# deltametric

##### Delta Metric

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
$$
\Delta(A,B) = \left[
\frac 1 {|W|}
\int_W
\left| \min(c, d(x,A)) - \min(c, d(x,B)) \right|^p \, {\rm d}x
\right]^{1/p}
$$
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

##### Examples

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

*Documentation reproduced from package spatstat, version 1.63-3, License: GPL (>= 2)*