Source code for spike.Algo.savitzky_golay

#!/usr/bin/env python 
# encoding: utf-8

"""
Smooth (and optionally differentiate) data with a Savitzky-Golay filter.

CODE from http://www.scipy.org/Cookbook/SavitzkyGolay
    adapted by M-A Delsuc, august 2011
"""
from __future__ import print_function
import numpy as np
import scipy.signal

[docs]def savitzky_golay(y, window_size, order, deriv=0): r"""Smooth (and optionally differentiate) data with a Savitzky-Golay filter. The Savitzky-Golay filter removes high frequency noise from data. It has the advantage of preserving the original shape and features of the signal better than other types of filtering approaches, such as moving averages techniques. Parameters ---------- y : array_like, shape (N,) the values of the time history of the signal. window_size : int the length of the window. Must be an odd integer number. order : int the order of the polynomial used in the filtering. Must be less than `window_size` - 1. deriv: int the order of the derivative to compute (default = 0 means only smoothing) Returns ------- ys : ndarray, shape (N) the smoothed signal (or it's n-th derivative). Notes ----- The Savitzky-Golay is a type of low-pass filter, particularly suited for smoothing noisy data. The main idea behind this approach is to make for each point a least-square fit with a polynomial of high order over a odd-sized window centered at the point. Examples -------- t = np.linspace(-4, 4, 500) y = np.exp( -t**2 ) + np.random.normal(0, 0.05, t.shape) ysg = savitzky_golay(y, window_size=31, order=4) import matplotlib.pyplot as plt plt.plot(t, y, label='Noisy signal') plt.plot(t, np.exp(-t**2), 'k', lw=1.5, label='Original signal') plt.plot(t, ysg, 'r', label='Filtered signal') plt.legend() plt.show() References ---------- .. [1] A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analytical Chemistry, 1964, 36 (8), pp 1627-1639. .. [2] Numerical Recipes 3rd Edition: The Art of Scientific Computing W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery Cambridge University Press ISBN-13: 9780521880688 """ m = sgolay_coef(window_size, order, deriv=deriv) return sgolay_comp(y, m, window_size)
[docs]def sgolay_coef(window_size, order, deriv=0): """compute savistki-golay coefficients""" try: window_size = np.abs(np.int(window_size)) order = np.abs(np.int(order)) except(ValueError): raise(ValueError("window_size and order have to be of type int")) # if window_size % 2 != 1 or window_size < 1: raise(TypeError("window_size size must be a positive odd number")) if window_size < order + 2: raise(TypeError("window_size is too small for the polynomials order")) order_range = range(order+1) half_window = (window_size -1) // 2 # precompute coefficients b = np.mat([[k**i for i in order_range] for k in range(-half_window, half_window+1)]) m = np.linalg.pinv(b).A[deriv] return m
[docs]def sgolay_comp(y, m, window_size): """apply savistki-golay filter on y from previously computed savistki-golay coefficients : m""" # pad the signal at the extremes with # values taken from the signal itself half_window = (window_size -1) // 2 firstvals = y[0] - np.abs( y[1:half_window+1][::-1] - y[0] ) lastvals = y[-1] + np.abs(y[-half_window-1:-1][::-1] - y[-1]) y = np.concatenate((firstvals, y, lastvals)) return np.convolve( m, y, mode='valid')
#####################################################
