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"""Some test functions for bivariate interpolation.
Most of these have been yoinked from ACM TOMS 792.
http://netlib.org/toms/792
"""
import numpy as np
from triangulate import Triangulation
class TestData(dict):
def __init__(self, *args, **kwds):
dict.__init__(self, *args, **kwds)
self.__dict__ = self
class TestDataSet(object):
def __init__(self, **kwds):
self.__dict__.update(kwds)
data = TestData(
franke100=TestDataSet(
x=np.array([ 0.0227035, 0.0539888, 0.0217008, 0.0175129, 0.0019029,
-0.0509685, 0.0395408, -0.0487061, 0.0315828, -0.0418785,
0.1324189, 0.1090271, 0.1254439, 0.093454 , 0.0767578,
0.1451874, 0.0626494, 0.1452734, 0.0958668, 0.0695559,
0.2645602, 0.2391645, 0.208899 , 0.2767329, 0.1714726,
0.2266781, 0.1909212, 0.1867647, 0.2304634, 0.2426219,
0.3663168, 0.3857662, 0.3832392, 0.3179087, 0.3466321,
0.3776591, 0.3873159, 0.3812917, 0.3795364, 0.2803515,
0.4149771, 0.4277679, 0.420001 , 0.4663631, 0.4855658,
0.4092026, 0.4792578, 0.4812279, 0.3977761, 0.4027321,
0.5848691, 0.5730076, 0.6063893, 0.5013894, 0.5741311,
0.6106955, 0.5990105, 0.5380621, 0.6096967, 0.5026188,
0.6616928, 0.6427836, 0.6396475, 0.6703963, 0.7001181,
0.633359 , 0.6908947, 0.6895638, 0.6718889, 0.6837675,
0.7736939, 0.7635332, 0.7410424, 0.8258981, 0.7306034,
0.8086609, 0.8214531, 0.729064 , 0.8076643, 0.8170951,
0.8424572, 0.8684053, 0.8366923, 0.9418461, 0.8478122,
0.8599583, 0.91757 , 0.8596328, 0.9279871, 0.8512805,
1.044982 , 0.9670631, 0.9857884, 0.9676313, 1.0129299,
0.965704 , 1.0019855, 1.0359297, 1.0414677, 0.9471506]),
y=np.array([-0.0310206, 0.1586742, 0.2576924, 0.3414014, 0.4943596,
0.5782854, 0.6993418, 0.7470194, 0.9107649, 0.996289 ,
0.050133 , 0.0918555, 0.2592973, 0.3381592, 0.4171125,
0.5615563, 0.6552235, 0.7524066, 0.9146523, 0.9632421,
0.0292939, 0.0602303, 0.2668783, 0.3696044, 0.4801738,
0.5940595, 0.6878797, 0.8185576, 0.9046507, 0.9805412,
0.0396955, 0.0684484, 0.2389548, 0.3124129, 0.4902989,
0.5199303, 0.6445227, 0.8203789, 0.8938079, 0.9711719,
-0.0284618, 0.1560965, 0.2262471, 0.3175094, 0.3891417,
0.5084949, 0.6324247, 0.7511007, 0.8489712, 0.9978728,
-0.0271948, 0.127243 , 0.2709269, 0.3477728, 0.4259422,
0.6084711, 0.6733781, 0.7235242, 0.9242411, 1.0308762,
0.0255959, 0.0707835, 0.2008336, 0.3259843, 0.4890704,
0.5096324, 0.669788 , 0.7759569, 0.9366096, 1.0064516,
0.0285374, 0.1021403, 0.1936581, 0.3235775, 0.4714228,
0.6091595, 0.6685053, 0.8022808, 0.847679 , 1.0512371,
0.0380499, 0.0902048, 0.2083092, 0.3318491, 0.4335632,
0.5910139, 0.6307383, 0.8144841, 0.904231 , 0.969603 ,
-0.01209 , 0.1334114, 0.2695844, 0.3795281, 0.4396054,
0.5044425, 0.6941519, 0.7459923, 0.8682081, 0.9801409])),
franke33=TestDataSet(
x=np.array([ 5.00000000e-02, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 1.00000000e-01, 1.00000000e-01,
1.50000000e-01, 2.00000000e-01, 2.50000000e-01,
3.00000000e-01, 3.50000000e-01, 5.00000000e-01,
5.00000000e-01, 5.50000000e-01, 6.00000000e-01,
6.00000000e-01, 6.00000000e-01, 6.50000000e-01,
7.00000000e-01, 7.00000000e-01, 7.00000000e-01,
