Volume 14, issue 6 (2014)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Pixelations of planar semialgebraic sets and shape recognition

Liviu I Nicolaescu and Brandon Rowekamp

Algebraic & Geometric Topology 14 (2014) 3345–3394
Abstract

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.

Keywords
semialgebraic sets, pixelations, normal cycle, total curvature, Morse theory
Mathematical Subject Classification 2010
Primary: 53A04
Secondary: 53C65, 58A35
References
Publication
Received: 5 August 2013
Revised: 22 April 2014
Accepted: 24 April 2014
Published: 15 January 2015
Authors
Liviu I Nicolaescu
Department of Mathematics
University of Notre Dame
255 Hurley
Notre Dame, IN 46556-4618
USA
http://www.nd.edu/~lnicolae/
Brandon Rowekamp
Department of Mathematics & Statistics
Minnesota State University, Mankato
273 Wissink Hall
Mankato, MN 56001
USA