Source code for colour.algebra.interpolation

#!/usr/bin/env python
# -*- coding: utf-8 -*-

"""
Interpolation
=============

Defines classes for interpolating variables.

-   :class:`LinearInterpolator`: 1-D function linear interpolation.
-   :class:`SpragueInterpolator`: 1-D function fifth-order polynomial
    interpolation.
-   :class:`CubicSplineInterpolator`: 1-D function cubic spline interpolation.
-   :class:`PchipInterpolator`: 1-D function piecewise cube Hermite
    interpolation.
"""

from __future__ import division, unicode_literals

import bisect
import numpy as np

from colour.utilities import (
    as_numeric,
    is_scipy_installed,
    is_uniform,
    steps,
    warning)

__author__ = 'Colour Developers'
__copyright__ = 'Copyright (C) 2013 - 2015 - Colour Developers'
__license__ = 'New BSD License - http://opensource.org/licenses/BSD-3-Clause'
__maintainer__ = 'Colour Developers'
__email__ = 'colour-science@googlegroups.com'
__status__ = 'Production'

__all__ = ['LinearInterpolator',
           'SpragueInterpolator',
           'CubicSplineInterpolator',
           'PchipInterpolator']


[docs]class LinearInterpolator(object): """ Linearly interpolates a 1-D function. Parameters ---------- x : ndarray Independent :math:`x` variable values corresponding with :math:`y` variable. y : ndarray Dependent and already known :math:`y` variable values to interpolate. Methods ------- __call__ Notes ----- This class is a wrapper around *numpy.interp* definition. See Also -------- SpragueInterpolator Examples -------- Interpolating a single numeric variable: >>> y = np.array([5.9200, ... 9.3700, ... 10.8135, ... 4.5100, ... 69.5900, ... 27.8007, ... 86.0500]) >>> x = np.arange(len(y)) >>> f = LinearInterpolator(x, y) >>> # Doctests ellipsis for Python 2.x compatibility. >>> f(0.5) # doctest: +ELLIPSIS 7.64... Interpolating an *array_like* variable: >>> f([0.25, 0.75]) array([ 6.7825, 8.5075]) """ def __init__(self, x=None, y=None): self.__x = None self.x = x self.__y = None self.y = y self.__validate_dimensions() @property def x(self): """ Property for **self.__x** private attribute. Returns ------- array_like self.__x """ return self.__x @x.setter
[docs] def x(self, value): """ Setter for **self.__x** private attribute. Parameters ---------- value : array_like Attribute value. """ if value is not None: value = np.atleast_1d(value).astype(np.float_) assert value.ndim == 1, ( '"x" independent variable must have exactly one dimension!') self.__x = value
@property def y(self): """ Property for **self.__y** private attribute. Returns ------- array_like self.__y """ return self.__y @y.setter
[docs] def y(self, value): """ Setter for **self.__y** private attribute. Parameters ---------- value : array_like Attribute value. """ if value is not None: value = np.atleast_1d(value).astype(np.float_) assert value.ndim == 1, ( '"y" dependent variable must have exactly one dimension!') self.__y = value
[docs] def __call__(self, x): """ Evaluates the interpolating polynomial at given point(s). Parameters ---------- x : numeric or array_like Point(s) to evaluate the interpolant at. Returns ------- float or ndarray Interpolated value(s). """ x = np.atleast_1d(x).astype(np.float_) xi = as_numeric(self.__evaluate(x)) return xi
