The Stability of the Apparent Contour of an Orientable 2-Manifold

Herbert Edelsbrunner, Dmitriy Morozov, Amit Patel.
Proceedings of the Workshop on Topological Methods in Data Analysis and Visualization, 2009.
PDF TopoInViz
The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f: M → R2 , is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming M is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.
Torus contour
Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. Quantifying transversality by measuring the robustness of intersections. Manuscript, Deptartment of Computer Science, Duke University, Durham, North Carolina, 2009.