Euler characteristic digital patterns

Figure 23. The 2D illustration of the digital pattern analysis for computing the Euler characteristic. The local curvature variables xe —1,0, +1 are assigned to the each lattice site at the boundaries of black pixels. The Euler characteristic is a sum of local curvature variables...

Figure 38. An example when the Euler characteristic cannot be correctly determined by using the digital pattern analysis, because the boundaries of the domains after the thresholding procedure cannot be specified.

In conclusion, the digital pattern method is very efficient computationally but fails to calculate the Euler characteristic for the cases similar to the one shown in Fig. 38. Such cases are typical, however, when the data are noisy or when the field values are distributed around the threshold value. 

The disagreement of those two methods can be used to characterize quantitatively the double-separated morphology. One should simply calculate the Euler characteristic once by using the Euler formula (130), and then calculate it a second time by using the digital pattern method. The difference between obtained quantities will give the number of double-separated structures. 

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