R: Delta Metric

deltametric {spatstat.geom} | R Documentation

## 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, ...)

### 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

Δ(A,B) = [ (1/|W|) * integral of |min(c, d(x,A))-min(c, d(x,B))|^p dx ]^(1/p)

where c ≥ 0 is a distance threshold parameter, 0 < p ≤ Inf 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.

- Δ(A,B)=0 if and only if A=B

- Δ(A,B)=Δ(B,A)

- Δ(A,C) ≤ Δ(A,B) + Δ(B,C)

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

If p=Inf and c=Inf 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.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

### 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)

[Package spatstat.geom version 2.3-2 Index]