Fantastic. Excellent work. This is something we have been missing for a long time. Travis -- (mobile phone of) Travis Oliphant Enthought, Inc. 1-512-536-1057 http://www.enthought.com On Jul 25, 2010, at 10:35 AM, Pauli Virtanen <pav@iki.fi> wrote:

Hi all,

I took the Qhull by the horns, and wrote a straightforward `griddata` implementation for working in N-D:

http://github.com/pv/scipy-work/tree/qhull http://github.com/pv/scipy-work/blob/qhull/scipy/interpolate/qhull.pyx

Sample (4-D): ------------------------------------------------------------- import numpy as np import matplotlib.pyplot as plt from scipy.interpolate import griddata

def f(x, y, z, u): return np.cos(x + np.cos(y + np.cos(z + np.cos(u))))

points = np.random.rand(2000, 4) values = f(*points.T)

p = np.linspace(0, 1, 1500) z = griddata(points, values, np.c_[p, p, p, p], method='linear')

plt.plot(p, z, '-', p, f(p, p, p, p), '-') plt.show() -------------------------------------------------------------

It performs the N-D Delaunay tesselation with Qhull, and after that does linear barycentric interpolation on the simplex containing each point. (The nearest-neighbor "interpolation" is, on the other hand, implemented using scipy.spatial.KDTree.)

I ended up writing some custom simplex-walking code for locating the simplex containing the points -- this is much faster than a brute-force search. However, it is slightly different from what `qh_findbestfacet` does. [If you're a computational geometry expert, see below...]

Speed-wise, Qhull appears to lose in 2-D to `scikits.delaunay` by a factor of about 10 in triangulation in my tests; this, however, includes the time taken by computing the barycentric coordinate transforms. Interpolation, however, seems to be faster.

***

I think this would be a nice addition to `scipy.optimize`. I'd like to get it in for 0.9.

Later on, we can specialize the `griddata` function to perform better in low dimensions (ndim=2, 3) and add more interpolation methods there; for example natural neighbors, interpolating splines, etc. In ndim > 3, the `griddata` is however already feature-equivalent to Octave and MATLAB.

Comments?

Pauli

PS. A question for computational geometry experts:

`qh_findbestfacet` finds the facet on the lower convex hull whose oriented hyperplane distance to the point lifted on the paraboloid is maximal --- however, this does not always give the simplex containing the point in the projected Delaunay tesselation. There's a counterexample in qhull.pyx:_find_simplex docstring.

On the other hand, Qhull documentation claims that this is the way to locate the Delaunay simplex containing a point. What gives?

It's clear, however, that if a simplex contains the point, then the hyperplane distance is positive. So currently, I just walk around positive-distance simplices checking the inside condition for each and hoping for the best. If nothing is found, I just fall back to brute force search.

This seems to work well in practice. However, if a point is outside the triangulation (but inside the bounding box), I have to fall back to a brute-force search. Is there a better option here?

***

Another sample (2-D, object-oriented interface): ------------------------------------------------------------- import time import numpy as np import matplotlib.pyplot as plt from scipy.interpolate import LinearNDInterpolator

# some random data x = np.array([(0,0), (0,1), (1, 1), (1, 0)], dtype=np.double) np.random.seed(0) xr = np.random.rand(200*200, 2).astype(np.double) x = np.r_[xr, x] y = np.arange(x.shape[0], dtype=np.double)

# evaluate on a grid nn = 500 xx = np.linspace(-0.1, 1.1, nn) yy = np.linspace(-0.1, 1.1, nn) xx, yy = np.broadcast_arrays(xx[:,None], yy[None,:])

xi = np.array([xx, yy]).transpose(1,2,0).copy() start = time.time() ip = LinearNDInterpolator(x, y) print "Triangulation", time.time() - start start = time.time() zi = ip(xi) print "Interpolation", time.time() - start

from scikits.delaunay import Triangulation

start = time.time() tri2 = Triangulation(x[:,0], x[:,1]) print "scikits.delaunay triangulation", time.time() - start

start = time.time() ip2 = tri2.linear_interpolator(y) zi2 = ip2[-0.1:1.1:500j,-0.1:1.1:500j].T print "scikits.delaunay interpolation", time.time() - start

print "rel-difference", np.nanmax(abs(zi - zi2))/np.nanmax(np.abs(zi))

plt.figure() plt.imshow(zi) plt.clim(y.min(), y.max()) plt.colorbar() plt.figure() plt.imshow(zi2) plt.clim(y.min(), y.max()) plt.colorbar() plt.show() -------------------------------------------------------------

-- Pauli Virtanen

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