├── GPLv3.txt ├── LICENSE ├── README.rst ├── fast_cubic_spline.py ├── fast_cubic_spline.pyx └── setup.py /GPLv3.txt: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. By contrast, 15 | the GNU General Public License is intended to guarantee your freedom to 16 | share and change all versions of a program--to make sure it remains free 17 | software for all its users. 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Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | Fast-Cubic-Spline-Python provides an implementation of 1D and 2D fast spline 2 | interpolation algorithm (Habermann and Kindermann 2007) in Python. 3 | Copyright (C) 2011, 2013 Joon H. Ro 4 | 5 | This file is part of Fast-Cubic-Spline-Python. 6 | 7 | Fast-Cubic-Spline-Python is free software: you can redistribute it and/or modify 8 | it under the terms of the GNU General Public License as published by 9 | the Free Software Foundation, either version 3 of the License, or 10 | (at your option) any later version. 11 | 12 | Fast-Cubic-Spline-Python is distributed in the hope that it will be useful, 13 | but WITHOUT ANY WARRANTY; without even the implied warranty of 14 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 15 | GNU General Public License for more details. 16 | 17 | You should have received a copy of the GNU General Public License 18 | along with this program. If not, see . 19 | -------------------------------------------------------------------------------- /README.rst: -------------------------------------------------------------------------------- 1 | ************ 2 | Introduction 3 | ************ 4 | 5 | :Date: Mar 28, 2013 6 | :Version: 0.1.0 7 | :Authors: Joon Ro, joonhyoung.ro[at]gmail.com 8 | :Web site: https://github.com/joonro/fast-cubic-spline-python 9 | :Copyright: This document has been placed in the public domain. 10 | :License: Fast-Cubic-Spline-Python is released under the GPLv3. 11 | 12 | 13 | Purpose 14 | ======= 15 | 16 | Fast-Cubic-Spline-Python provides an implementation of fast spline 17 | interpolation algorithm of `Habermann and Kindermann (2007)`_ in Python. While 18 | higher dimensional interpolation is also possible with this code, currently 19 | only 1D and 2D examples are provided. 20 | 21 | Calculation of spline coefficients are in NumPy, and actual interpolation 22 | routine is coded in Cython. This is advantageous since if your main routine is 23 | coded in Cython, once you have coefficients, you can call interpolation 24 | functions without any Python overhead. 25 | 26 | ***** 27 | Usage 28 | ***** 29 | 30 | Run the main module for an example: 31 | 32 | .. code-block:: sh 33 | 34 | $ python fast_cubic_spline.py 35 | 36 | 37 | ************ 38 | Installation 39 | ************ 40 | 41 | Dependencies 42 | ============ 43 | 44 | * Python 45 | * Cython (http://cython.org) 46 | 47 | Compiling Cython Module 48 | ======================= 49 | 50 | .. code-block:: sh 51 | 52 | $ python setup.py build_ext --inplace 53 | 54 | References 55 | ========== 56 | 57 | Habermann, C., & Kindermann, F. (2007). Multidimensional Spline Interpolation: 58 | Theory and Applications. Computational Economics, 30(2), 153–169. 59 | 60 | 61 | .. _Habermann and Kindermann (2007): http://www.springerlink.com/index/10.1007/s10614-007-9092-4 62 | -------------------------------------------------------------------------------- /fast_cubic_spline.py: -------------------------------------------------------------------------------- 1 | # Fast-Cubic-Spline-Python provides an implementation of 1D and 2D fast spline 2 | # interpolation algorithm (Habermann and Kindermann 2007) in Python. 