7.50000000e-01, 7.50000000e-01, 7.50000000e-01,
8.00000000e-01, 8.00000000e-01, 8.50000000e-01,
9.00000000e-01, 9.00000000e-01, 9.50000000e-01,
1.00000000e+00, 1.00000000e+00, 1.00000000e+00]),
y=np.array([ 4.50000000e-01, 5.00000000e-01, 1.00000000e+00,
0.00000000e+00, 1.50000000e-01, 7.50000000e-01,
3.00000000e-01, 1.00000000e-01, 2.00000000e-01,
3.50000000e-01, 8.50000000e-01, 0.00000000e+00,
1.00000000e+00, 9.50000000e-01, 2.50000000e-01,
6.50000000e-01, 8.50000000e-01, 7.00000000e-01,
2.00000000e-01, 6.50000000e-01, 9.00000000e-01,
1.00000000e-01, 3.50000000e-01, 8.50000000e-01,
4.00000000e-01, 6.50000000e-01, 2.50000000e-01,
3.50000000e-01, 8.00000000e-01, 9.00000000e-01,
0.00000000e+00, 5.00000000e-01, 1.00000000e+00])),
lawson25=TestDataSet(
x=np.array([ 0.1375, 0.9125, 0.7125, 0.225 , -0.05 , 0.475 , 0.05 ,
0.45 , 1.0875, 0.5375, -0.0375, 0.1875, 0.7125, 0.85 ,
0.7 , 0.275 , 0.45 , 0.8125, 0.45 , 1. , 0.5 ,
0.1875, 0.5875, 1.05 , 0.1 ]),
y=np.array([ 0.975 , 0.9875 , 0.7625 , 0.8375 , 0.4125 , 0.6375 ,
-0.05 , 1.0375 , 0.55 , 0.8 , 0.75 , 0.575 ,
0.55 , 0.4375 , 0.3125 , 0.425 , 0.2875 , 0.1875 ,
-0.0375 , 0.2625 , 0.4625 , 0.2625 , 0.125 , -0.06125, 0.1125 ])),
random100=TestDataSet(
x=np.array([ 0.0096326, 0.0216348, 0.029836 , 0.0417447, 0.0470462,
0.0562965, 0.0646857, 0.0740377, 0.0873907, 0.0934832,
0.1032216, 0.1110176, 0.1181193, 0.1251704, 0.132733 ,
0.1439536, 0.1564861, 0.1651043, 0.1786039, 0.1886405,
0.2016706, 0.2099886, 0.2147003, 0.2204141, 0.2343715,
0.240966 , 0.252774 , 0.2570839, 0.2733365, 0.2853833,
0.2901755, 0.2964854, 0.3019725, 0.3125695, 0.3307163,
0.3378504, 0.3439061, 0.3529922, 0.3635507, 0.3766172,
0.3822429, 0.3869838, 0.3973137, 0.4170708, 0.4255588,
0.4299218, 0.4372839, 0.4705033, 0.4736655, 0.4879299,
0.494026 , 0.5055324, 0.5162593, 0.5219219, 0.5348529,
0.5483213, 0.5569571, 0.5638611, 0.5784908, 0.586395 ,
0.5929148, 0.5987839, 0.6117561, 0.6252296, 0.6331381,
0.6399048, 0.6488972, 0.6558537, 0.6677405, 0.6814074,
0.6887812, 0.6940896, 0.7061687, 0.7160957, 0.7317445,
0.7370798, 0.746203 , 0.7566957, 0.7699998, 0.7879347,
0.7944014, 0.8164468, 0.8192794, 0.8368405, 0.8500993,
0.8588255, 0.8646496, 0.8792329, 0.8837536, 0.8900077,
0.8969894, 0.9044917, 0.9083947, 0.9203972, 0.9347906,
0.9434519, 0.9490328, 0.9569571, 0.9772067, 0.9983493]),
y=np.array([ 0.3083158, 0.2450434, 0.8613847, 0.0977864, 0.3648355,
0.7156339, 0.5311312, 0.9755672, 0.1781117, 0.5452797,
0.1603881, 0.7837139, 0.9982015, 0.6910589, 0.104958 ,
0.8184662, 0.7086405, 0.4456593, 0.1178342, 0.3189021,
0.9668446, 0.7571834, 0.2016598, 0.3232444, 0.4368583,
0.8907869, 0.064726 , 0.5692618, 0.2947027, 0.4332426,
0.3347464, 0.7436284, 0.1066265, 0.8845357, 0.515873 ,
0.9425637, 0.4799701, 0.1783069, 0.114676 , 0.8225797,
0.2270688, 0.4073598, 0.887508 , 0.7631616, 0.9972804,
0.4959884, 0.3410421, 0.249812 , 0.6409007, 0.105869 ,
0.5411969, 0.0089792, 0.8784268, 0.5515874, 0.4038952,
0.1654023, 0.2965158, 0.3660356, 0.0366554, 0.950242 ,
0.2638101, 0.9277386, 0.5377694, 0.7374676, 0.4674627,
