\name{deltametric}
\Rdversion{1.1}
\alias{deltametric}
\title{
Delta Metric
}
\description{
Computes the discrepancy between two sets \eqn{A} and \eqn{B}
according to Baddeley's delta-metric.
}
\usage{
deltametric(A, B, p = 2, c = Inf, ...)
}
\arguments{
\item{A,B}{
The two sets which will be compared.
Windows (objects of class \code{"owin"}),
point patterns (objects of class \code{"ppp"})
or line segment patterns (objects of class \code{"psp"}).
}
\item{p}{
Index of the \eqn{L^p} metric.
Either a positive numeric value, or \code{Inf}.
}
\item{c}{
Distance threshold.
Either a positive numeric value, or \code{Inf}.
}
\item{\dots}{
Arguments passed to \code{\link{as.mask}} to determine the
pixel resolution of the distance maps computed by \code{\link{distmap}}.
}
}
\details{
Baddeley (1992a, 1992b) defined a distance
between two sets \eqn{A} and \eqn{B} contained in a space \eqn{W} by
\deqn{
\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}
}{
Delta(A,B) = [ (1/|W|) * integral of |min(c, d(x,A))-min(c, d(x,B))|^p dx ]^(1/p)
}
where \eqn{c \ge 0}{c >= 0} is a distance threshold parameter,
\eqn{0 < p \le \infty}{0 < p <= Inf} is the exponent parameter,
and \eqn{d(x,A)} denotes the
shortest distance from a point \eqn{x} to the set \eqn{A}.
Also \code{|W|} denotes the area or volume of the containing space \eqn{W}.
This is defined so that it is a \emph{metric}, i.e.
\itemize{
\item \eqn{\Delta(A,B)=0}{Delta(A,B)=0} if and only if \eqn{A=B}
\item \eqn{\Delta(A,B)=\Delta(B,A)}{Delta(A,B)=Delta(B,A)}
\item \eqn{\Delta(A,C) \le \Delta(A,B) + \Delta(B,C)}{Delta(A,C) <=
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 \eqn{p=\infty}{p=Inf} and \eqn{c=\infty}{c=Inf} the Delta metric
is equal to the Hausdorff metric.
The algorithm uses \code{\link{distmap}} to compute the distance maps
\eqn{d(x,A)} and \eqn{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 \code{\dots} passed
to \code{\link{as.mask}}.
}
\value{
A numeric value.
}
\references{
Baddeley, A.J. (1992a)
Errors in binary images and an \eqn{L^p} version of the Hausdorff metric.
\emph{Nieuw Archief voor Wiskunde} \bold{10}, 157--183.
Baddeley, A.J. (1992b)
An error metric for binary images.
In W. Foerstner and S. Ruwiedel (eds)
\emph{Robust Computer Vision}. Karlsruhe: Wichmann.
Pages 59--78.
}
\author{
Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\seealso{
\code{\link{distmap}}
}
\examples{
X <- runifpoint(20)
Y <- runifpoint(10)
deltametric(X, Y, p=1,c=0.1)
}
\keyword{spatial}
\keyword{math}