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ƒ } | | | | | | f S( s Convert a 2-tuple of slices to start,stop,steps for x and y.
key -- (slice(ystart,ystop,ystep), slice(xtart, xstop, xstep))
For now, the only accepted step values are imaginary integers (interpreted
in the same way numpy.mgrid, etc. do).
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*c B s& e Z d Z e i d „ Z d „ Z RS( sð Interpolate a function defined on the nodes of a triangulation by
using the planes defined by the three function values at each corner of
the triangles.
LinearInterpolator(triangulation, z, default_value=numpy.nan)
triangulation -- Triangulation instance
z -- the function values at each node of the triangulation
default_value -- a float giving the default value should the interpolating
point happen to fall outside of the convex hull of the triangulation
At the moment, the only regular rectangular grids are supported for
interpolation.
vals = interp[ystart:ystop:ysteps*1j, xstart:xstop:xsteps*1j]
vals would then be a (ysteps, xsteps) array containing the interpolated
values. These arguments are interpreted the same way as numpy.mgrid.
Attributes:
planes -- (ntriangles, 3) array of floats specifying the plane for each
triangle.
Linear Interpolation
--------------------
Given the Delauany triangulation (or indeed *any* complete triangulation) we
can interpolate values inside the convex hull by locating the enclosing
triangle of the interpolation point and returning the value at that point of
the plane defined by the three node values.
f = planes[tri,0]*x + planes[tri,1]*y + planes[tri,2]
The interpolated function is C0 continuous across the convex hull of the
input points. It is C1 continuous across the convex hull except for the
nodes and the edges of the triangulation.
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Interpolate a function defined on the nodes of a triangulation by
the natural neighbors method.
NNInterpolator(triangulation, z, default_value=numpy.nan)
triangulation -- Triangulation instance
z -- the function values at each node of the triangulation
default_value -- a float giving the default value should the interpolating
point happen to fall outside of the convex hull of the triangulation
At the moment, the only regular rectangular grids are supported for
interpolation.
vals = interp[ystart:ystop:ysteps*1j, xstart:xstop:xsteps*1j]
vals would then be a (ysteps, xsteps) array containing the interpolated
values. These arguments are interpreted the same way as numpy.mgrid.
Natural Neighbors Interpolation
-------------------------------
One feature of the Delaunay triangulation is that for each triangle, its
circumcircle contains no other point (although in degenerate cases, like
squares, other points may be *on* the circumcircle). One can also construct
what is called the Voronoi diagram from a Delaunay triangulation by
connecting the circumcenters of the triangles to those of their neighbors to
form a tesselation of irregular polygons covering the plane and containing
only one node from the triangulation. Each point in one node's Voronoi
polygon is closer to that node than any other node.
To compute the Natural Neighbors interpolant, we consider adding the
interpolation point to the triangulation. We define the natural neighbors of
this point as the set of nodes participating in Delaunay triangles whose
circumcircles contain the point. To restore the Delaunay-ness of the
triangulation, one would only have to alter those triangles and Voronoi
polygons. The new Voronooi diagram would have a polygon around the inserted
point. This polygon would "steal" area from the original Voronoi polygons.
For each node i in the natural neighbors set, we compute the area stolen
from its original Voronoi polygon, stolen[i]. We define the natural
neighbors coordinates
phi[i] = stolen[i] / sum(stolen,axis=0)
We then use these phi[i] to weight the corresponding function values from
the input data z to compute the interpolated value.
The interpolated surface is C1-continuous except at the nodes themselves
across the convex hull of the input points. One can find the set of points
that a given node will affect by computing the union of the areas covered by
the circumcircles of each Delaunay triangle that node participates in.
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