Physics of auroral phenomena : proceedings of the 37th Annual seminar, Apatity, 25 - 28 February, 2014 / [ed. board: A. G. Yahnin, N. V. Semenova]. - Апатиты : Изд-во Кольского научного центра РАН, 2014. - 125 с. : ил., табл.

В. V. Kozelov 3. Simple numerical example To illustrate the above definitions by numerical examples let we consider the test data set generated by function / ( * ) = 50 (sin —sin— + s in— + + 130 ... 1 v J V. 8 400 40 / (4 ) Here xE{0,1,...,400}, s - random number (Gaussian noise with zero mean and unit standard deviation). Now we define subgraph o f a function as a set of points that lies between the graph of the fimction and above the X axis of the graph SG(f)={(x,y)eK*R 10<у<Л*)} (5) Fig. 2 illustrates the basic MM operations by applications to this data set. In Fig. 2 (a,b) the subgraph SG(f) considered as a set A, and horizontal segment [0,10] is taken as structuring element B. Fig. 2(c) illustrates filtering by open-close filter with segments [0,4] (red) and [0,16] (blue). One can see that the red line is higher the blue one when the graph has a peak which is not thinner in X axis than 4. The red line is lower the blue one when the graph has a ‘valley’ of the correspondent range of scales. Using the filtering with sequentially increasing structuring element it is possible to obtain a morphological spectrum, as it is illustrated by Fig. 3. The original data set is shown in panel ‘a ’. The filter (3) has been applied sequentially with structuring elements [0, ?]] and [0, t2], where t2=t\+l, /]6{1,...,100}. Location of “peaks” and “valleys” at correspondent scale is shown in panel ‘b’ by white and black segments. Positions of center of each segment are shown in panel ‘c ’ (Really, an operation called thinning has been applied.). Now we can calculate intervals between points on each scale and to obtain a statistical distribution presented in panel ‘d \ Integration this distribution over all scales gives us the integrated morphological spectrum shown in panel ‘e \ One can see that despite the short data set and random noise, the spectrum obviously demonstrates two main periods (16 and 80 data points). Figure 2. Application of MM operations to data set generated by Eq.4 (see text). Ю0ШИИИ ' r woi 200 coordinate 80 Figure 3. Construction of the MM spectrum for data set generated by Eq.4: a - data set; b - results of filtering at different scales (white - maximums, black - minimums); с - results of thinning operation applied to panel (b); d - spectra of intervals between elements in panel (c) at different scales; e - integrated MM spectrum. erosion 43