3 | # Copyright (C) 2012, 2013 Joon H. Ro 4 | 5 | # This file is part of Fast-Cubic-Spline-Python. 6 | 7 | # Fast-Cubic-Spline-Python is free software: you can redistribute it and/or modify 8 | # it under the terms of the GNU General Public License as published by 9 | # the Free Software Foundation, either version 3 of the License, or 10 | # (at your option) any later version. 11 | 12 | # Fast-Cubic-Spline-Python is distributed in the hope that it will be useful, 13 | # but WITHOUT ANY WARRANTY; without even the implied warranty of 14 | # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 15 | # GNU General Public License for more details. 16 | 17 | # You should have received a copy of the GNU General Public License 18 | # along with this program. If not, see . 19 | 20 | from __future__ import division 21 | 22 | from numpy import arange, array, linspace, ones, zeros 23 | from scipy.linalg import solve_banded 24 | 25 | import fast_cubic_spline as _spline 26 | 27 | def cal_coefs(a, b, y, c=None, alpha=0, beta=0): 28 | ''' 29 | Return spline coefficients 30 | 31 | Parameters 32 | ---------- 33 | a : float 34 | lower bound of the grid. 35 | b : float 36 | upper bound of the grid. 37 | y : ndarray 38 | actual function value at grid points. 39 | c : (y.shape[0] + 2, ) ndarray, optional 40 | ndarry to be written 41 | alpha : float 42 | Second-order derivative at a. Default is 0. 43 | beta : float 44 | Second-order derivative at b. Default is 0. 45 | 46 | Returns 47 | ------- 48 | out : ndarray 49 | Array of coefficients. 50 | ''' 51 | n = y.shape[0] - 1 52 | h = (b - a)/n 53 | 54 | if c is None: 55 | c = zeros((n + 3, )) 56 | ifreturn = True 57 | else: 58 | assert(c.shape[0] == n + 3) 59 | ifreturn = False 60 | 61 | c[1] = 1/6 * (y[0] - (alpha * h**2)/6) 62 | c[n + 1] = 1/6 * (y[n] - (beta * h**2)/6) 63 | 64 | # ab matrix here is just compressed banded matrix 65 | ab = ones((3, n - 1)) 66 | ab[0, 0] = 0 67 | ab[1, :] = 4 68 | ab[-1, -1] = 0 69 | 70 | B = y[1:-1].copy() 71 | B[0] -= c[1] 72 | B[-1] -= c[n + 1] 73 | 74 | c[2:-2] = solve_banded((1, 1), ab, B) 75 | 76 | c[0] = alpha * h**2/6 + 2 * c[1] - c[2] 77 | c[-1] = beta * h**2/6 + 2 * c[-2] - c[-3] 78 | 79 | if ifreturn: 80 | return(c) 81 | 82 | # aliases 83 | interpolate = _spline.interpolate 84 | interpolate_2d = _spline.interpolate_2d 85 | 86 | if __name__ == '__main__': 87 | 88 | # 1D interpolation 89 | f = lambda x: x**2 90 | 91 | a = -1 92 | b = 1 93 | n = 49 # there are n + 1 grid points (0,..., n) 94 | 95 | h = (b - a) / n 96 | grid = arange(n + 1) * h + a 97 | 98 | y = f(grid) 99 | alpha = 0 100 | beta = 0 101 | 102 | c = cal_coefs(a, b, y) 103 | 104 | grid_hat = linspace(a, b, 100) 105 | fhat = array([interpolate(x, a, b, c) for x in grid_hat]) 106 | 107 | from matplotlib import pyplot as plt 108 | line_actual = plt.plot(grid_hat, f(grid_hat), label='actual') 109 | line_approx = plt.plot(grid_hat, fhat, '-.', label='interpolated') 110 | plt.setp(line_actual, linewidth=1, linestyle='--') 111 | plt.setp(line_approx, linewidth=2, linestyle='-.') 