laslett

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Laslett's Transform

Apply Laslett's Transform to a spatial region, returning the original and transformed regions, and the original and transformed positions of the lower tangent points. This is a diagnostic for the Boolean model.

Keywords
manip, spatial
Usage
laslett(X, …, verbose = FALSE, plotit = TRUE, discretise = FALSE,
type=c("lower", "upper", "left", "right"))
Arguments
X

Spatial region to be transformed. A window (object of class "owin") or a logical-valued pixel image (object of class "im").

Graphics arguments to control the plot (passed to plot.laslett when plotit=TRUE) or arguments determining the pixel resolution (passed to as.mask).

verbose

Logical value indicating whether to print progress reports.

plotit

Logical value indicating whether to plot the result.

discretise

Logical value indicating whether polygonal windows should first be converted to pixel masks before the Laslett transform is computed. This should be set to TRUE for very complicated polygons.

type

Type of tangent points to be detected. This also determines the direction of contraction in the set transformation. Default is type="lower".

Details

This function finds the lower tangent points of the spatial region X, then applies Laslett's Transform to the space, and records the transformed positions of the lower tangent points.

Laslett's transform is a diagnostic for the Boolean Model. A test of the Boolean model can be performed by applying a test of CSR to the transformed tangent points. See the Examples.

The rationale is that, if the region X was generated by a Boolean model with convex grains, then the lower tangent points of X, when subjected to Laslett's transform, become a Poisson point process (Cressie, 1993, section 9.3.5; Molchanov, 1997; Barbour and Schmidt, 2001).

Intuitively, Laslett's transform is a way to account for the fact that tangent points of X cannot occur inside X. It treats the interior of X as empty space, and collapses this empty space so that only the exterior of X remains. In this collapsed space, the tangent points are completely random.

Formally, Laslett's transform is a random (i.e. data-dependent) spatial transformation which maps each spatial location $(x,y)$ to a new location $(x',y)$ at the same height $y$. The transformation is defined so that $x'$ is the total uncovered length of the line segment from $(0,y)$ to $(x,y)$, that is, the total length of the parts of this segment that fall outside the region X.

In more colourful terms, suppose we use an abacus to display a pixellated version of X. Each wire of the abacus represents one horizontal line in the pixel image. Each pixel lying outside the region X is represented by a bead of the abacus; pixels inside X are represented by the absence of a bead. Next we find any beads which are lower tangent points of X, and paint them green. Then Laslett's Transform is applied by pushing all beads to the left, as far as possible. The final locations of all the beads provide a new spatial region, inside which is the point pattern of tangent points (marked by the green-painted beads).

If plotit=TRUE (the default), a before-and-after plot is generated, showing the region X and the tangent points before and after the transformation. This plot can also be generated by calling plot(a) where a is the object returned by the function laslett.

If the argument type is given, then this determines the type of tangents that will be detected, and also the direction of contraction in Laslett's transform. The computation is performed by first rotating X, applying Laslett's transform for lower tangent points, then rotating back.

There are separate algorithms for polygonal windows and pixellated windows (binary masks). The polygonal algorithm may be slow for very complicated polygons. If this happens, setting discretise=TRUE will convert the polygonal window to a binary mask and invoke the pixel raster algorithm.

Value

A list, which also belongs to the class "laslett" so that it can immediately be printed and plotted.

The list elements are:

oldX:

the original dataset X;

TanOld:

a point pattern, whose window is Frame(X), containing the lower tangent points of X;

TanNew:

a point pattern, whose window is the Laslett transform of Frame(X), and which contains the Laslett-transformed positions of the tangent points;

Rect:

a rectangular window, which is the largest rectangle lying inside the transformed set;

df:

a data frame giving the locations of the tangent points before and after transformation.

type:

character string specifying the type of tangents.

References

Barbour, A.D. and Schmidt, V. (2001) On Laslett's Transform for the Boolean Model. Advances in Applied Probability 33(1), 1--5.

Cressie, N.A.C. (1993) Statistics for spatial data, second edition. John Wiley and Sons.

Molchanov, I. (1997) Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley.

plot.laslett

• laslett
Examples
# NOT RUN {
a <- laslett(heather\$coarse)
with(a, clarkevans.test(TanNew[Rect], correction="D", nsim=39))
X <- discs(runifpoint(15) %mark% 0.2, npoly=16)
b <- laslett(X, type="left")
b
# }
Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)

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