Universal Natural Shapes: From Unifying Shape Description to Simple Methods for Shape Analysis and Boundary Value Problems

Johan Gielis (1, 2, 3) , Diego Caratelli (4) , Yohan Fougerolle (5) , Paolo Emilio Ricci (6) , Ilia Tavkelidze (7) , T. Gerats (8)

1 genicap
2 Institute for Wetland and Water Research
3 Department Biosciences Engineering
4 International Research Centre for Telecommunications and Radar
5 Le2i - Laboratoire Electronique, Informatique et Image [UMR6306]
6 Campus Bio-Medico University, Rome, Italy
7 I. Vekua Institute of Applied Mathematics
8 Institute for Wetland and Water Research

Gielis curves and surfaces can describe a wide range of natural shapes and they have been used in various studies in biology and physics as descriptive tool. This has stimulated the generalization of widely used computational methods. Here we show that proper normalization of the Levenberg-Marquardt algorithm allows for efficient and robust reconstruction of Gielis curves, including self-intersecting and asymmetric curves, without increasing the overall complexity of the algorithm. Then, we show how complex curves of k-type can be constructed and how solutions to the Dirichlet problem for the Laplace equation on these complex domains can be derived using a semi-Fourier method. In all three methods, descriptive and computational power and efficiency is obtained in a surprisingly simple way.

Mots clés: Bounday Value Problem; Gielis Curves; Laplace equation

Domaines: Vision par ordinateur et reconnaissance de formes [cs.CV]; Equations aux dérivées partielles [math.AP]

Johan Gielis, Diego Caratelli, Yohan Fougerolle, Paolo Emilio Ricci, Ilia Tavkelidze, et al.. Universal Natural Shapes: From Unifying Shape Description to Simple Methods for Shape Analysis and Boundary Value Problems. PLoS ONE, 2012, 7 (9), pp.e29324. ⟨10.1371/journal.pone.0029324⟩. ⟨hal-00736577⟩