Distance measurements on processes of flats

2003 ◽  
Vol 35 (1) ◽  
pp. 70-95 ◽  
Author(s):  
Daniel Hug ◽  
Günter Last ◽  
Wolfgang Weil
Keyword(s):  
Stochastic Geometry ◽  
Boolean Model ◽  
Distribution Functions ◽  
Random Sets ◽  
Stationary Processes ◽  
Boolean Models ◽  
Distance Measurements ◽  
Contact Distribution ◽  
The Individual ◽  
Nearest Points

Distance measurements are useful tools in stochastic geometry. For a Boolean modelZin ℝd, the classical contact distribution functions allow the estimation of important geometric parameters ofZ. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random setsZwhich are generated as the union sets of Poisson processesXofk-flats,k∈ {0, …,d-1}, and study distances from a fixed point or a fixed convex body toZ. In addition, we also consider the distances from a given flat or a flag consisting of flats to the individual members ofXand investigate the associated process of nearest points in the flats ofX. In particular, we discuss to which extent the directional distribution ofXis determined by this point process. Some of our results are presented for more general stationary processes of flats.

Distance measurements on processes of flats

2003 ◽  
Vol 35 (01) ◽  
pp. 70-95 ◽  
Author(s):  
Daniel Hug ◽  
Günter Last ◽  
Wolfgang Weil
Keyword(s):  
Stochastic Geometry ◽  
Boolean Model ◽  
Distribution Functions ◽  
Random Sets ◽  
Stationary Processes ◽  
Boolean Models ◽  
Distance Measurements ◽  
Contact Distribution ◽  
The Individual ◽  
Nearest Points

Distance measurements are useful tools in stochastic geometry. For a Boolean modelZin ℝd, the classical contact distribution functions allow the estimation of important geometric parameters ofZ. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random setsZwhich are generated as the union sets of Poisson processesXofk-flats,k∈ {0, …,d-1}, and study distances from a fixed point or a fixed convex body toZ. In addition, we also consider the distances from a given flat or a flag consisting of flats to the individual members ofXand investigate the associated process of nearest points in the flats ofX. In particular, we discuss to which extent the directional distribution ofXis determined by this point process. Some of our results are presented for more general stationary processes of flats.

scholarly journals ON THE ESTIMATION OF DISTANCE DISTRIBUTION FUNCTIONS FOR POINT PROCESSES AND RANDOM SETS

2011 ◽  
Vol 20 (1) ◽  
pp. 65 ◽  
Author(s):  
Dietrich Stoyan ◽  
Helga Stoyan ◽  
André Tscheschel ◽  
Torsten Mattfeldt
Keyword(s):  
Distribution Function ◽  
Point Processes ◽  
Distribution Functions ◽  
Distance Distribution ◽  
Random Sets ◽  
Boolean Models ◽  
Closed Set ◽  
Stationary Point Process ◽  
Contact Distribution ◽  
Nearest Neighbour Distance Distribution

This paper discusses various estimators for the nearest neighbour distance distribution function D of a stationary point process and for the quadratic contact distribution function Hq of a stationary random closed set. It recommends the use of Hanisch's estimator of D, which is of Horvitz-Thompson type, and the minussampling estimator of Hq. This recommendation is based on simulations for Poisson processes and Boolean models.

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1999 ◽  
Vol 31 (02) ◽  
pp. 283-314 ◽  
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
The Individual ◽  
Individual Grains

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝ d generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

scholarly journals Generalized contact distributions of inhomogeneous Boolean models

2002 ◽  
Vol 34 (01) ◽  
pp. 21-47 ◽  
Author(s):  
Daniel Hug ◽  
Günter Last ◽  
Wolfgang Weil
Keyword(s):  
Convex Body ◽  
Large Class ◽  
Random Vector ◽  
Boundary Point ◽  
Conditional Distribution ◽  
Boolean Model ◽  
Spatial Density ◽  
Boolean Models ◽  
Grain Distribution ◽  
Contact Distribution

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝ d and a gauge body B ⊂ ℝ d , such a generalized contact distribution is the conditional distribution of the random vector (d B (L,Z),u B (L,Z),p B (L,Z),l B (L,Z)) given that Z∩L = ∅, where Z is a Boolean model, d B (L,Z) is the distance of L from Z with respect to B, p B (L,Z) is the boundary point in L realizing this distance (if it exists uniquely), u B (L,Z) is the corresponding boundary point of B (if it exists uniquely) and l B (L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

scholarly journals Local Digital Estimators of Intrinsic Volumes for Boolean Models and in the Design-Based Setting

2014 ◽  
Vol 46 (01) ◽  
pp. 35-58 ◽  
Author(s):  
Anne Marie Svane
Keyword(s):  
High Resolution ◽  
Linear Equations ◽  
Boolean Model ◽  
Random Sets ◽  
Weighted Sums ◽  
Boolean Models ◽  
Explicit Formulas ◽  
Intrinsic Volumes ◽  
Fixed Set ◽  
Digital Algorithms

