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.

Value

  • 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.

See Also

distmap

Aliases
  • deltametric
Examples
X <- runifpoint(20)
  Y <- runifpoint(10)
  deltametric(X, Y, p=1,c=0.1)
Documentation reproduced from package spatstat, version 1.21-0, License: GPL (>= 2)

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