112 | plt.legend() 113 | plt.show() 114 | 115 | if not "2D interpolation": 116 | # 2D interpolation 117 | f2d = lambda x, z: x**2 + 2*x + 1 + z ** 0.5 + 3 * z 118 | 119 | a1, a2 = 0, 0 120 | b1, b2 = 1, 1 121 | n1, n2 = 49, 39 # n + 1 grid points (0,..., n) 122 | 123 | h1, h2 = (b1 - a1)/n1, (b2 - a2)/n2 124 | grid_x = arange(n1 + 1) * h1 + a1 125 | grid_z = arange(n2 + 1) * h2 + a2 126 | 127 | y = zeros((n1 + 1, n2 + 1)) 128 | 129 | for i, x in enumerate(grid_x): 130 | for j, z in enumerate(grid_z): 131 | y[i, j] = f2d(x, z) 132 | 133 | alpha = 0 134 | beta = 0 135 | 136 | c_tmp = zeros((n1 + 3, n2 + 1)) 137 | cal_coefs(a1, b1, y, c_tmp) 138 | 139 | c = zeros((n1 + 3, n2 + 3)) 140 | # NOTE: here you have to pass c_tmp.T and c.T 141 | cal_coefs(a2, b2, c_tmp.T, c.T) 142 | 143 | fhat = zeros((n1 + 1, n2 + 1)) 144 | for i, x in enumerate(grid_x): 145 | for j, z in enumerate(grid_z): 146 | fhat[i, j] = interpolate_2d(x, z, a1, b1, a2, b2, c) 147 | 148 | real_val = zeros((n1 + 1, n2 + 1)) 149 | for i, x in enumerate(grid_x): 150 | for j, z in enumerate(grid_z): 151 | real_val[i, j] = f2d(x, z) 152 | 153 | from mayavi import mlab 154 | 155 | def draw_3d(grid_x, grid_y, fval, title='pi'): 156 | mlab.figure() 157 | mlab.surf(grid_x, grid_y, fval)#, warp_scale="auto") 158 | mlab.axes(xlabel='x', ylabel='z', zlabel=title) 159 | mlab.orientation_axes(xlabel='x', ylabel='z', zlabel=title) 160 | mlab.title(title) 161 | 162 | draw_3d(grid_x, grid_z, fhat, title='interpolated') 163 | draw_3d(grid_x, grid_z, real_val, title='real') 164 | mlab.show() 165 | -------------------------------------------------------------------------------- /fast_cubic_spline.pyx: -------------------------------------------------------------------------------- 1 | # Fast-Cubic-Spline-Python provides an implementation of 1D and 2D fast spline 2 | # interpolation algorithm (Habermann and Kindermann 2007) in Python. 3 | # Copyright (C) 2012, 2013 Joon H. Ro 4 | 5 | # This file is part of Fast-Cubic-Spline-Python. 6 | 7 | # Fast-Cubic-Spline-Python is free software: you can redistribute it and/or modify 8 | # it under the terms of the GNU General Public License as published by 9 | # the Free Software Foundation, either version 3 of the License, or 10 | # (at your option) any later version. 11 | 12 | # Fast-Cubic-Spline-Python is distributed in the hope that it will be useful, 13 | # but WITHOUT ANY WARRANTY; without even the implied warranty of 14 | # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 15 | # GNU General Public License for more details. 16 | 17 | # You should have received a copy of the GNU General Public License 18 | # along with this program. If not, see . 19 | 20 | 21 | #cython: boundscheck = False 22 | #cython: wraparound = False 23 | #cython: cdivision = True 24 | 25 | from __future__ import division 26 | import numpy as np 27 | cimport numpy as np 28 | cimport cython 29 | 30 | from libc.math cimport fabs, fmin 31 | from cython.parallel import prange 32 | 33 | # "ctypedef" assigns a corresponding compile-time type to DTYPE_t. For 34 | # every type in the numpy module there's a corresponding compile-time 35 | # type with a _t-suffix. 