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\name{vgram.matrix}
\alias{vgram.matrix}
\alias{plot.vgram.matrix}
\title{
  Computes a variogram from an image 
}
\description{
Computes a variogram for an image taking into account different directions 
and returning summary information about the differences in each of these 
directions.  
}
\usage{
vgram.matrix(dat, R = NULL, dx = NULL, dy = NULL)

\method{plot}{vgram.matrix}(x,...)
}
\arguments{
\item{dat}{
Either a matrix spacing of rows and columns are assumed to have the same 
distance or a list in image format with components x, y and z. 
}
\item{R}{
Maximum radius for finding variogram differences assuming that the grid
points are spaced one unit a part. Default is go out to a 
radius of \code{5*max( c(dx, dy) )}.  
}

\item{dx}{
The spacing of grid points on the X axis. This is used to calculate the 
correct distance between grid points. If dx is not equal to dy then the 
collapse argument must be FALSE. If an image object is passed and dx and dy are not specified they will be calculated from the x and y components of the image
list.
}
\item{dy}{ The spacing of grid points on the Y axis.
See additional notes for dx.}

\item{x}{Returned object from vgram.matrix}

\item{\dots}{ Arguments for image.plot}

}
\value{
  
An object of class vgram.matrix with the following components:  d, a vector of distances for 
the differences, 
and vgram, the variogram values. This is the traditional variogram 
ignoring direction.  

d.full, a vector of distances for all possible shifts up distance R, 
 ind, a two column matrix giving the x and y increment used to compute
the shifts,  and vgram.full, the variogram at each of these 
separations. Also computed is vgram.robust, Cressie's version of a robust 
variogram statistic.

Also returned is the component N the number of differences found for each 
separation csae. 

}

\details{
For the "full" case 
the statistics can summarize  departures from 
isotropy by separating the variogram differences according to 
orientation. For small R this runs efficiently because the differences are 
found by sub-setting the image matrix.  

For example, suppose that a row of the ind matrix is 
(2,3). The variogram value associated with this row is the mean of  
the differences (1/2)*(X(i,j)- X( i+2,j+3))**2 for all i and j. 
(Here X(.,.) are the values for the spatial field.)  In this example  
d= sqrt(13) and there will be another entry with the same distance 
but 
corresponding to the direction (3,2). 
plot.vgram.matrix attempts to organize all the different directions into a 
coherent image plot.
}
\seealso{
\code{\link{vgram}} 
}
\examples{
# variogram for Lennon image.
data(lennon)
out<-vgram.matrix( lennon) 

plot( out$d, out$vgram, xlab="separation distance", ylab="variogram") 
# image plot of vgram values by direction.  

# look at different directions 
out<-vgram.matrix( lennon, R=8)  

plot( out$d, out$vgram) 
# add in different orientations 
points( out$d.full, out$vgram.full, col="red")

#image plot of variogram values for different directions. 
set.panel(1,1)
plot.vgram.matrix( out)
# John Lennon appears remarkably isotropic!

}
\keyword{spatial}
% docclass is function
% Converted by Sd2Rd version 1.21.