scholarly journals Confidence Sets for the Aumann Mean of a Random Closed Set

2004 ◽  
pp. 298-307
Author(s):  
Raffaello Seri ◽  
Christine Choirat
Keyword(s):  
Confidence Sets ◽  
Closed Set ◽  
Random Closed Set

Limit theorems for convex hulls of random sets

1993 ◽  
Vol 25 (02) ◽  
pp. 395-414 ◽  
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Convex Hulls ◽  
Stable Sets ◽  
Limiting Distributions ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set ◽  
Simple Limit

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn , and independent copies A 1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

scholarly journals Spectral theory for random closed sets and estimating the covariance via frequency space

2003 ◽  
Vol 35 (03) ◽  
pp. 603-613 ◽  
Author(s):  
Karsten Koch ◽  
Joachim Ohser ◽  
Katja Schladitz
Keyword(s):  
Power Spectrum ◽  
Spectral Theory ◽  
Real World ◽  
Second Order ◽  
Mathematical Basis ◽  
Closed Set ◽  
Frequency Space ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. The definition and a proof of the existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khinchin theorem for the power spectrum are used to two ends. First, well-known second-order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second-order characteristics in frequency space. Examples show that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real-world counterparts.

scholarly journals ON THE SPECIFIC AREA OF INHOMOGENEOUS BOOLEAN MODELS. EXISTENCE RESULTS AND APPLICATIONS

2011 ◽  
Vol 29 (2) ◽  
pp. 111 ◽  
Author(s):  
Elena Villa
Keyword(s):  
Measure Theory ◽  
Geometric Measure Theory ◽  
Specific Area ◽  
Boolean Models ◽  
Closed Set ◽  
Geometric Measure ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set ◽  
The Mean

The problem of the evaluation of the so-called specific area of a random closed set, in connection with its mean boundary measure, is mentioned in the classical book by Matheron on random closed sets (Matheron, 1975, p. 50); it is still an open problem, in general. We offer here an overview of some recent results concerning the existence of the specific area of inhomogeneous Boolean models, unifying results from geometric measure theory and from stochastic geometry. A discussion of possible applications to image analysis concerning the estimation of the mean surface density of random closed sets, and, in particular, to material science concerning birth-and-growth processes, is also provided.

scholarly journals 3D IMAGE ANALYSIS OF OPEN FOAMS USING RANDOM TESSELLATIONS

2011 ◽  
Vol 25 (2) ◽  
pp. 87 ◽  
Author(s):  
Claudia Lautensack ◽  
Tetyana Sych
Keyword(s):  
Image Analysis ◽  
Real Data ◽  
Suitable Model ◽  
Deterministic Models ◽  
Closed Set ◽  
Mean Values ◽  
Random Closed Set ◽  
Random Tessellations ◽  
Mean Size

Volume image analysis provides a number of methods for the characterization of the microstructure of open foams. Mean values of characteristics of the edge system are measured directly from the volume image. Further characteristics like the intensity and mean size of the cells are obtained using model assumptions where the edge system of the foam is interpreted as a realization of a random closed set. Macroscopically homogeneous random tessellations provide a suitable model for foam structures. However, their cells often lack the degree of regularity observed in real data. In this respect some deterministic models seem to be closer to realistic structures, although they do not capture the microscopic heterogeneity of real foams. In this paper, the influence of the model choice on the obtained mean values is studied. Moreover, a method for reconstruction of the cells of an open foam from its edge system is described and tested for the tessellations under consideration.

Surprising optimal estimators for the area fraction

1999 ◽  
Vol 31 (4) ◽  
pp. 995-1001 ◽  
Author(s):  
Katja Schladitz
Keyword(s):  
Area Fraction ◽  
Area Measurement ◽  
Point Counting ◽  
Observation Window ◽  
Closed Set ◽  
Minimal Variance ◽  
Random Closed Set ◽  
The Mean ◽  
Optimal Estimators ◽  
Better Than

For a random closed set X and a compact observation window W the mean coverage fraction of W can be estimated by measuring the area of W covered by X. Jensen and Gundersen, and Baddeley and Cruz-Orive described cases where a point counting estimator is more efficient than area measurement. We give two other examples, where at first glance unnatural estimators are not only better than the area measurement but by Grenander's Theorem have minimal variance. Whittle's Theorem is used to show that the point counting estimator in the original Jensen-Gundersen paradox is optimal for large randomly translated discs.

Surprising optimal estimators for the area fraction

1999 ◽  
Vol 31 (04) ◽  
pp. 995-1001
Author(s):  
Katja Schladitz
Keyword(s):  
Area Fraction ◽  
Area Measurement ◽  
Point Counting ◽  
Observation Window ◽  
Closed Set ◽  
Minimal Variance ◽  
Random Closed Set ◽  
The Mean ◽  
Optimal Estimators ◽  
Better Than

For a random closed set X and a compact observation window W the mean coverage fraction of W can be estimated by measuring the area of W covered by X. Jensen and Gundersen, and Baddeley and Cruz-Orive described cases where a point counting estimator is more efficient than area measurement. We give two other examples, where at first glance unnatural estimators are not only better than the area measurement but by Grenander's Theorem have minimal variance. Whittle's Theorem is used to show that the point counting estimator in the original Jensen-Gundersen paradox is optimal for large randomly translated discs.

scholarly journals Spectral theory for random closed sets and estimating the covariance via frequency space

2003 ◽  
Vol 35 (3) ◽  
pp. 603-613 ◽  
Author(s):  
Karsten Koch ◽  
Joachim Ohser ◽  
Katja Schladitz
Keyword(s):  
Power Spectrum ◽  
Spectral Theory ◽  
Real World ◽  
Second Order ◽  
Mathematical Basis ◽  
Closed Set ◽  
Frequency Space ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. The definition and a proof of the existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khinchin theorem for the power spectrum are used to two ends. First, well-known second-order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second-order characteristics in frequency space. Examples show that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real-world counterparts.

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