0.9186109, 0.0416884, 0.1291029, 0.6763676, 0.8444238,
0.3273328, 0.1893879, 0.0645923, 0.0180147, 0.8904992,
0.4160648, 0.4688995, 0.2174508, 0.5734231, 0.8853319,
0.8018436, 0.6388941, 0.8931002, 0.1000558, 0.2789506,
0.9082948, 0.3259159, 0.8318747, 0.0508513, 0.970845 ,
0.5120548, 0.2859716, 0.9581641, 0.6183429, 0.3779934,
0.4010423, 0.9478657, 0.7425486, 0.8883287, 0.549675 ])),
uniform9=TestDataSet(
x=np.array([ 1.25000000e-01, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 1.25000000e-01, 1.25000000e-01,
1.25000000e-01, 1.25000000e-01, 1.25000000e-01,
1.25000000e-01, 1.25000000e-01, 1.25000000e-01,
2.50000000e-01, 2.50000000e-01, 2.50000000e-01,
2.50000000e-01, 2.50000000e-01, 2.50000000e-01,
2.50000000e-01, 2.50000000e-01, 2.50000000e-01,
3.75000000e-01, 3.75000000e-01, 3.75000000e-01,
3.75000000e-01, 3.75000000e-01, 3.75000000e-01,
3.75000000e-01, 3.75000000e-01, 3.75000000e-01,
5.00000000e-01, 5.00000000e-01, 5.00000000e-01,
5.00000000e-01, 5.00000000e-01, 5.00000000e-01,
5.00000000e-01, 5.00000000e-01, 5.00000000e-01,
6.25000000e-01, 6.25000000e-01, 6.25000000e-01,
6.25000000e-01, 6.25000000e-01, 6.25000000e-01,
6.25000000e-01, 6.25000000e-01, 6.25000000e-01,
7.50000000e-01, 7.50000000e-01, 7.50000000e-01,
7.50000000e-01, 7.50000000e-01, 7.50000000e-01,
7.50000000e-01, 7.50000000e-01, 7.50000000e-01,
8.75000000e-01, 8.75000000e-01, 8.75000000e-01,
8.75000000e-01, 8.75000000e-01, 8.75000000e-01,
8.75000000e-01, 8.75000000e-01, 8.75000000e-01,
1.00000000e+00, 1.00000000e+00, 1.00000000e+00,
1.00000000e+00, 1.00000000e+00, 1.00000000e+00,
1.00000000e+00, 1.00000000e+00, 1.00000000e+00]),
y=np.array([ 0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00,
0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00,
0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00,
0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00,
0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00,
0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00,
0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00,
0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00,
0.00000000e+00, 1.25000000e-01, 2.50000000e-01,
3.75000000e-01, 5.00000000e-01, 6.25000000e-01,
7.50000000e-01, 8.75000000e-01, 1.00000000e+00])),
)
def constant(x, y):
return np.ones(x.shape, x.dtype)
constant.title = 'Constant'
def xramp(x, y):
return x
xramp.title = 'X Ramp'
def yramp(x, y):
return y
yramp.title = 'Y Ramp'
def exponential(x, y):
x = x*9
y = y*9
x1 = x+1.0
x2 = x-2.0
x4 = x-4.0
x7 = x-7.0
y1 = x+1.0
y2 = y-2.0
y3 = y-3.0
y7 = y-7.0
f = (0.75 * np.exp(-(x2*x2+y2*y2)/4.0) +
0.75 * np.exp(-x1*x1/49.0 - y1/10.0) +
0.5 * np.exp(-(x7*x7 + y3*y3)/4.0) -
0.2 * np.exp(-x4*x4 -y7*y7))
return f
exponential.title = 'Exponential and Some Gaussians'
def cliff(x, y):
f = np.tanh(9.0*(y-x) + 1.0)/9.0
return f
cliff.title = 'Cliff'
def saddle(x, y):
f = (1.25 + np.cos(5.4*y))/(6.0 + 6.0*(3*x-1.0)**2)
return f
saddle.title = 'Saddle'
def gentle(x, y):
f = np.exp(-5.0625*((x-0.5)**2+(y-0.5)**2))/3.0
return f
gentle.title = 'Gentle Peak'
def steep(x, y):
f = np.exp(-20.25*((x-0.5)**2+(y-0.5)**2))/3.0