def __evaluate(self, x): """ Performs the interpolating polynomial evaluation at given points. Parameters ---------- x : ndarray Points to evaluate the interpolant at. Returns ------- ndarray Interpolated points values. """ self.__validate_dimensions() self.__validate_interpolation_range(x) return np.interp(x, self.__x, self.__y) def __validate_dimensions(self): """ Validates variables dimensions to be the same. """ if len(self.__x) != len(self.__y): raise ValueError( ('"x" independent and "y" dependent variables have different ' 'dimensions: "{0}", "{1}"').format(len(self.__x), len(self.__y))) def __validate_interpolation_range(self, x): """ Validates given point to be in interpolation range. """ below_interpolation_range = x < self.__x[0] above_interpolation_range = x > self.__x[-1] if below_interpolation_range.any(): raise ValueError('"{0}" is below interpolation range.'.format(x)) if above_interpolation_range.any(): raise ValueError('"{0}" is above interpolation range.'.format(x))
[docs]class SpragueInterpolator(object): """ Constructs a fifth-order polynomial that passes through :math:`y` dependent variable. Sprague (1880) method is recommended by the *CIE* for interpolating functions having a uniformly spaced independent variable. Parameters ---------- x : array_like Independent :math:`x` variable values corresponding with :math:`y` variable. y : array_like Dependent and already known :math:`y` variable values to interpolate. Methods ------- __call__ See Also -------- LinearInterpolator Notes ----- The minimum number :math:`k` of data points required along the interpolation axis is :math:`k=6`. References ---------- .. [1] CIE TC 1-38. (2005). 9.2.4 Method of interpolation for uniformly spaced independent variable. In CIE 167:2005 Recommended Practice for Tabulating Spectral Data for Use in Colour Computations (pp. 1–27). ISBN:978-3-901-90641-1 .. [2] Westland, S., Ripamonti, C., & Cheung, V. (2012). Interpolation Methods. In Computational Colour Science Using MATLAB (2nd ed., pp. 29–37). ISBN:978-0-470-66569-5 Examples -------- Interpolating a single numeric variable: >>> y = np.array([5.9200, ... 9.3700, ... 10.8135, ... 4.5100, ... 69.5900, ... 27.8007, ... 86.0500]) >>> x = np.arange(len(y)) >>> f = SpragueInterpolator(x, y) >>> f(0.5) # doctest: +ELLIPSIS 7.2185025... Interpolating an *array_like* variable: >>> f([0.25, 0.75]) # doctest: +ELLIPSIS array([ 6.7295161..., 7.8140625...]) """ SPRAGUE_C_COEFFICIENTS = np.array( [[884, -1960, 3033, -2648, 1080, -180], [508, -540, 488, -367, 144, -24], [-24, 144, -367, 488, -540, 508], [-180, 1080, -2648, 3033, -1960, 884]]) """ Defines the coefficients used to generate extra points for boundaries interpolation. SPRAGUE_C_COEFFICIENTS : array_like, (4, 6) References ---------- .. [3] CIE TC 1-38. (2005). Table V. Values of the c-coefficients of Equ.s 6 and 7. In CIE 167:2005 Recommended Practice for Tabulating Spectral Data for Use in Colour Computations (p. 19). ISBN:978-3-901-90641-1 """ def __init__(self, x=None, y=None): self.__xp = None self.__yp = None self.__x = None self.x = x self.__y = None self.y = y self.__validate_dimensions() @property def x(self): """ Property for **self.__x** private attribute. Returns ------- array_like self.__x """ return self.__x @x.setter
[docs] def x(self, value): """ Setter for **self.__x** private attribute. Parameters ---------- value : array_like Attribute value. """ if value is not None: value = np.atleast_1d(value).astype(np.float_) assert value.ndim == 1, ( '"x" independent variable must have exactly one dimension!') assert is_uniform(value), ( '"x" independent variable is not uniform!') value_steps = steps(value)[0] xp1 = value[0] - value_steps * 2 xp2 = value[0] - value_steps xp3 = value[-1] + value_steps xp4 = value[-1] + value_steps * 2 self.__xp = np.concatenate(((xp1, xp2), value, (xp3, xp4))) self.__x = value
@property def y(self): """ Property for **self.__y** private attribute. Returns ------- array_like self.__y """ return self.__y @y.setter
