Complexity and Synergetics by Stefan C. Müller Peter J. Plath Günter Radons & Armin Fuchs

Author:Stefan C. Müller, Peter J. Plath, Günter Radons & Armin Fuchs
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

By removing such a free pair from a complex , a subcomplex is created. Two complexes and are equivalent if a sequence of collapses (or the contrary: expansions) can transform one into the other, which is nothing else but a continuous morphing, previously called a homotopy.

Now imagine a 2D grid with scalar data that can be visualized as a landscape with mountains and valleys with their specific height. The landscape is filled with water. As long as the water level is below the lowest valley, the landscape is dry. Once the water starts filling the lowest valley, a lake is formed and the topology of the water surface changes. Once the water level exceeds a saddle between two hills and thereby combines two valleys such that two lakes join, the topology changes again. As long as no new valleys are filled and the water level does not exceed mountains, the topology does not change. This also works in the case of a 3D grid with the scalar values of neighboring vertices. Consider the lower level cuts, which contain all cells with values lower than a threshold. The idea is to analyse how the topology of these lower level cuts changes as the threshold value varies, and then use these critical values as isovalues.

Definition 3

(Lower star) The star of a cell in a cubical complex is the set of all cofaces of , the lower star of a cell in a cubical complex is the set of all cofaces of with scalar value less or equal to the scalar value of .

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