1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
|
#!/usr/bin/env python
"""rbf - Radial basis functions for interpolation/smoothing scattered Nd data.
Written by John Travers <jtravs@gmail.com>, February 2007
Based closely on Matlab code by Alex Chirokov
Additional, large, improvements by Robert Hetland
Permission to use, modify, and distribute this software is given under the
terms of the SciPy (BSD style) license. See LICENSE.txt that came with
this distribution for specifics.
NO WARRANTY IS EXPRESSED OR IMPLIED. USE AT YOUR OWN RISK.
Copyright (c) 2006-2007, Robert Hetland <hetland@tamu.edu>
Copyright (c) 2007, John Travers <jtravs@gmail.com>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the following
disclaimer in the documentation and/or other materials provided
with the distribution.
* Neither the name of Robert Hetland nor the names of any
contributors may be used to endorse or promote products derived
from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
"""
from numpy import sqrt, log, asarray, newaxis, all, dot, float64, eye
import scipy.linalg
class Rbf(object):
""" A class for radial basis function approximation/interpolation of
n-dimensional scattered data.
"""
def _euclidean_norm(self, x1, x2):
return sqrt( ((x1 - x2)**2).sum(axis=0) )
def _function(self, r):
if self.function.lower() == 'multiquadric':
return sqrt((1.0/self.epsilon*r)**2 + 1)
elif self.function.lower() == 'inverse multiquadric':
return 1.0/sqrt((1.0/self.epsilon*r)**2 + 1)
elif self.function.lower() == 'gausian':
return exp(-(self.epsilon*r)**2)
elif self.function.lower() == 'cubic':
return r**3
elif self.function.lower() == 'quintic':
return r**5
elif self.function.lower() == 'thin-plate':
return r**2 * log(r)
else:
raise ValueError, 'Invalid basis function name'
def __init__(self, *args, **kwargs):
""" Constructor for Rbf class.
Inputs:
x, y, z, ..., d
Where x, y, z, ... are the coordinates of the nodes
and d is the array of values at the nodes
function the radial basis function, based on the radius, r, given
by the norm (defult is Euclidean distance); the default
is 'multiquadratic'.
'multiquadric': sqrt((self.epsilon*r)**2 + 1)
'inverse multiquadric': 1.0/sqrt((self.epsilon*r)**2 + 1)
'gausian': exp(-(self.epsilon*r)**2)
'cubic': r**3
'quintic': r**5
'thin-plate': r**2 * log(r)
epsilon adjustable constant for gaussian or multiquadrics
functions - defaults to approximate average distance
between nodes (which is a good start)
smooth values greater than zero increase the smoothness
of the approximation.
0 is for interpolation (default), the function will
always go through the nodal points in this case.
norm A function that returns the 'distance' between two points,
with inputs as arrays of positions (x, y, z, ...), and an
output as an array of distance. E.g, the default is
def euclidean_norm(self, x1, x2):
return sqrt( ((x1 - x2)**2).sum(axis=0) )
which is called with x1 = x1[ndims, newaxis, :] and
x2 = x2[ndims, :, newaxis] such that the result is a
symetric, square matrix of the distances between each point
to each other point.
Outputs:
Interpolator object rbfi that returns interpolated values at new positions:
>>> rbfi = Rbf(x, y, z, d) # radial basis function interpolator instance
>>> di = rbfi(xi, yi, zi) # interpolated values
"""
self.xi = asarray([asarray(a, dtype=float64).flatten() for a in args[:-1]])
self.N = self.xi.shape[-1]
self.di = asarray(args[-1], dtype=float64).flatten()
assert [x.size==self.di.size for x in self.xi], \
'All arrays must be equal length'
self.norm = kwargs.pop('norm', self._euclidean_norm)
r = self._call_norm(self.xi, self.xi)
self.epsilon = kwargs.pop('epsilon', r.mean())
self.function = kwargs.pop('function', 'multiquadric')
self.smooth = kwargs.pop('smooth', 0.0)
self.A = self._function(r) - eye(self.N)*self.smooth
self.nodes = scipy.linalg.solve(self.A, self.di)
def _call_norm(self, x1, x2):
if len(x1.shape) == 1:
x1 = x1[newaxis, :]
if len(x2.shape) == 1:
x2 = x2[newaxis, :]
x1 = x1[..., :, newaxis]
x2 = x2[..., newaxis, :]
return self.norm(x1, x2)
def __call__(self, *args):
assert all([x.shape == y.shape \
for x in args \
for y in args]), 'Array lengths must be equal'
shp = args[0].shape
self.xa = asarray([a.flatten() for a in args], dtype=float64)
r = self._call_norm(self.xa, self.xi)
return dot(self._function(r), self.nodes).reshape(shp)
|
