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
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
|
"""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
Some additional alterations by Travis Oliphant
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 __future__ import division, print_function, absolute_import
import sys
import numpy as np
from scipy import linalg
from scipy._lib.six import callable, get_method_function, get_function_code
from scipy.special import xlogy
__all__ = ['Rbf']
class Rbf(object):
"""
Rbf(*args)
A class for radial basis function approximation/interpolation of
n-dimensional scattered data.
Parameters
----------
*args : arrays
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 : str or callable, optional
The radial basis function, based on the radius, r, given by the norm
(default is Euclidean distance); the default is 'multiquadric'::
'multiquadric': sqrt((r/self.epsilon)**2 + 1)
'inverse': 1.0/sqrt((r/self.epsilon)**2 + 1)
'gaussian': exp(-(r/self.epsilon)**2)
'linear': r
'cubic': r**3
'quintic': r**5
'thin_plate': r**2 * log(r)
If callable, then it must take 2 arguments (self, r). The epsilon
parameter will be available as self.epsilon. Other keyword
arguments passed in will be available as well.
epsilon : float, optional
Adjustable constant for gaussian or multiquadrics functions
- defaults to approximate average distance between nodes (which is
a good start).
smooth : float, optional
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 : callable, optional
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::
def euclidean_norm(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 matrix of the
distances from each point in x1 to each point in x2.
Examples
--------
>>> from scipy.interpolate import Rbf
>>> x, y, z, d = np.random.rand(4, 50)
>>> rbfi = Rbf(x, y, z, d) # radial basis function interpolator instance
>>> xi = yi = zi = np.linspace(0, 1, 20)
>>> di = rbfi(xi, yi, zi) # interpolated values
>>> di.shape
(20,)
"""
def _euclidean_norm(self, x1, x2):
return np.sqrt(((x1 - x2)**2).sum(axis=0))
def _h_multiquadric(self, r):
return np.sqrt((1.0/self.epsilon*r)**2 + 1)
def _h_inverse_multiquadric(self, r):
return 1.0/np.sqrt((1.0/self.epsilon*r)**2 + 1)
def _h_gaussian(self, r):
return np.exp(-(1.0/self.epsilon*r)**2)
def _h_linear(self, r):
return r
def _h_cubic(self, r):
return r**3
def _h_quintic(self, r):
return r**5
def _h_thin_plate(self, r):
return xlogy(r**2, r)
# Setup self._function and do smoke test on initial r
def _init_function(self, r):
if isinstance(self.function, str):
self.function = self.function.lower()
_mapped = {'inverse': 'inverse_multiquadric',
'inverse multiquadric': 'inverse_multiquadric',
'thin-plate': 'thin_plate'}
if self.function in _mapped:
self.function = _mapped[self.function]
func_name = "_h_" + self.function
if hasattr(self, func_name):
self._function = getattr(self, func_name)
else:
functionlist = [x[3:] for x in dir(self) if x.startswith('_h_')]
raise ValueError("function must be a callable or one of " +
", ".join(functionlist))
self._function = getattr(self, "_h_"+self.function)
elif callable(self.function):
allow_one = False
if hasattr(self.function, 'func_code') or \
hasattr(self.function, '__code__'):
val = self.function
allow_one = True
elif hasattr(self.function, "im_func"):
val = get_method_function(self.function)
elif hasattr(self.function, "__call__"):
val = get_method_function(self.function.__call__)
else:
raise ValueError("Cannot determine number of arguments to function")
argcount = get_function_code(val).co_argcount
if allow_one and argcount == 1:
self._function = self.function
elif argcount == 2:
if sys.version_info[0] >= 3:
self._function = self.function.__get__(self, Rbf)
else:
import new
self._function = new.instancemethod(self.function, self,
Rbf)
else:
raise ValueError("Function argument must take 1 or 2 arguments.")
a0 = self._function(r)
if a0.shape != r.shape:
raise ValueError("Callable must take array and return array of the same shape")
return a0
def __init__(self, *args, **kwargs):
self.xi = np.asarray([np.asarray(a, dtype=np.float_).flatten()
for a in args[:-1]])
self.N = self.xi.shape[-1]
self.di = np.asarray(args[-1]).flatten()
if not all([x.size == self.di.size for x in self.xi]):
raise ValueError("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', None)
if self.epsilon is None:
# default epsilon is the "the average distance between nodes" based
# on a bounding hypercube
dim = self.xi.shape[0]
ximax = np.amax(self.xi, axis=1)
ximin = np.amin(self.xi, axis=1)
edges = ximax-ximin
edges = edges[np.nonzero(edges)]
self.epsilon = np.power(np.prod(edges)/self.N, 1.0/edges.size)
self.smooth = kwargs.pop('smooth', 0.0)
self.function = kwargs.pop('function', 'multiquadric')
# attach anything left in kwargs to self
# for use by any user-callable function or
# to save on the object returned.
for item, value in kwargs.items():
setattr(self, item, value)
self.A = self._init_function(r) - np.eye(self.N)*self.smooth
self.nodes = linalg.solve(self.A, self.di)
def _call_norm(self, x1, x2):
if len(x1.shape) == 1:
x1 = x1[np.newaxis, :]
if len(x2.shape) == 1:
x2 = x2[np.newaxis, :]
x1 = x1[..., :, np.newaxis]
x2 = x2[..., np.newaxis, :]
return self.norm(x1, x2)
def __call__(self, *args):
args = [np.asarray(x) for x in args]
if not all([x.shape == y.shape for x in args for y in args]):
raise ValueError("Array lengths must be equal")
shp = args[0].shape
xa = np.asarray([a.flatten() for a in args], dtype=np.float_)
r = self._call_norm(xa, self.xi)
return np.dot(self._function(r), self.nodes).reshape(shp)
|
