The Stability of the Contour of an Orientable 2-Manifold
Jul 30, 2009 4126 views 
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Think of the view of the boundary of a solid shape as a projection of a 2-manifold to R^2. Its apparent contour is the projection of the critical points. Generalizing the projection to smooth mappings of a 2-manifold to R^2, we get the contour as the image of the points at which the derivative is not surjective. Measuring difference with the erosion distance (the Hausdorff distance between the complements), we prove that the contour is stable. Along the way, we introduce the by now well established method of persistent homology, including the stability of its diagrams, as well as an extension using zigzag modules. Joint work with Dmitriy Morozov and Amit Patel.