Geodesic
Pixelations of planar semialgebraic sets and shape recognition
Nicolaescu, Liviu1 ; Rowekamp, Brandon2
1Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556-4618, USA
2Department of Mathematics & Statistics, Minnesota State University, Mankato, 273 Wissink Hall, Mankato, MN 56001, USA
Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3345-3394
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We describe an algorithm that associates to each positive real number ε and each finite collection Cε of planar pixels of size ε a planar piecewise linear set Sε with the following property: If Cε is the collection of pixels of size ε that touch a given compact semialgebraic set S, then the normal cycle of Sε converges in the sense of currents to the normal cycle of S. In particular, in the limit we can recover the homotopy type of S and its geometric invariants such as area, perimeter and curvature measures. At its core, this algorithm is a discretization of stratified Morse theory.

DOI : 10.2140/agt.2014.14.3345
Classification : 53A04, 53C65, 58A35
Keywords: semialgebraic sets, pixelations, normal cycle, total curvature, Morse theory

Nicolaescu, Liviu  1   ; Rowekamp, Brandon  2

1 Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556-4618, USA
2 Department of Mathematics & Statistics, Minnesota State University, Mankato, 273 Wissink Hall, Mankato, MN 56001, USA 
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Nicolaescu, Liviu; Rowekamp, Brandon. Pixelations of planar semialgebraic sets and shape recognition. Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3345-3394. doi: 10.2140/agt.2014.14.3345

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