ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O
tokens: be expected to be smaller than both N and N̄ p . Indeed, a maximum likelihood fit υ ML ( z max ) of the PDF given by Eq. ( 13 ) to the set of simulated { z max } with N p as free parameter provides N̄ p = 2210. The right panel of Fig. 3 shows that something similar holds for the GDM since the theoretical value N * = 17 684 is greater than the number Ñ ∗ = 15330 corresponding to the maximum likelihood fit g ML ( z max ). Here, it is worth stressing that υ Th ( z max ) and g Th ( z max ), and hence the corresponding CDFs Υ Th ( z max ) and G Th ( z max ) shown in Fig. 4 are almost identical. The same holds for υ ML ( z max ) and g ML ( z max ) and the corresponding CDFs Υ ML ( z max ) and G ML ( z max ) again shown in Fig. 4 . As expected the two approaches are practically equivalent, given that both methods are essentially based on the same assumptions. Moreover, in Fig. 4 all these CDFs appear close to the sample CDF. This indicates that the value of the parameters N p and N * is not of critical importance. Fig. 3. Left panel : histogram H ( z max ) vs. the theoretical PDF υ Th ( z max ), mean PDF υ Mean ( z max ) and the maximum likelihood PDF υ ML ( z max ) of the value z max 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 × 10 3 GRF’s of size 500 × 500 pixels. υ Th ( z max ) has been computed using Eqs. ( 18 ) and ( 8 ) with N p obtained as explained in Vio17, whereas υ mean ( z max ) has been computed with N p given by the mean number of peaks in the simulated GRF’s. In the case of υ ML ( z max ) the number N p comes out from a maximum likelihood method (see text). Right panel : corresponding g Th ( z max ), computed by means of Eq. ( 27 ) with N * given by the ratio of the total area of the GRF to the window area πl 2 with (see text). Also the PDF g ML ( z max ) has been computed with N * resulting from a maximum likelihood method. To notice that υ Th ( z max ) and g Th ( z max ) are almost identical. The same holds for υ ML ( z max ) and g ML ( z max ). Fig. 4. Cumulative distribution functions corresponding to the PDFs in Fig. 3 . A further comparison of the two methods concerns the computational burden. In this respect the GDM is superior in situations where the autocorrelation function has circular symmetry (i.e., it is characterized by only a scale parameter), since the more expensive step consists in the computation of the quantities σ j , j = 0, 1, 2, which requires the numerical computation of three one-dimensional integrals. On the other hand, PAM requires the numerical computation of a number of integrals equal to the number of peaks selected for the detection. In general the difference in computational time is on the order of a few seconds even in the case of large images. The computational superiority of GDM vanishes in the presence of autocorrelation functions when they are not