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cell-complex-examples - Computational Topology(Jeff...

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Computational Topology (Jeff Erickson) Examples of Cell Complexes Arithmétique ! algèbre ! géométrie ! trinité grandiose ! triangle lumineux ! Celui qui ne vous a pas connues est un insensé ! Il mériterait l’épreuve des plus grands supplices ; car, il ya du mépris aveugle dans son insouciance ignorante . . . . [Arithmetic! Algebra! Geometry! Grandiose trinity! Luminous triangle! Whoever has not known you is a fool! He deserves the most intense torture, for there is blind contempt in his ignorant indifference . . . . ] — Le Comte de Lautréamont [Isidore Lucien Ducasse], Chant Deuxième, Les Chants de Maldoror (1869) À bas Euclide ! Mort aux triangles ! [Down with Euclid! Death to triangles!] — Jean Dieudonné, keynote address at the Royaumont Seminar (1959) 15 Examples of Cell Complexes 15.1 Proximity Complexes of Point Clouds Point clouds are an increasingly common representation for complex geometric objects or domains. For many applications, instead of storing an explicit description of the domain, either because the object is too complex or because it is simply unknown, it may be sufficient to store a representative sample of points from the object. Typical sources of point-could data are scanners (such as digital cameras, laser range-finders, LIDAR, medical imaging systems, and telescopes), edge- and feature-detection algorithms from computer vision, locations of sensors and ad-hoc network devices, Monte Carlo sampling and integration algorithms, and training data for machine learning systems. By themselves, point clouds have no interesting topology. However, there are several natural ways to impose topological structure onto a point cloud, intuitively by ‘connecting’ points that are sufficiently ‘close’. If the underlying domain is sufficiently ‘nice’ and the point sample is sufficiently ‘dense’, we can recover important topological features of the underlying domain. 15.1.1 Aleksandrov- ˇ Cech Complexes: Nerves and Unions Let P be a set of points in some metric space S , and let " be a positive real number. Typically, but not universally, the point set P is finite and the underlying space S is the Euclidean space R d . The Aleksandrov- ˇ Cech complex A ˇ C " ( P ) is the intersection complex or nerve of the set of balls of radius " centered at points in P . That is, k + 1 points in P define a k -simplex in A ˇ C " ( P ) if and only if the " -balls centered at those points have a non-empty common intersection, or equivalently, if those points lie inside a ball of radius " . Formally, the Aleksandrov- ˇ Cech complex is an abstract simplicial complex; its simplices can overlap arbitrarily and can have arbitrarily high dimension. Aleksandrov- ˇ Cech complexes were developed independently by Pavel Aleksandrov [ 6 ] and Eduard ˇ Cech [ 16 ] 1 ; despite Aleksandrov’s earlier work, they are more commonly known as ˇ Cech complexes .
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