No Access

Compensated convexity and Hausdorff stable extraction of intersections for smooth manifolds

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Instituto de Estructuras and Departamento de Mecánica, Universidad Nacional de Tucumán, Av. Independencia 1800, San Miguel de Tucumán, Argentina

Search for more papers by this author

, and

Elaine Crooks

Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK

Search for more papers by this author

https://doi.org/10.1142/S0218202515500207Cited by:7 (Source: Crossref)

Abstract

We apply compensated convex transforms to define a multiscale Hausdorff stable method to extract intersections between smooth compact manifolds represented by their characteristic functions or as point clouds embedded in ℝⁿ. We prove extraction results on intersections of smooth compact manifolds and for points of high curvature. As a result of the Hausdorff–Lipschitz continuity of our transforms, we show that our method is stable against dense sampling of smooth manifolds with noise. Examples of explicitly calculated prototype models for some simple cases are presented, which are also used in the proofs of our main results. Numerical experiments in two- and three-dimensional space, and applications to geometric objects are also shown.

Keywords:

AMSC: 52A41, 49J52, 62H35, 14J17, 58K25, 53A05, 65D17

References

L. Ambrosio and P. Tilli , Topics on Analysis in Metric Spaces ( Oxford Univ. Press , 2004 ) . Google Scholar
H. Attouch and D. Aze , Ann. Inst. H. Poincaré Anal. Non Linéaire 10 , 289 ( 1993 ) . Crossref, Web of Science, Google Scholar
H. Busemann , The Geometry of Geodesics ( Academic Press , 1955 ) . Google Scholar
P. Cannarsa and C. Sinestrari , Semiconcave Functions, Hamilton–Jacobi Equations and Optimal Control ( Birkhäuser , 2004 ) . Crossref, Google Scholar
J. Cao et al. , IET Image Process. 6 , 863 ( 2012 ) . Crossref, Web of Science, Google Scholar
J. Cheeger and D. G. Ebin , Comparison Theorems in Riemannian Geometry ( North-Holland , 1975 ) . Google Scholar
M. G. Crandall , H. Ishii and P.-L. Lions , Bull. Amer. Math. Soc. 27 , 1 ( 1992 ) . Crossref, Web of Science, Google Scholar
K. Demarsin et al. , Comput.-Aided Des. 39 , 276 ( 2007 ) . Crossref, Web of Science, Google Scholar
J.-B. Hiriart-Urruty and C. Lemaréchal , Fundamentals of Convex Analysis ( Springer , 2001 ) . Crossref, Google Scholar
P. T. Jackway, Morphological scale-space, IAPR Int. Conf. Pattern Recognition (IEEE Computer Society Press, 1992) pp. 252–255. Google Scholar
P. T. Jackway and M. Deriche , IEEE Trans. Pattern Anal. Mach. Intell. 18 , 38 ( 1996 ) . Crossref, Web of Science, Google Scholar
S. Kobayashi and K. Nomizu , Foundations of Differential Geometry II ( Interscience Wiley and Sons , 1969 ) . Google Scholar
J. M. Lasry and P. L. Lions , Israel Math. J. 55 , 257 ( 1986 ) . Crossref, Web of Science, Google Scholar
Y. Lucet , SIAM J. Optim. 20 , 216 ( 2009 ) . Crossref, Web of Science, Google Scholar
J.-J. Moreau , Bull. Soc. Math. France 93 , 273 ( 1965 ) . Crossref, Google Scholar
J.-J. Moreau , Fonctionnelles Convexes ( Collége de France , 1966 ) . Google Scholar
N. M. Patrikalakis and T. Maekawa , Shape Interrogation for Computer-Aided Design and Manufacturing ( Springer , 2002 ) . Crossref, Google Scholar
R. T. Rockafellar , Convex Analysis ( Princeton Univ. Press , 1966 ) . Google Scholar
J. Serra , Image Analysis and Mathematical Morphology ( Academic Press , 1982 ) . Google Scholar
J. Serra and L. Vincent , Circuits Systems Signal Process. 11 , 47 ( 1992 ) . Crossref, Web of Science, Google Scholar
P. Soille , Morphological Image Analysis , 2nd edn. ( Springer , 2004 ) . Crossref, Google Scholar
R. Van Den Boomgaard, Mathematical morphology: Extensions towards computer vision, Ph.D. Thesis, University of Amsterdam (1992) . Google Scholar
R. Van den Boomgaard and A. W. M. Smelders, The morphological structure of images, IAPR Int. Conf. on Pattern Recognition (IEEE Computer Society Press, 1992) pp. 268–271. Google Scholar
C. Weber, S. Hahmann and H. Hagen, Sharp feature detection in point clouds, IEEE Shape Modeling Int. Conf. (SMI,) (IEEE Computer Society, Aix-en-Provence, France, 2010) pp. 175–186. Google Scholar
K. Zhang , Ann. Inst. H. Poincaré Anal. Non Linéaire 25 , 743 ( 2008 ) . Crossref, Web of Science, Google Scholar
K. Zhang , J. Nonlinear Convex Anal. 9 , 379 ( 2008 ) . Web of Science, Google Scholar
K. Zhang, E. C. M Crooks and A. Orlando, Compensated convexity transforms and numerical algorithms, in preparation . Google Scholar
K. Zhang, A Orlando and E. C. M. Crooks, Compensated convexity and Hausdorff stable geometric singularity extraction, to appear in Math. Models Methods Appl. Sci., DOI:10.1142/S0218202515500189 . Google Scholar