The Annals of Statistics

Nonparametric estimation of surface integrals

Raúl Jiménez and J. E. Yukich

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The estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics, with powerful applications to physics and image analysis, among other fields. Provided that one can determine whether random shots hit the body, Cuevas et al. [Ann. Statist. 35 (2007) 1031–1051] estimate the boundary measure (the boundary length for planar sets and the surface area for 3-dimensional objects) via the consideration of shots at a box containing the body. The statistics considered by these authors, as well as those in subsequent papers, are based on the estimation of Minkowski content and depend on a smoothing parameter which must be carefully chosen. For the same sampling scheme, we introduce a new approach which bypasses this issue, providing strongly consistent estimators of both the boundary measure and the surface integrals of scalar functions, provided one can collect the function values at the sample points. Examples arise in experiments in which the density of the body can be measured by physical properties of the impacts, or in situations where such quantities as temperature and humidity are observed by randomly distributed sensors. Our method is based on random Delaunay triangulations and involves a simple procedure for surface reconstruction from a dense cloud of points inside and outside the body. We obtain basic asymptotics of the estimator, perform simulations and discuss, via Google Earth’s data, an application to the image analysis of the Aral Sea coast and its cliffs.

Article information

Ann. Statist., Volume 39, Number 1 (2011), 232-260.

First available in Project Euclid: 3 December 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Surface estimation boundary measure Delaunay triangulation stabilization methods


Jiménez, Raúl; Yukich, J. E. Nonparametric estimation of surface integrals. Ann. Statist. 39 (2011), no. 1, 232--260. doi:10.1214/10-AOS837.

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  • [1] Amenta, N., Bern, J. and Eppstein, D. (1998). The Crust and the β-skeleton: Combinatorial curve reconstruction. Graph. Model. Image Process. 60 125–135.
  • [2] Armendáriz, I., Cuevas, A. and Fraiman, R. (2009). Nonparametric estimation of baundary measures and related functional: Asymptotics results. Adv. in Appl. Probab. 41 311–322.
  • [3] Baryshnikov, Y. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 213–253.
  • [4] Baryshnikov, Y., Penrose, M. and Yukich, J. E. (2009). Gaussian limits for generalized spacings. Ann. Appl. Probab. 19 158–185.
  • [5] Boissonnat, J.-D. and Cazals, F. (2000). Natural coordinates of points on a surface. In Proceedings of the 16th Annual ACM Symposium on Computational Geometry 223–232. ACM, New York.
  • [6] Chalker, T. K., Godbole, A. P., Hitczenko, P., Radcliff, J. and Ruehr, O. G. (1999). On the size of a random sphere of influence graph. Adv. in Appl. Probab. 31 596–609.
  • [7] Choi, H. I., Choi, S. W. and Moon, H. P. (1997). Mathematical theory of medial axis transform. Pacific J. Math. 181 57–88.
  • [8] Cuevas, A., Fraiman, R. and Rodríguez-Casal, A. (2007). A nonparametric approach to the estimation of lengths and surface areas. Ann. Statist. 35 1031–1051.
  • [9] Cuevas, A. and Fraiman, R. (2010). Set estimation. In New Perspectives in Stochastic Geometry (I. Molchanov and W. Kendall, eds.) 374–397. Oxford Univ. Press, Oxford.
  • [10] Cuevas, A., Fraiman, R. and Györfi, L. (2010). Towards a universally consistent estimator of the Minkowski content. Preprint.
  • [11] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd ed. Springer, New York.
  • [12] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd ed. Springer, New York.
  • [13] Evans, I. S. (1972). General geomorphometry, derivatives of altitude, and descriptive statistics. In Spatial Analysis in Geomorphology (R. J. Chorley, ed.) 17–90. Mathuen & Co., London.
  • [14] Federer, H. (1969). Geometric Measure Theory. Springer, New York.
  • [15] Kirkpatrick, D. G. and Radke, J. D. (1998). A framework for computational morphology. In Computational Geometry (G. Toussaint ed.) 217–248. North-Holland, Amsterdam.
  • [16] Mattila, P. (1999). Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge Univ. Press, Cambridge.
  • [17] Möller, J. (1994). Lectures on Random Voronoi Tessellations. Lectures Notes in Statistics 87. Springer, New York.
  • [18] Pateiro-López, B. and Rodríguez-Casal, A. (2008). Length and surface area estimation under convexity type restrictions. Adv. in Appl. Probab. 40 348–358.
  • [19] Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Probab. 12 989–1035.
  • [20] Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 1124–1150.
  • [21] Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277–303.
  • [22] Pike, R. J. and Wilson S. E. (1971). Elevation-relief ratio, hypsometric integral, and geomorphic area-altitude analysis. Geol. Soc. Amer. Bull. 82 1079–1084.
  • [23] Schreiber, T. and Yukich, J. E. (2008). Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. Ann. Probab. 36 363–396.
  • [24] Small, C. (1996). The Statistical Theory of Shape. Springer, New York.