Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On estimating the perimeter using the alpha-shape

Ery Arias-Castro and Alberto Rodríguez-Casal

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Abstract

We consider the problem of estimating the perimeter of a smooth domain in the plane based on a sample from the uniform distribution over the domain. We study the performance of the estimator defined as the perimeter of the alpha-shape of the sample. Some numerical experiments corroborate our theoretical findings.

Résumé

Nous considérons le problème de l’estimation du périmètre d’un domaine à bord lisse dans le plan basé sur un échantillon tiré de la loi uniforme ayant pour support le domaine en question. Nous étudions la performance de l’estimateur défini par le périmètre de la forme-alpha (« alpha-shape ») de l’échantillon. Des expériences numériques confirment notre théorie.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1051-1068.

Dates
Received: 19 June 2015
Revised: 31 January 2016
Accepted: 13 February 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624030

Digital Object Identifier
doi:10.1214/16-AIHP747

Mathematical Reviews number (MathSciNet)
MR3689960

Zentralblatt MATH identifier
06775431

Subjects
Primary: 62G99: None of the above, but in this section 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Perimeter estimation $\alpha$-shape $r$-convex hull Rolling condition Sets with positive reach

Citation

Arias-Castro, Ery; Rodríguez-Casal, Alberto. On estimating the perimeter using the alpha-shape. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1051--1068. doi:10.1214/16-AIHP747. https://projecteuclid.org/euclid.aihp/1500624030


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References

  • [1] L. Ambrosio, A. Colesanti and E. Villa. Outer Minkowski content for some classes of closed sets. Math. Ann. 342 (4) (2008) 727–748.
  • [2] G. Biau, B. Cadre and B. Pelletier. A graph-based estimator of the number of clusters. ESAIM Probab. Stat. 11 (2007) 272–280.
  • [3] H. Bräker and T. Hsing. On the area and perimeter of a random convex hull in a bounded convex set. Probab. Theory Related Fields 111 (4) (1998) 517–550.
  • [4] B. Cadre. Kernel estimation of density level sets. J. Multivariate Anal. 97 (4) (2006) 999–1023.
  • [5] G. Carlsson. Topology and data. Bull. Amer. Math. Soc. (N.S.) 46 (2) (2009) 255–308.
  • [6] F. Chazal and A. Lieutier. Weak feature size and persistant homology: Computing homology of solids in $\mathbb{R}^{n}$ from noisy data samples. In Computational Geometry (SCG’05) 255–262. ACM, New York, 2005.
  • [7] A. Cuevas, R. Fraiman and A. Rodríguez-Casal. A nonparametric approach to the estimation of lengths and surface areas. Ann. Statist. 35 (3) (2007) 1031–1051.
  • [8] A. Cuevas and R. Fraiman. Set estimation. In New Perspectives in Stochastic Geometry 374–397. Oxford Univ. Press, Oxford, 2010.
  • [9] A. Cuevas, R. Fraiman and B. Pateiro-López. On statistical properties of sets fulfilling rolling-type conditions. Adv. in Appl. Probab. 44 (2) (2012) 311–329.
  • [10] H. Edelsbrunner. Alpha shapes-a survey. In Tessellations in the Sciences, 2010.
  • [11] H. Edelsbrunner, D. G. Kirkpatrick and R. Seidel. On the shape of a set of points in the plane. IEEE Trans. Inform. Theory 29 (4) (1983) 551–559.
  • [12] H. Federer. Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418–491.
  • [13] R. Jiménez and J. E. Yukich. Nonparametric estimation of surface integrals. Ann. Statist. 39 (1) (2011) 232–260.
  • [14] J.-C. Kim and A. Korostelëv. Estimation of smooth functionals in image models. Math. Methods Statist. 9 (2) (2000) 140–159.
  • [15] A. P. Korostelëv and A. B. Tsybakov. Minimax Theory of Image Reconstruction. Lecture Notes in Statistics 82. Springer-Verlag, New York, 1993.
  • [16] J. M. Lee. Introduction to Topological Manifolds, 2nd edition. Graduate Texts in Mathematics 202. Springer, New York, 2011.
  • [17] E. Levina and P. Bickel. Maximum likelihood estimation of intrinsic dimension. In Advances in Neural Information Processing Systems 17 777–784. MIT Press, Cambridge, MA, 2005.
  • [18] E. Mammen and A. B. Tsybakov. Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 (2) (1995) 502–524.
  • [19] J.-M. Morvan. Generalized Curvatures. Springer, Berlin, 2008.
  • [20] P. Niyogi, S. Smale and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39 (1–3) (2008) 419–441.
  • [21] B. Pateiro-Lopez. Set estimation under convexity type restrictions. Ph.D. thesis, Universidad de Santiago de Compostela, 2008.
  • [22] B. Pateiro-López and A. Rodríguez-Casal. Length and surface area estimation under smoothness restrictions. Adv. in Appl. Probab. 40 (2) (2008) 348–358.
  • [23] B. Pateiro-López and A. Rodríguez-Casal. Surface area estimation under convexity type assumptions. J. Nonparametr. Stat. 21 (6) (2009) 729–741.
  • [24] B. Pateiro-López and A. Rodrıguez-Casal. Generalizing the convex hull of a sample: The R package alphahull. J. Stat. Softw. 34 (5) (2010) 1–28.
  • [25] B. Pateiro-López and A. Rodríguez-Casal. Recovering the shape of a point cloud in the plane. TEST 22 (1) (2013) 19–45.
  • [26] J. Perkal. Sur les ensembles $\varepsilon$-convexes. Colloq. Math. 4 (1956) 1–10.
  • [27] W. Polonik. Measuring mass concentrations and estimating density contour clusters – an excess mass approach. Ann. Statist. 23 (3) (1995) 855–881.
  • [28] M. Reitzner. Random polytopes. In New Perspectives in Stochastic Geometry 45–76. Oxford Univ. Press, Oxford, 2010.
  • [29] A. Rényi and R. Sulanke. Über die konvexe Hülle von $n$ zufällig gewählten Punkten. II. Z. Wahrsch. Verw. Gebiete 3 (1964) 138–147.
  • [30] V. Robins. Towards computing homology from finite approximations. In Proceedings of the 14th Summer Conference on General Topology and Its Applications (Brookville, NY, 1999) 503–532, Topology Proc. 24, 1999.
  • [31] A. Rodríguez Casal. Set estimation under convexity type assumptions. Ann. Inst. Henri Poincaré Probab. Stat. 43 (6) (2007) 763–774.
  • [32] A. Rodríguez-Casal and P. Saavedra-Nieves. A fully data-driven method for estimating the shape of a point cloud, 2014. Available at arXiv:1404.7397.
  • [33] A. Singh, C. Scott and R. Nowak. Adaptive Hausdorff estimation of density level sets. Ann. Statist. 37 (5B) (2009) 2760–2782.
  • [34] A. B. Tsybakov. On nonparametric estimation of density level sets. Ann. Statist. 25 (3) (1997) 948–969.
  • [35] G. Walther. Granulometric smoothing. Ann. Statist. 25 (6) (1997) 2273–2299.
  • [36] G. Walther. On a generalization of Blaschke’s rolling theorem and the smoothing of surfaces. Math. Methods Appl. Sci. 22 (4) (1999) 301–316.
  • [37] A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom. 33 (2) (2005) 249–274.