# deltametric

0th

Percentile

##### Delta Metric

Computes the discrepancy between two sets $$A$$ and $$B$$ according to Baddeley's delta-metric.

Keywords
spatial, math
##### Usage
deltametric(A, B, p = 2, c = Inf, ...)
##### Arguments
A,B

The two sets which will be compared. Windows (objects of class "owin"), point patterns (objects of class "ppp") or line segment patterns (objects of class "psp").

p

Index of the $$L^p$$ metric. Either a positive numeric value, or Inf.

c

Distance threshold. Either a positive numeric value, or Inf.

Arguments passed to as.mask to determine the pixel resolution of the distance maps computed by distmap.

##### Details

Baddeley (1992a, 1992b) defined a distance between two sets $$A$$ and $$B$$ contained in a space $$W$$ by $$\Delta(A,B) = \left[ \frac 1 {|W|} \int_W \left| \min(c, d(x,A)) - \min(c, d(x,B)) \right|^p \, {\rm d}x \right]^{1/p}$$ where $$c \ge 0$$ is a distance threshold parameter, $$0 < p \le \infty$$ is the exponent parameter, and $$d(x,A)$$ denotes the shortest distance from a point $$x$$ to the set $$A$$. Also |W| denotes the area or volume of the containing space $$W$$.

This is defined so that it is a metric, i.e.

• $$\Delta(A,B)=0$$ if and only if $$A=B$$

• $$\Delta(A,B)=\Delta(B,A)$$

• $$\Delta(A,C) \le \Delta(A,B) + \Delta(B,C)$$

It is topologically equivalent to the Hausdorff metric (Baddeley, 1992a) but has better stability properties in practical applications (Baddeley, 1992b).

If $$p=\infty$$ and $$c=\infty$$ the Delta metric is equal to the Hausdorff metric.

The algorithm uses distmap to compute the distance maps $$d(x,A)$$ and $$d(x,B)$$, then approximates the integral numerically. The accuracy of the computation depends on the pixel resolution which is controlled through the extra arguments … passed to as.mask.

A numeric value.

##### References

Baddeley, A.J. (1992a) Errors in binary images and an $$L^p$$ version of the Hausdorff metric. Nieuw Archief voor Wiskunde 10, 157--183.

Baddeley, A.J. (1992b) An error metric for binary images. In W. Foerstner and S. Ruwiedel (eds) Robust Computer Vision. Karlsruhe: Wichmann. Pages 59--78.

distmap

• deltametric
##### Examples
# NOT RUN {
X <- runifpoint(20)
Y <- runifpoint(10)
deltametric(X, Y, p=1,c=0.1)
# }

Documentation reproduced from package spatstat, version 1.49-0, License: GPL (>= 2)

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