Preservation of Topological Properties of a Simple Closed Curve under Digitalization

In many applications of computer vision and image processing, one has to infer the properties of a real scene from its digitalized image. Therefore, preservation of different types of properties under digitalization is of importance. In this paper we have studied the effect of digitalization on the topological properties of simple closed curves, with the aim of determining the conditions under which these properties are preserved. Considering a square digitalizing grid, we have shown that for simple closed curves with a finite number of extrema, there exists an upper bound on the grid spacing for the preservation of the topological properties. Moreover, the properties are preserved for all translations of the grid except for a set of measure zero.

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