36 | 37 | ''' 38 | Cubic spline interpolation using Habermann and Kindermann (2007)'s algorithm 39 | ''' 40 | #---------------------------------------------------------------------- 41 | cdef double Pi(double t) nogil: 42 | cdef: 43 | double abs_t = fabs(t) 44 | if abs_t <= 1: 45 | return(4 - 6 * abs_t**2 + 3 * abs_t **3) 46 | elif abs_t <= 2: 47 | return((2 - abs_t)**3) 48 | else: 49 | return(0) 50 | 51 | #---------------------------------------------------------------------- 52 | cdef double u(double x, int k, double a, double h) nogil: 53 | return(Pi((x - a)/h - (k - 2))) 54 | 55 | #---------------------------------------------------------------------- 56 | def interpolate(double x, 57 | double a, double b, 58 | double[:] c, 59 | ): 60 | ''' 61 | Return interpolated function value at x 62 | 63 | Parameters 64 | ---------- 65 | x : float 66 | The value where the function will be approximated at 67 | a : double 68 | Lower bound of the grid 69 | b : double 70 | Upper bound of the grid 71 | c : ndarray 72 | Coefficients of spline 73 | 74 | Returns 75 | ------- 76 | out : float 77 | Approximated function value at x 78 | ''' 79 | return(_interpolate(x, a, b, c)) 80 | 81 | cdef double _interpolate(double x, 82 | double a, double b, 83 | double[:] c, 84 | ) nogil: 85 | 86 | cdef: 87 | int n = c.shape[0] - 3 88 | double h = (b - a)/n 89 | int l = ((x - a)//h) + 1 90 | int m = (fmin(l + 3, n + 3)) 91 | int i1 92 | double s = 0 93 | 94 | for i1 in xrange(l, m + 1): 95 | s += c[i1 - 1] * u(x, i1, a, h) 96 | 97 | return(s) 98 | 99 | #---------------------------------------------------------------------- 100 | def interpolate_2d(double x, double y, 101 | double a1, double b1, 102 | double a2, double b2, 103 | double[:, :] c, 104 | ): 105 | ''' 106 | Return interpolated function value at x 107 | 108 | Parameters 109 | ---------- 110 | x, y : float 111 | The values where the function will be approximated at 112 | a1, b1 : double 113 | Lower and upper bounds of the grid for x 114 | a2, b2 : double 115 | Lower and upper bounds of the grid for y 116 | c : ndarray 117 | Coefficients of spline 118 | 119 | Returns 120 | ------- 121 | out : float 122 | Approximated function value at (x, y) 123 | ''' 124 | return(_interpolate_2d(x, y, a1, b1, a2, b2, c)) 125 | 126 | cdef double _interpolate_2d(double x, double y, 127 | double a1, double b1, 128 | double a2, double b2, 129 | double[:, :] c, 130 | ) nogil: 131 | cdef: 132 | int n1 = c.shape[0] - 3 133 | int n2 = c.shape[1] - 3 134 | double h1 = (b1 - a1)/n1 135 | double h2 = (b2 - a2)/n2 136 | int l1 = ((x - a1)//h1) + 1 137 | int l2 = ((y - a2)//h2) + 1 138 | int m1 = (fmin(l1 + 3, n1 + 3)) 139 | int m2 = (fmin(l2 + 3, n2 + 3)) 140 | int i1, i2 141 | double s = 0 142 | double u_x, u_y 143 | 144 | for i1 in xrange(l1, m1 + 1): 145 | u_x = u(x, i1, a1, h1) 146 | for i2 in xrange(l2, m2 + 1): 147 | u_y = u(y, i2, a2, h2) 148 | s += c[i1 - 1, i2 - 1] * u_x * u_y 149 | 150 | return(s) 151 | -------------------------------------------------------------------------------- /setup.py: -------------------------------------------------------------------------------- 1 | from distutils.core import setup 2 | from Cython.Distutils import Extension 3 | from Cython.Distutils import build_ext 4 | import os 5 | import numpy 6 | 7 | ext_modules = [] 8 | 9 | ext_modules.append(Extension("fast_cubic_spline", 10 | ["fast_cubic_spline.pyx"], 11 | libraries=["m"], 12 | extra_compile_args=['-fopenmp'], 13 | extra_link_args=['-fopenmp'], 14 | ) 15 | ) 16 | 17 | setup(include_dirs=[numpy.get_include()], 18 | cmdclass={'build_ext': build_ext}, 19 | ext_modules=ext_modules, 20 | ) 21 | --------------------------------------------------------------------------------