# 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$A=B$
- $\Delta(A,B)=\Delta(B,A)$
- $\Delta(A,C) \le \Delta(A,B) + \Delta(B,C)$

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

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

*Documentation reproduced from package spatstat, version 1.19-2, License: GPL (>= 2)*