Diagnosis of stochastic fields by the mathematical morphology and computational topology methods

The paper focuses on the diagnosis of extended observations arising in different sciences. In many cases, the nonlinearity and heterogeneity of natural processes produce fields with stochastic properties. The evolution of the fields is represented by a sequence of patterns (maps). The direct generalization of the Takens algorithm (Lect. Notes Math. 898 (1981) 366) to the patterns leads to computational difficulties. To extract dynamical information characterizing the spatial-temporal chaos, one can use topological complexity of patterns. The Morphological Image Analysis allows extracting the geometry and topology of the maps by mean of the Minkowski functionals (Phys. Rep. 347 (2001) 461). Then, the time series of the functionals can be used to reconstruct the universal model by the embedding techniques. Moreover, it is possible to extract additional information concerning dynamic scenario by studying a change of a connectedness of the map using different resolutions. The change is estimated by the index of disconnectedness which is equivalent to the box dimension for simple sets. The application of these methods to seismology, ecology, and solar physics is demonstrated.