# rbdd: Simulation of Boolean Model of Deterministic Discs In lacunaritycovariance: Gliding Box Lacunarity and Other Metrics for 2D Random Closed Sets

 rbdd R Documentation

## Simulation of Boolean Model of Deterministic Discs

### Description

Functions for simulating a Boolean model with grains that are discs of fixed constant radius (the abbreviation 'bdd' is short for Boolean model with Deterministic Discs). A Boolean model is a two stage model, first the locations (called germs) of grains are randomly distributed according to a Poisson point process, then a random grain is placed on each germ independently. Introductions to Boolean models are available in many stochastic geometry books (Chiu et al., 2013). Also described here are functions for calculating the coverage probability, germ intensity, and covariance from model parameters for a Boolean model with deterministic discs.

### Usage

```rbdd(lambda, discr, window, seed = NULL)

bddcoverageprob(lambda, discr)

bddlambda(coverp, discr)

bdddiscr(coverp, lambda)

bddcovar.iso(r, lambda, discr)

bddcovar(xrange, yrange, eps, lambda, discr)
```

### Arguments

 `lambda` Intensity of the germ process (which is a Poisson point process) `discr` Radius of the discs `window` The window to simulate in (an `owin` object) `seed` Optional input (default in NULL). Is an integer passed to `set.seed`. Used to reproduce patterns exactly. `coverp` Coverage probability of the Boolean model `r` is the radius to calculate covariance `xrange` range of x values for `bddcovar` `yrange` range of y values for `bddcovar` `eps` list of length 2 of the steps between samples points in x and y respectively for `bddcovar`. If eps is of length 1 then the steps between sample points in the x and y directions will both be equal to eps.

### Value

See Functions section.

### Functions

• `rbdd`: Returns an `owin` that is a set generated by simulating a Boolean model with specified intensity and disc radius. The window information is not contained in this object. If the simulated set is empty then an empty `owin` object is returned. The point process of germs is generated using spatstat's `rpoispp`.

• `bddcoverageprob`: Returns the true coverage probability given the intensity and disc radius.

• `bddlambda`: Returns the germ intensity given coverage probability and disc radius.

• `bdddiscr`: Returns the disc radius given coverage probability and germ intensity.

• `bddcovar.iso`: Returns the true covariance of points separated by a distance `r` given the intensity, `lambda` and disc radius `discr` of the model.

• `bddcovar`: Returns an image of the covariance as calculated from disc radius and intensity.

### WARNING

The returned object of `rbdd` is an `owin` specifying the realisation of the Boolean model within the simulation window. The simulation window is not included, thus the object returned by `rbdd` can have much smaller extent than the simulation window (e.g. when the simulated set is empty).

### References

Chiu, S.N., Stoyan, D., Kendall, W.S. and Mecke, J. (2013) Stochastic Geometry and Its Applications, 3rd ed. Chichester, United Kingdom: John Wiley & Sons.

### Examples

```# Simulate Boolean model with discs of radius 10.
# The coverage probability is very close to 0.5.
discr <- 10
w <- owin(xrange = c(0, 100), c(0, 100))
lambda <- 2.2064E-3
xi <- rbdd(lambda, discr, w)

# Compute properties of Boolean model from parameters
cp <- bddcoverageprob(lambda, discr)
cvc <- bddcovar(c(-10, 10), c(-10, 10), c(0.2, 0.2), lambda, discr)
```

lacunaritycovariance documentation built on March 18, 2022, 5:20 p.m.