Gradient algorithms for polygonal approximation of convex contours

The subjects of this paper are descent algorithms to optimally approximate a strictly convex contour with a polygon. This classic geometric problem is relevant in interpolation theory and data compression, and has potential applications in robotic sensor networks. We design gradient descent laws for intuitive performance metrics such as the area of the inner, outer, and ''outer minus inner'' approximating polygons. The algorithms position the polygon vertices based on simple feedback ideas and on limited nearest-neighbor interaction.

[1]  Alain Sarlette,et al.  Synchronization and balancing on the N-torus , 2007, Syst. Control. Lett..

[2]  Timothy W. McLain,et al.  Cooperative forest fire surveillance using a team of small unmanned air vehicles , 2006, Int. J. Syst. Sci..

[3]  F. Bullo,et al.  Distributed algorithms for polygonal approximation of convex contours , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[4]  Demetri Terzopoulos,et al.  Snakes: Active contour models , 2004, International Journal of Computer Vision.

[5]  J. P. Lasalle.  The stability and control of discrete processes , 1986 .

[6]  T. Banchoff,et al.  Differential Geometry of Curves and Surfaces , 2010 .

[7]  Andrew Vogt,et al.  A Geometric Method for Approximating Convex Arcs , 1980 .

[8]  Andrea L. Bertozzi,et al.  Tracking Environmental Level Sets with Autonomous Vehicles , 2004 .

[9]  Nieuw Archief voor Wiskunde , 1892 .

[10]  Rafael Fierro,et al.  Mobile robotic sensors for perimeter detection and tracking. , 2007, ISA transactions.

[11]  Robert E. Mahony,et al.  Convergence of the Iterates of Descent Methods for Analytic Cost Functions , 2005, SIAM J. Optim..

[12]  Naomi Ehrich Leonard,et al.  Generating contour plots using multiple sensor platforms , 2005, Proceedings 2005 IEEE Swarm Intelligence Symposium, 2005. SIS 2005..

[13]  Jörg M. Wills,et al.  Convexity and its applications , 1983 .

[14]  Hans Wittich.  Über eine Extremalaufgabe der konformen Abbildung , 1949 .

[15]  P. Gruber.  Approximation of convex bodies , 1983 .

[16]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.