[docs]def savitzky_golay2D ( z, window_size, order, derivative=None): """ realises Savitzky-Golay smoothing in 2D. see savitzky_golay() for more information """ # number of terms in the polynomial expression n_terms = ( order + 1 ) * ( order + 2) / 2.0 if window_size % 2 == 0: raise ValueError('window_size must be odd') if window_size**2 < n_terms: raise ValueError('order is too high for the window size') half_size = window_size // 2 # exponents of the polynomial. # p(x,y) = a0 + a1*x + a2*y + a3*x^2 + a4*y^2 + a5*x*y + ... # this line gives a list of two item tuple. Each tuple contains # the exponents of the k-th term. First element of tuple is for x # second element for y. # Ex. exps = [(0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...] exps = [ (k-n, n) for k in range(order+1) for n in range(k+1) ] # coordinates of points ind = np.arange(-half_size, half_size+1, dtype=np.float64) dx = np.repeat( ind, window_size ) dy = np.tile( ind, [window_size, 1]).reshape(window_size**2, ) # build matrix of system of equation A = np.empty( (window_size**2, len(exps)) ) for i, exp in enumerate( exps ): A[:,i] = (dx**exp[0]) * (dy**exp[1]) # pad input array with appropriate values at the four borders new_shape = z.shape[0] + 2*half_size, z.shape[1] + 2*half_size Z = np.zeros( (new_shape) ) # top band band = z[0, :] Z[:half_size, half_size:-half_size] = band - np.abs( np.flipud( z[1:half_size+1, :] ) - band ) # bottom band band = z[-1, :] Z[-half_size:, half_size:-half_size] = band + np.abs( np.flipud( z[-half_size-1:-1, :] ) -band ) # left band band = np.tile( z[:,0].reshape(-1,1), [1,half_size]) Z[half_size:-half_size, :half_size] = band - np.abs( np.fliplr( z[:, 1:half_size+1] ) - band ) # right band band = np.tile( z[:,-1].reshape(-1,1), [1,half_size] ) Z[half_size:-half_size, -half_size:] = band + np.abs( np.fliplr( z[:, -half_size-1:-1] ) - band ) # central band Z[half_size:-half_size, half_size:-half_size] = z # top left corner band = z[0,0] Z[:half_size,:half_size] = band - np.abs( np.flipud(np.fliplr(z[1:half_size+1,1:half_size+1]) ) - band ) # bottom right corner band = z[-1,-1] Z[-half_size:,-half_size:] = band + np.abs( np.flipud(np.fliplr(z[-half_size-1:-1,-half_size-1:-1]) ) - band ) # top right corner band = Z[half_size,-half_size:] Z[:half_size,-half_size:] = band - np.abs( np.flipud(Z[half_size+1:2*half_size+1,-half_size:]) - band ) # bottom left corner band = Z[-half_size:,half_size].reshape(-1,1) Z[-half_size:,:half_size] = band - np.abs( np.fliplr(Z[-half_size:, half_size+1:2*half_size+1]) - band ) # solve system and convolve if derivative is None: m = np.linalg.pinv(A)[0].reshape((window_size, -1)) return scipy.signal.fftconvolve(Z, m, mode='valid') elif derivative == 'col': c = np.linalg.pinv(A)[1].reshape((window_size, -1)) return scipy.signal.fftconvolve(Z, -c, mode='valid') elif derivative == 'row': r = np.linalg.pinv(A)[2].reshape((window_size, -1)) return scipy.signal.fftconvolve(Z, -r, mode='valid') elif derivative == 'both': c = np.linalg.pinv(A)[1].reshape((window_size, -1)) r = np.linalg.pinv(A)[2].reshape((window_size, -1)) return scipy.signal.fftconvolve(Z, -r, mode='valid'), scipy.signal.fftconvolve(Z, -c, mode='valid')
[docs]def test_sg(): ######## test 1D ##################### # create some sample twoD data import matplotlib.pylab as plt x = np.linspace(-3,3,100) z = np.exp( -(x**2)) # add noise zn = z + np.random.normal( 0, 0.1, z.shape ) # filter it zf = savitzky_golay( zn, window_size=29, order=4) zfd = savitzky_golay( zn, window_size=29, order=4, deriv=1) # do some plotting plt.subplot(4,2,1) plt.plot(z) plt.subplot(4,2,3) plt.plot(zn) plt.subplot(4,2,5) plt.plot(zf) plt.subplot(4,2,7) plt.plot(zfd) ######## test 2D ##################### # create some sample twoD data x = np.linspace(-3,3,100) y = np.linspace(-3,3,100) X, Y = np.meshgrid(x,y) Z = np.exp( -(X**2+Y**2)) # add noise Zn = Z + np.random.normal( 0, 0.1, Z.shape ) # filter it Zf = savitzky_golay2D( Zn, window_size=29, order=4) Zfd = savitzky_golay2D( Zn, window_size=29, order=4, derivative="row") # do some plotting plt.subplot(4,2,2) plt.contour(Z) plt.subplot(4,2,4) plt.contour(Zn) plt.subplot(4,2,6) plt.contour(Zf) plt.subplot(4,2,8) plt.contour(Zfd) plt.show()
if __name__ == "__main__": test_sg()