks in the simulated GRF’s. In the case of υML(zmax) the number Np comes out from a maximum likelihood method (see text). Right panel: corresponding gTh(zmax), computed by means of Eq. (27) with N* given by the ratio of the total area of the GRF to the window area πl2 with (see text). Also the PDF gML(zmax) has been computed with N* resulting from a maximum likelihood method. To notice that υTh(zmax) and gTh(zmax) are almost identical. The same holds for υML(zmax) and gML(zmax).
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