| Title:
|
On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane (English) |
| Author:
|
Mrkvička, Tomáš |
| Author:
|
Rataj, Jan |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
45 |
| Issue:
|
6 |
| Year:
|
2009 |
| Pages:
|
931-945 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
random closed set |
| Keyword:
|
convex ring |
| Keyword:
|
curvature measure |
| Keyword:
|
intrinsic volume |
| MSC:
|
60D05 |
| MSC:
|
62G05 |
| MSC:
|
62G07 |
| MSC:
|
62G20 |
| MSC:
|
65C60 |
| idZBL:
|
Zbl 1186.62050 |
| idMR:
|
MR2650074 |
| . |
| Date available:
|
2010-06-02T19:25:09Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140028 |
| . |
| Reference:
|
[1] R. Klette, A. Rosenfeld: Digital Geometry.Elsevier, New York 2004. Zbl 1064.68090, MR 2095127 |
| Reference:
|
[2] T. Mrkvička, J. Rataj: On estimation of intrinsic volume densities of stationary random closed sets.Stoch. Proc. Appl. 118 (2008), 2, 213–231. MR 2376900, 10.1016/j.spa.2007.04.004 |
| Reference:
|
[3] T. Mrkvička: Estimation of intrinsic volume via parallel sets in plane and space.Inzynieria Materialowa 4 (2008), 392–395. |
| Reference:
|
[4] X.-X. Nguyen, H. Zessin: Ergodic theorems for spatial processes.Z. Wahrsch. Verw. Gebiete 48 (1979), 133–158 Zbl 0397.60080, MR 0534841, 10.1007/BF01886869 |
| Reference:
|
[5] J. Ohser, F. Mücklich: Statistical Analysis of Microstructures in Materials Science.Wiley, Chichester 2000. |
| Reference:
|
[6] J. Rataj: Estimation of intrinsic volumes from parallel neighbourhoods.Rend. Circ. Mat. Palermo, Ser. II, Suppl. 77 (2006), 553–563. Zbl 1101.62084, MR 2245722 |
| Reference:
|
[7] V. Schmidt, E. Spodarev: Joint estimators for the specific intrinsic volumes of stationary random sets.Stoch. Proc. Appl. 115 (2005), 959–981. Zbl 1075.60006, MR 2138810, 10.1016/j.spa.2004.12.007 |
| Reference:
|
[8] R. Schneider, W. Weil: Stochastische Geometrie.Teubner, Stuttgart 2000. Zbl 0964.52009, MR 1794753 |
| . |
