Choi and Hall (2001) proposed a procedure for
data sharpening of spatial point patterns.
This procedure is appropriate for earthquake epicentres
and other point patterns which are believed to exhibit
strong concentrations of points along a curve. Data sharpening
causes such points to concentrate more tightly along the curve.
If the original data points are
\(X_1, \ldots, X_n\)
then the sharpened points are
$$
\hat X_i = \frac{\sum_j X_j k(X_j-X_i)}{\sum_j k(X_j - X_i)}
$$
where \(k\) is a smoothing kernel in two dimensions.
Thus, the new point \(\hat X_i\) is a
vector average of the nearby points \(X[j]\).
The function sharpen is generic. It currently has only one
method, for two-dimensional point patterns (objects of class
"ppp").
If sigma is given, the smoothing kernel is the
isotropic two-dimensional Gaussian density with standard deviation
sigma in each axis. If varcov is given, the smoothing
kernel is the Gaussian density with variance-covariance matrix
varcov.
The data sharpening procedure tends to cause the point pattern
to contract away from the boundary of the window. That is,
points \(X_i\) that lie `quite close to the edge of the window
of the point pattern tend to be displaced inward.
If edgecorrect=TRUE then the algorithm is modified to
correct this vector bias.