[docs] def y(self, value): """ Setter for **self.__y** private attribute. Parameters ---------- value : array_like Attribute value. """ if value is not None: value = np.atleast_1d(value).astype(np.float_) assert value.ndim == 1, ( '"y" dependent variable must have exactly one dimension!') assert len(value) >= 6, ( '"y" dependent variable values count must be in domain [6:]!') yp1 = np.ravel((np.dot( self.SPRAGUE_C_COEFFICIENTS[0], np.array(value[0:6]).reshape((6, 1)))) / 209)[0] yp2 = np.ravel((np.dot( self.SPRAGUE_C_COEFFICIENTS[1], np.array(value[0:6]).reshape((6, 1)))) / 209)[0] yp3 = np.ravel((np.dot( self.SPRAGUE_C_COEFFICIENTS[2], np.array(value[-6:]).reshape((6, 1)))) / 209)[0] yp4 = np.ravel((np.dot( self.SPRAGUE_C_COEFFICIENTS[3], np.array(value[-6:]).reshape((6, 1)))) / 209)[0] self.__yp = np.concatenate(((yp1, yp2), value, (yp3, yp4))) self.__y = value
[docs] def __call__(self, x): """ Evaluates the interpolating polynomial at given point(s). Parameters ---------- x : numeric or array_like Point(s) to evaluate the interpolant at. Returns ------- numeric or ndarray Interpolated value(s). """ try: return np.array([self.__evaluate(element) for element in x]) except TypeError: return self.__evaluate(x)
def __evaluate(self, x): """ Performs the interpolating polynomial evaluation at given point. Parameters ---------- x : numeric Point to evaluate the interpolant at. Returns ------- float Interpolated point values. """ self.__validate_dimensions() self.__validate_interpolation_range(x) if x in self.__x: return self.__y[np.where(self.__x == x)][0] i = bisect.bisect(self.__xp, x) - 1 X = (x - self.__xp[i]) / (self.__xp[i + 1] - self.__xp[i]) r = self.__yp a0p = r[i] a1p = ((2 * r[i - 2] - 16 * r[i - 1] + 16 * r[i + 1] - 2 * r[i + 2]) / 24) a2p = ((-r[i - 2] + 16 * r[i - 1] - 30 * r[i] + 16 * r[i + 1] - r[i + 2]) / 24) a3p = ((-9 * r[i - 2] + 39 * r[i - 1] - 70 * r[i] + 66 * r[i + 1] - 33 * r[i + 2] + 7 * r[i + 3]) / 24) a4p = ((13 * r[i - 2] - 64 * r[i - 1] + 126 * r[i] - 124 * r[i + 1] + 61 * r[i + 2] - 12 * r[i + 3]) / 24) a5p = ((-5 * r[i - 2] + 25 * r[i - 1] - 50 * r[i] + 50 * r[i + 1] - 25 * r[i + 2] + 5 * r[i + 3]) / 24) y = (a0p + a1p * X + a2p * X ** 2 + a3p * X ** 3 + a4p * X ** 4 + a5p * X ** 5) return y def __validate_dimensions(self): """ Validates variables dimensions to be the same. """ if len(self.__x) != len(self.__y): raise ValueError( ('"x" independent and "y" dependent variables have different ' 'dimensions: "{0}", "{1}"').format(len(self.__x), len(self.__y))) def __validate_interpolation_range(self, x): """ Validates given point to be in interpolation range. """ below_interpolation_range = x < self.__x[0] above_interpolation_range = x > self.__x[-1] if below_interpolation_range.any(): raise ValueError('"{0}" is below interpolation range.'.format(x)) if above_interpolation_range.any(): raise ValueError('"{0}" is above interpolation range.'.format(x))
if is_scipy_installed(): from scipy.interpolate import PchipInterpolator, interp1d
[docs] class CubicSplineInterpolator(interp1d): """ Interpolates a 1-D function using cubic spline interpolation. Notes ----- This class is a wrapper around *scipy.interpolate.interp1d* class. """ def __init__(self, *args, **kwargs): # TODO: Implements proper wrapper to ensure return values # consistency and avoid having to cast to numeric in # :meth:`SpectralPowerDistribution.interpolate` method. super(CubicSplineInterpolator, self).__init__( kind='cubic', *args, **kwargs)
else: warning(('"scipy.interpolate.PchipInterpolator" and ' '"scipy.interpolate.interp1d" interpolators are not available, ' 'using "LinearInterpolator" instead!')) PchipInterpolator = CubicSplineInterpolator = LinearInterpolator