## Interpolation and Extrapolation

In many areas ranging from cartography to molecular imaging and modeling, one finds the need to
fit a function or surface to a collection of scattered data points.
Interpolation refers to finding values for points between the given
points (i.e. inside their convex hull); if the function should extend
over a wider region of the input domain the problem is instead referred
to as extrapolation. The method chosen should depend on the properties
one desires the resulting surface to have; many interpolation methods
are based on Voronoi diagrams and Delaunay
triangulations. A piecewise constant approximation, used in
rainfall estimation, can be found by simply choosing the function value
in each Voronoi cell to be that of the cell's generating site. This is
a analog for continuous domains of the nearest neighbor rule from
machine learning theory, which has been generalized e.g. to voting among
three nearest neighbors; one could similarly define median-of-three
interpolation schemes (still piecewise constant, but less susceptable to
erroneous data), however I know of no application in which such schemes
have been tried. The Delaunay triangulation itself provides a piecewise
linear continuous function, defined within the convex hull of the input,
that minimizes a certain energy functional. "Natural neighbor
interpolation" is defined for each point x by adding x as a site to the
Voronoi diagram of the original sites, and averaging the sites' values
weighted by the fraction of the cell for x previously covered by each
other cell. This somewhat complicated procedure results in a continuous
function smooth everywhere except at the original sites.
At the 1995 ARO-MSI Worksh. on Comp. Geom.,
Ken
Clarkson
described a similar technique that results in a function
smooth everywhere.
- Applications of computational geometry.
John Hershberger describes some geometric problems arising in his work
at Mentor graphics including interpolation of thermal data, minimum spanning trees, and breakout
routing in PC board design.

- Convex hulls and interpolation. Ken Clarkson describes some implementation
details of algorithms for convex hulls, alpha shapes, Voronoi diagrams,
and natural
neighbor interpolation.

- Delaunay interpolation.
Various people discuss the pros and cons of using Delaunay triangulation
for data interpolatation.

- The
DoD groundwater modeling system uses
natural neighbor interpolation for groundwater plume characterization.

- Geophysical
applications of natural neighbors.
A group at ANU Canberra uses Voronoi diagram based interpolation
for a range of geophysical problems.

- Mosaic / stained glass graphic effect. One gets interesting
visual effects by taking a random sample of pixels from a bitmap image,
computing the Voronoi diagram of the sample, and filling each cell
with the corresponding sample's color. This appears to be what
Photoshop does in its "Pixelate->Crystalize" filter;
it can be thought of as a form of piecewise constant function interpolation.

- Visualization of three different interpolation methods: k-nearest neighbors, natural neighbors, and gamma-observable neighbors. Java applet by Michaël Aupetit.

Part of
Geometry in Action,
a collection of applications of computational geometry.

David Eppstein,
Theory Group,
ICS,
UC Irvine.

Semi-automatically
filtered
from a common source file.