In order to estimate the specific intrinsic volumes of a planar Boolean model from a binary image, we consider local digital algorithms based on weighted sums of 2×2 configuration counts. For Boolean models with balls as grains, explicit formulas for the bias of such algorithms are derived, resulting in a set of linear equations that the weights must satisfy in order to minimize the bias in high resolution. These results generalize to larger classes of random sets, as well as to the design-based situation, where a fixed set is observed on a stationary isotropic lattice. Finally, the formulas for the bias obtained for Boolean models are applied to existing algorithms in order to compare their accuracy.

scholarly journals A TIME-OPTIMAL ALGORITHM FOR THE ESTIMATION OF CONTACT DISTRIBUTION FUNCTIONS OF RANDOM SETS

2011 ◽  
Vol 23 (3) ◽  
pp. 177 ◽  
Author(s):  
Johannes Mayer
Keyword(s):  
Distribution Function ◽  
Optimal Algorithm ◽  
Linear Time ◽  
Distribution Functions ◽  
Distance Distribution ◽  
Random Sets ◽  
Distance Distribution Function ◽  
Structuring Element ◽  
Time Optimal ◽  
Contact Distribution

This paper presents a linear-time and therefore time-optimal algorithm for the estimation of distance distribution functions and contact distribution functions of random sets. The distance distribution function is the area fraction of a dilated set, where this function depends on the size of the structuring element used for the dilation. Furthermore, contact distribution functions are related to distance distribution functions. Minussampling estimators are used for the estimation.

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1999 ◽  
Vol 31 (2) ◽  
pp. 283-314 ◽  
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
The Individual ◽  
Individual Grains

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝd generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

scholarly journals Generalized contact distributions of inhomogeneous Boolean models

2002 ◽  
Vol 34 (1) ◽  
pp. 21-47 ◽  
Author(s):  
Daniel Hug ◽  
Günter Last ◽  
Wolfgang Weil
Keyword(s):  
Convex Body ◽  
Large Class ◽  
Random Vector ◽  
Boundary Point ◽  
Conditional Distribution ◽  
Boolean Model ◽  
Spatial Density ◽  
Boolean Models ◽  
Grain Distribution ◽  
Contact Distribution

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd, such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z),uB(L,Z),pB(L,Z),lB(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, dB(L,Z) is the distance of L from Z with respect to B, pB(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), uB(L,Z) is the corresponding boundary point of B (if it exists uniquely) and lB(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

scholarly journals The Boolean model in the Shannon regime: three thresholds and related asymptotics

2016 ◽  
Vol 53 (4) ◽  
pp. 1001-1018 ◽  
Author(s):  
Venkat Anantharam ◽  
François Baccelli
Keyword(s):  
Stochastic Geometry ◽  
Volume Fraction ◽  
Boolean Model ◽  
Boolean Models ◽  
Large Deviations Principle ◽  
Percolation Probability ◽  
Whole Space ◽  
Probability Threshold ◽  
The Mean ◽  
Model Features

Abstract Consider a family of Boolean models, indexed by integers n≥1. The nth model features a Poisson point process in ℝn of intensity e{nρn}, and balls of independent and identically distributed radii distributed like X̅n√n. Assume that ρn→ρ as n→∞, and that X̅n satisfies a large deviations principle. We show that there then exist the three deterministic thresholds τd, the degree threshold, τp, the percolation probability threshold, and τv, the volume fraction threshold, such that, asymptotically as n tends to ∞, we have the following features. (i) For ρ<τd, almost every point is isolated, namely its ball intersects no other ball; (ii) for τd<ρ<τp, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless there is no percolation; (iii) for τp<ρ<τv, the volume fraction is 0 and nevertheless percolation occurs; (iv) for τd<ρ<τv, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless the volume fraction is 0; (v) for ρ>τv, the whole space is covered. The analysis of this asymptotic regime is motivated by problems in information theory, but it could be of independent interest in stochastic geometry. The relations between these three thresholds and the Shannon‒Poltyrev threshold are discussed.

Asymptotic geometry of high-density smooth-grained Boolean models in bounded domains

2003 ◽  
Vol 35 (04) ◽  
pp. 913-936
Author(s):  
Tomasz Schreiber
Keyword(s):  
Convex Sets ◽  
Asymptotic Properties ◽  
Boolean Model ◽  
Moderate Deviation ◽  
Random Sets ◽  
Boolean Models ◽  
Support Functions ◽  
Original Process ◽  
Multidimensional Ball ◽  
Asymptotic Geometry

The purpose of the paper is to study the asymptotic geometry of a smooth-grained Boolean model (X [t]) t≥0 restricted to a bounded domain as the intensity parameter t goes to ∞. Our approach is based on investigating the asymptotic properties as t → ∞ of the random sets X [t;β], β≥0, defined as the Gibbsian modifications of X [t] with the Hamiltonian given by βtμ(·), where μ is a certain normalized measure on the setting space. We show that our model exhibits a phase transition at a certain critical value of the inverse temperature β and we prove that at higher temperatures the behaviour of X [t;β] is qualitatively very similar to that of X [t] but it becomes essentially different in the low-temperature region. From these facts we derive information about the asymptotic properties of the original process X [t]. The results obtained include large- and moderate-deviation principles. We conclude the paper with an example application of our methods to analyse the asymptotic moderate-deviation properties of convex hulls of large uniform samples on a multidimensional ball. To translate the above problem to the Boolean model setting considered we use an appropriate representation of convex sets in terms of their support functions.

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