Measuring regularity of convex polygons

We propose several measures to evaluate to which extent the shape of a given convex polygon is close to be regular, focusing on a range of characteristics of regularity: optimal ratio area-perimeter, equality of angles and edge lengths, regular fitting, angular and areal symmetry. We prove that our measures satisfy a number of reasonable requirements that guarantee them to be well defined, and provide efficient algorithms for their computation. All these algorithms have been implemented, and we provide experimental results on all the proposed measures.

[1]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[2]  Micha Sharir,et al.  Largest Placement of One Convex Polygon Inside Another , 1998, Discret. Comput. Geom..

[3]  David Shaw,et al.  Regular polygon detection , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[4]  Henk J. A. M. Heijmans,et al.  Similarity and Symmetry Measures for Convex Shapes Using Minkowski Addition , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[5]  Jean Cardinal,et al.  Where to build a temple, and where to dig to find one , 2006 .

[6]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[7]  Laurence Boxer.  Computing deviations from convexity in polygons , 1993, Pattern Recognit. Lett..

[8]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

[9]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[10]  Marco Campani,et al.  Robust method for road sign detection and recognition , 1996, Image Vis. Comput..

[11]  Iosif Pinelis.  A characterization of the convexity of cyclic polygons in terms of the central angles , 2006 .

[12]  Iosif Pinelis,et al.  Cyclic polygons with given edge lengths: Existence and uniqueness , 2005 .

[13]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[14]  Paul L. Rosin,et al.  A new convexity measure for polygons , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.