return f
steep.title = 'Steep Peak'
def sphere(x, y):
circle = 64-81*((x-0.5)**2 + (y-0.5)**2)
f = np.where(circle >= 0, np.sqrt(np.clip(circle,0,100)) - 0.5, 0.0)
return f
sphere.title = 'Sphere'
def trig(x, y):
f = 2.0*np.cos(10.0*x)*np.sin(10.0*y) + np.sin(10.0*x*y)
return f
trig.title = 'Cosines and Sines'
def gauss(x, y):
x = 5.0-10.0*x
y = 5.0-10.0*y
g1 = np.exp(-x*x/2)
g2 = np.exp(-y*y/2)
f = g1 + 0.75*g2*(1 + g1)
return f
gauss.title = 'Gaussian Peak and Gaussian Ridges'
def cloverleaf(x, y):
ex = np.exp((10.0-20.0*x)/3.0)
ey = np.exp((10.0-20.0*y)/3.0)
logitx = 1.0/(1.0+ex)
logity = 1.0/(1.0+ey)
f = (((20.0/3.0)**3 * ex*ey)**2 * (logitx*logity)**5 *
(ex-2.0*logitx)*(ey-2.0*logity))
return f
cloverleaf.title = 'Cloverleaf'
def cosine_peak(x, y):
circle = np.hypot(80*x-40.0, 90*y-45.)
f = np.exp(-0.04*circle) * np.cos(0.15*circle)
return f
cosine_peak.title = 'Cosine Peak'
allfuncs = [exponential, cliff, saddle, gentle, steep, sphere, trig, gauss, cloverleaf, cosine_peak]
class LinearTester(object):
name = 'Linear'
def __init__(self, xrange=(0.0, 1.0), yrange=(0.0, 1.0), nrange=101, npoints=250):
self.xrange = xrange
self.yrange = yrange
self.nrange = nrange
self.npoints = npoints
rng = np.random.RandomState(1234567890)
self.x = rng.uniform(xrange[0], xrange[1], size=npoints)
self.y = rng.uniform(yrange[0], yrange[1], size=npoints)
self.tri = Triangulation(self.x, self.y)
def replace_data(self, dataset):
self.x = dataset.x
self.y = dataset.y
self.tri = Triangulation(self.x, self.y)
def interpolator(self, func):
z = func(self.x, self.y)
return self.tri.linear_extrapolator(z, bbox=self.xrange+self.yrange)
def plot(self, func, interp=True, plotter='imshow'):
import matplotlib as mpl
from matplotlib import pylab as pl
if interp:
lpi = self.interpolator(func)
z = lpi[self.yrange[0]:self.yrange[1]:complex(0,self.nrange),
self.xrange[0]:self.xrange[1]:complex(0,self.nrange)]
else:
y, x = np.mgrid[self.yrange[0]:self.yrange[1]:complex(0,self.nrange),
self.xrange[0]:self.xrange[1]:complex(0,self.nrange)]
z = func(x, y)
z = np.where(np.isinf(z), 0.0, z)
extent = (self.xrange[0], self.xrange[1],
self.yrange[0], self.yrange[1])
pl.ioff()
pl.clf()
pl.hot() # Some like it hot
if plotter == 'imshow':
pl.imshow(np.nan_to_num(z), interpolation='nearest', extent=extent, origin='lower')
elif plotter == 'contour':
Y, X = np.ogrid[self.yrange[0]:self.yrange[1]:complex(0,self.nrange),
self.xrange[0]:self.xrange[1]:complex(0,self.nrange)]
pl.contour(np.ravel(X), np.ravel(Y), z, 20)
x = self.x
y = self.y
lc = mpl.collections.LineCollection(np.array([((x[i], y[i]), (x[j], y[j]))
for i, j in self.tri.edge_db]), colors=[(0,0,0,0.2)])
ax = pl.gca()
ax.add_collection(lc)
if interp:
title = '%s Interpolant' % self.name
else:
title = 'Reference'
if hasattr(func, 'title'):
pl.title('%s: %s' % (func.title, title))
else:
pl.title(title)
pl.show()
pl.ion()
class NNTester(LinearTester):
name = 'Natural Neighbors'
def interpolator(self, func):
z = func(self.x, self.y)
return self.tri.nn_extrapolator(z, bbox=self.xrange+self.yrange)
