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"""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. """
""" 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,)
"""
return np.sqrt(((x1 - x2)**2).sum(axis=0))
return np.sqrt((1.0/self.epsilon*r)**2 + 1)
return 1.0/np.sqrt((1.0/self.epsilon*r)**2 + 1)
return np.exp(-(1.0/self.epsilon*r)**2)
return r
return r**3
return r**5
return xlogy(r**2, r)
# Setup self._function and do smoke test on initial 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
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) 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.nodes = linalg.solve(self.A, self.di)
def A(self): # this only exists for backwards compatibility: self.A was available # and, at least technically, public. r = self._call_norm(self.xi, self.xi) return self._init_function(r) - np.eye(self.N)*self.smooth
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)
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) |