spatstat.geom (version 3.2-5)

deltametric: Delta Metric

Description

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

Usage

deltametric(A, B, p = 2, c = Inf, ...)

Value

A numeric value.

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.

Author

Adrian Baddeley

and Rolf Turner

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.

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

Examples

  X <- runifrect(20)
  Y <- runifrect(10)
  deltametric(X, Y, p=1,c=0.1)