def plotallfuncs(allfuncs=allfuncs):
from matplotlib import pylab as pl
pl.ioff()
nnt = NNTester(npoints=1000)
lpt = LinearTester(npoints=1000)
for func in allfuncs:
print func.title
nnt.plot(func, interp=False, plotter='imshow')
pl.savefig('%s-ref-img.png' % func.func_name)
nnt.plot(func, interp=True, plotter='imshow')
pl.savefig('%s-nn-img.png' % func.func_name)
lpt.plot(func, interp=True, plotter='imshow')
pl.savefig('%s-lin-img.png' % func.func_name)
nnt.plot(func, interp=False, plotter='contour')
pl.savefig('%s-ref-con.png' % func.func_name)
nnt.plot(func, interp=True, plotter='contour')
pl.savefig('%s-nn-con.png' % func.func_name)
lpt.plot(func, interp=True, plotter='contour')
pl.savefig('%s-lin-con.png' % func.func_name)
pl.ion()
def plot_dt(tri, colors=None):
import matplotlib as mpl
from matplotlib import pylab as pl
if colors is None:
colors = [(0,0,0,0.2)]
lc = mpl.collections.LineCollection(np.array([((tri.x[i], tri.y[i]), (tri.x[j], tri.y[j]))
for i, j in tri.edge_db]), colors=colors)
ax = pl.gca()
ax.add_collection(lc)
pl.draw_if_interactive()
def plot_vo(tri, colors=None):
import matplotlib as mpl
from matplotlib import pylab as pl
if colors is None:
colors = [(0,1,0,0.2)]
lc = mpl.collections.LineCollection(np.array(
[(tri.circumcenters[i], tri.circumcenters[j])
for i in xrange(len(tri.circumcenters))
for j in tri.triangle_neighbors[i] if j != -1]),
colors=colors)
ax = pl.gca()
ax.add_collection(lc)
pl.draw_if_interactive()
def plot_cc(tri, edgecolor=None):
import matplotlib as mpl
from matplotlib import pylab as pl
if edgecolor is None:
edgecolor = (0,0,1,0.2)
dxy = (np.array([(tri.x[i], tri.y[i]) for i,j,k in tri.triangle_nodes])
- tri.circumcenters)
r = np.hypot(dxy[:,0], dxy[:,1])
ax = pl.gca()
for i in xrange(len(r)):
p = mpl.patches.Circle(tri.circumcenters[i], r[i], resolution=100, edgecolor=edgecolor,
facecolor=(1,1,1,0), linewidth=0.2)
ax.add_patch(p)
pl.draw_if_interactive()
def quality(func, mesh, interpolator='nn', n=33):
"""Compute a quality factor (the quantity r**2 from TOMS792).
interpolator must be in ('linear', 'nn').
"""
fz = func(mesh.x, mesh.y)
tri = Triangulation(mesh.x, mesh.y)
intp = getattr(tri, interpolator+'_extrapolator')(fz, bbox=(0.,1.,0.,1.))
Y, X = np.mgrid[0:1:complex(0,n),0:1:complex(0,n)]
Z = func(X, Y)
iz = intp[0:1:complex(0,n),0:1:complex(0,n)]
#nans = np.isnan(iz)
#numgood = n*n - np.sum(np.array(nans.flat, np.int32))
numgood = n*n
SE = (Z - iz)**2
SSE = np.sum(SE.flat)
meanZ = np.sum(Z.flat) / numgood
SM = (Z - meanZ)**2
SSM = np.sum(SM.flat)
r2 = 1.0 - SSE/SSM
print func.func_name, r2, SSE, SSM, numgood
return r2
def allquality(interpolator='nn', allfuncs=allfuncs, data=data, n=33):
results = {}
kv = data.items()
kv.sort()
for name, mesh in kv:
reslist = results.setdefault(name, [])
for func in allfuncs:
reslist.append(quality(func, mesh, interpolator, n))
return results
def funky():
x0 = np.array([0.25, 0.3, 0.5, 0.6, 0.6])
y0 = np.array([0.2, 0.35, 0.0, 0.25, 0.65])
tx = 0.46
ty = 0.23
t0 = Triangulation(x0, y0)
t1 = Triangulation(np.hstack((x0, [tx])), np.hstack((y0, [ty])))
return t0, t1