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 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.

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.

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.

On projected thick sections of stationary point processes and Boolean models

1996 ◽  
Vol 28 (2) ◽  
pp. 335-335
Author(s):  
Markus Kiderlen
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Linear Subspace ◽  
Isotropic Case ◽  
Thick Section ◽  
Boolean Models ◽  
Dimensional Linear Subspace ◽  
Stationary Point Process ◽  
Stationary Point Processes

For a stationary point process X of convex particles in ℝd the projected thick section process X(L) on a q-dimensional linear subspace L is considered. Formulae connecting geometric functionals, e.g. the quermass densities of X and X(L), are presented. They generalize the classical results of Miles (1976) and Davy (1976) which hold only in the isotropic case.

scholarly journals PARAMETER ESTIMATION IN NON-HOMOGENEOUS BOOLEAN MODELS: AN APPLICATION TO PLANT DEFENSE RESPONSE

2014 ◽  
Vol 33 (2) ◽  
pp. 27
Author(s):  
Maria Angeles Gallego ◽  
Maria Victoria Ibanez ◽  
Amelia Simó
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Plant Immunity ◽  
Simulated Data ◽  
Region Of Interest ◽  
R Package ◽  
Boolean Model ◽  
Boolean Models ◽  
Data Set ◽  
Stationary Point Process

Many medical and biological problems require to extract information from microscopical images. Boolean models have been extensively used to analyze binary images of random clumps in many scientific fields. In this paper, a particular type of Boolean model with an underlying non-stationary point process is considered. The intensity of the underlying point process is formulated as a fixed function of the distance to a region of interest. A method to estimate the parameters of this Boolean model is introduced, and its performance is checked in two different settings. Firstly, a comparative study with other existent methods is done using simulated data. Secondly, the method is applied to analyze the longleaf data set, which is a very popular data set in the context of point processes included in the R package spatstat. Obtained results show that the new method provides as accurate estimates as those obtained with more complex methods developed for the general case. Finally, to illustrate the application of this model and this method, a particular type of phytopathological images are analyzed. These images show callose depositions in leaves of Arabidopsis plants. The analysis of callose depositions, is very popular in the phytopathological literature to quantify activity of plant immunity.

scholarly journals COMPUTATION OF THE PERIMETER OF MEASURABLE SETS VIA THEIR COVARIOGRAM. APPLICATIONS TO RANDOM SETS

2011 ◽  
Vol 30 (1) ◽  
pp. 39 ◽  
Author(s):  
Bruno Galerne
Keyword(s):  
Unit Volume ◽  
Lipschitz Constant ◽  
Directional Derivative ◽  
Random Sets ◽  
Directional Derivatives ◽  
Boolean Models ◽  
Closed Set ◽  
Finite Perimeter ◽  
Measurable Set ◽  
Grain Distribution

The covariogram of a measurable set A ⊂ Rd is the function gA which to each y ∈ Rd associates the Lebesgue measure of A ∩ (y + A). This paper proves two formulas. The first equates the directional derivatives at the origin of gA to the directional variations of A. The second equates the average directional derivative at the origin of gA to the perimeter of A. These formulas, previously known with restrictions, are proved for any measurable set. As a by-product, it is proved that the covariogram of a set A is Lipschitz if and only if A has finite perimeter, the Lipschitz constant being half the maximal directional variation. The two formulas have counterparts for mean covariogram of random sets. They also permit to compute the expected perimeter per unit volume of any stationary random closed set. As an illustration, the expected perimeter per unit volume of stationary Boolean models having any grain distribution is computed.

On projected thick sections of stationary point processes and Boolean models

1996 ◽  
Vol 28 (02) ◽  
pp. 335
Author(s):  
Markus Kiderlen
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Linear Subspace ◽  
Isotropic Case ◽  
Thick Section ◽  
Boolean Models ◽  
Dimensional Linear Subspace ◽  
Stationary Point Process ◽  
Stationary Point Processes

For a stationary point process X of convex particles in ℝ d the projected thick section process X(L) on a q-dimensional linear subspace L is considered. Formulae connecting geometric functionals, e.g. the quermass densities of X and X(L), are presented. They generalize the classical results of Miles (1976) and Davy (1976) which hold only in the isotropic case.

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