On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane

Tomáš Mrkvička; Jan Rataj

Kybernetika (2009)

  • Volume: 45, Issue: 6, page 931-945
  • ISSN: 0023-5954

Abstract

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A method of estimation of intrinsic volume densities for stationary random closed sets in ℝ d based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the discrete approximation, which differs from the standard techniques used for measuring parallel sets in image analysis. A method of reducing the bias is proposed and tested on simulated data.

How to cite

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Mrkvička, Tomáš, and Rataj, Jan. "On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane." Kybernetika 45.6 (2009): 931-945. <http://eudml.org/doc/37686>.

@article{Mrkvička2009,
abstract = {A method of estimation of intrinsic volume densities for stationary random closed sets in $\mathbb \{R\}^d$ based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the discrete approximation, which differs from the standard techniques used for measuring parallel sets in image analysis. A method of reducing the bias is proposed and tested on simulated data.},
author = {Mrkvička, Tomáš, Rataj, Jan},
journal = {Kybernetika},
keywords = {random closed set; convex ring; curvature measure; intrinsic volume; random closed set; convex ring; curvature measure; intrinsic volume},
language = {eng},
number = {6},
pages = {931-945},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane},
url = {http://eudml.org/doc/37686},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Mrkvička, Tomáš
AU - Rataj, Jan
TI - On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane
JO - Kybernetika
PY - 2009
PB - Institute of Information Theory and Automation AS CR
VL - 45
IS - 6
SP - 931
EP - 945
AB - A method of estimation of intrinsic volume densities for stationary random closed sets in $\mathbb {R}^d$ based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the discrete approximation, which differs from the standard techniques used for measuring parallel sets in image analysis. A method of reducing the bias is proposed and tested on simulated data.
LA - eng
KW - random closed set; convex ring; curvature measure; intrinsic volume; random closed set; convex ring; curvature measure; intrinsic volume
UR - http://eudml.org/doc/37686
ER -

References

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  1. R. Klette, A. Rosenfeld, Digital Geometry, Elsevier, New York 2004. (2004) Zbl1064.68090MR2095127
  2. T. Mrkvička, J. Rataj, 10.1016/j.spa.2007.04.004, Stoch. Proc. Appl. 118 (2008), 2, 213–231. (2008) MR2376900DOI10.1016/j.spa.2007.04.004
  3. T. Mrkvička, Estimation of intrinsic volume via parallel sets in plane and space, Inzynieria Materialowa 4 (2008), 392–395. (2008) 
  4. X.-X. Nguyen, H. Zessin, 10.1007/BF01886869, Z. Wahrsch. Verw. Gebiete 48 (1979), 133–158 (1979) Zbl0397.60080MR0534841DOI10.1007/BF01886869
  5. J. Ohser, F. Mücklich, Statistical Analysis of Microstructures in Materials Science, Wiley, Chichester 2000. (2000) 
  6. J. Rataj, Estimation of intrinsic volumes from parallel neighbourhoods, Rend. Circ. Mat. Palermo, Ser. II, Suppl. 77 (2006), 553–563. (2006) Zbl1101.62084MR2245722
  7. V. Schmidt, E. Spodarev, 10.1016/j.spa.2004.12.007, Stoch. Proc. Appl. 115 (2005), 959–981. (2005) Zbl1075.60006MR2138810DOI10.1016/j.spa.2004.12.007
  8. R. Schneider, W. Weil, Stochastische Geometrie, Teubner, Stuttgart 2000. (2000) Zbl0964.52009MR1794753

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