Savitzky Golay Filtering
The Savitzky Golay filter is a particular type of low-pass filter, well adapted for data smoothing. For further information see: http://www.wire.tu-bs.de/OLDWEB/mameyer/cmr/savgol.pdf (or http://www.dalkescientific.com/writings/NBN/data/savitzky_golay.py for a pre-numpy implementation).
1 def savitzky_golay(y, window_size, order, deriv=0, rate=1): 2 r"""Smooth (and optionally differentiate) data with a Savitzky-Golay filter. 3 The Savitzky-Golay filter removes high frequency noise from data. 4 It has the advantage of preserving the original shape and 5 features of the signal better than other types of filtering 6 approaches, such as moving averages techniques. 7 Parameters 8 ---------- 9 y : array_like, shape (N,) 10 the values of the time history of the signal. 11 window_size : int 12 the length of the window. Must be an odd integer number. 13 order : int 14 the order of the polynomial used in the filtering. 15 Must be less then `window_size` - 1. 16 deriv: int 17 the order of the derivative to compute (default = 0 means only smoothing) 18 Returns 19 ------- 20 ys : ndarray, shape (N) 21 the smoothed signal (or it's n-th derivative). 22 Notes 23 ----- 24 The Savitzky-Golay is a type of low-pass filter, particularly 25 suited for smoothing noisy data. The main idea behind this 26 approach is to make for each point a least-square fit with a 27 polynomial of high order over a odd-sized window centered at 28 the point. 29 Examples 30 -------- 31 t = np.linspace(-4, 4, 500) 32 y = np.exp( -t**2 ) + np.random.normal(0, 0.05, t.shape) 33 ysg = savitzky_golay(y, window_size=31, order=4) 34 import matplotlib.pyplot as plt 35 plt.plot(t, y, label='Noisy signal') 36 plt.plot(t, np.exp(-t**2), 'k', lw=1.5, label='Original signal') 37 plt.plot(t, ysg, 'r', label='Filtered signal') 38 plt.legend() 39 plt.show() 40 References 41 ---------- 42 ..  A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of 43 Data by Simplified Least Squares Procedures. Analytical 44 Chemistry, 1964, 36 (8), pp 1627-1639. 45 ..  Numerical Recipes 3rd Edition: The Art of Scientific Computing 46 W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery 47 Cambridge University Press ISBN-13: 9780521880688 48 """ 49 import numpy as np 50 from math import factorial 51 52 try: 53 window_size = np.abs(np.int(window_size)) 54 order = np.abs(np.int(order)) 55 except ValueError, msg: 56 raise ValueError("window_size and order have to be of type int") 57 if window_size % 2 != 1 or window_size < 1: 58 raise TypeError("window_size size must be a positive odd number") 59 if window_size < order + 2: 60 raise TypeError("window_size is too small for the polynomials order") 61 order_range = range(order+1) 62 half_window = (window_size -1) // 2 63 # precompute coefficients 64 b = np.mat([[k**i for i in order_range] for k in range(-half_window, half_window+1)]) 65 m = np.linalg.pinv(b).A[deriv] * rate**deriv * factorial(deriv) 66 # pad the signal at the extremes with 67 # values taken from the signal itself 68 firstvals = y - np.abs( y[1:half_window+1][::-1] - y ) 69 lastvals = y[-1] + np.abs(y[-half_window-1:-1][::-1] - y[-1]) 70 y = np.concatenate((firstvals, y, lastvals)) 71 return np.convolve( m[::-1], y, mode='valid')
In lines 61-62 the coefficients of the local least-square polynomial fit are pre-computed. These will be used later at line 68, where they will be correlated with the signal. To prevent spurious results at the extremes of the data, the signal is padded at both ends with its mirror image, (lines 65-67).
CD-spectrum of a protein. Black: raw data. Red: filter applied
A wrapper for cyclic voltammetry data
One of the most popular applications of S-G filter, apart from smoothing UV-VIS and IR spectra, is smoothing of curves obtained in electroanalytical experiments. In cyclic voltammetry, voltage (being the abcissa) changes like a triangle wave. And in the signal there are cusps at the turning points (at switching potentials) which should never be smoothed. In this case, Savitzky-Golay smoothing should be done piecewise, ie. separately on pieces monotonic in x:
def savitzky_golay_piecewise(xvals, data, kernel=11, order =4): turnpoint=0 last=len(xvals) if xvals>xvals : #x is increasing? for i in range(1,last) : #yes if xvals[i]<xvals[i-1] : #search where x starts to fall turnpoint=i break else: #no, x is decreasing for i in range(1,last) : #search where it starts to rise if xvals[i]>xvals[i-1] : turnpoint=i break if turnpoint==0 : #no change in direction of x return savitzky_golay(data, kernel, order) else: #smooth the first piece firstpart=savitzky_golay(data[0:turnpoint],kernel,order) #recursively smooth the rest rest=savitzky_golay_piecewise(xvals[turnpoint:], data[turnpoint:], kernel, order) return numpy.concatenate((firstpart,rest))
Two dimensional data smoothing and least-square gradient estimate
Savitsky-Golay filters can also be used to smooth two dimensional data affected by noise. The algorithm is exactly the same as for the one dimensional case, only the math is a bit more tricky. The basic algorithm is as follow:
- for each point of the two dimensional matrix extract a sub-matrix, centered at that point and with a size equal to an odd number "window_size".
- for this sub-matrix compute a least-square fit of a polynomial surface, defined as
p(x,y) = a0 + a1*x + a2*y + a3*x2 + a4*y2 + a5*x*y + ... . Note that x and y are equal to zero at the central point.
- replace the initial central point with the value computed with the fit.
Note that because the fit coefficients are linear with respect to the data spacing, they can pre-computed for efficiency. Moreover, it is important to appropriately pad the borders of the data, with a mirror image of the data itself, so that the evaluation of the fit at the borders of the data can happen smoothly.
Here is the code for two dimensional filtering.
1 def sgolay2d ( z, window_size, order, derivative=None): 2 """ 3 """ 4 # number of terms in the polynomial expression 5 n_terms = ( order + 1 ) * ( order + 2) / 2.0 6 7 if window_size % 2 == 0: 8 raise ValueError('window_size must be odd') 9 10 if window_size**2 < n_terms: 11 raise ValueError('order is too high for the window size') 12 13 half_size = window_size // 2 14 15 # exponents of the polynomial. 16 # p(x,y) = a0 + a1*x + a2*y + a3*x^2 + a4*y^2 + a5*x*y + ... 17 # this line gives a list of two item tuple. Each tuple contains 18 # the exponents of the k-th term. First element of tuple is for x 19 # second element for y. 20 # Ex. exps = [(0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...] 21 exps = [ (k-n, n) for k in range(order+1) for n in range(k+1) ] 22 23 # coordinates of points 24 ind = np.arange(-half_size, half_size+1, dtype=np.float64) 25 dx = np.repeat( ind, window_size ) 26 dy = np.tile( ind, [window_size, 1]).reshape(window_size**2, ) 27 28 # build matrix of system of equation 29 A = np.empty( (window_size**2, len(exps)) ) 30 for i, exp in enumerate( exps ): 31 A[:,i] = (dx**exp) * (dy**exp) 32 33 # pad input array with appropriate values at the four borders 34 new_shape = z.shape + 2*half_size, z.shape + 2*half_size 35 Z = np.zeros( (new_shape) ) 36 # top band 37 band = z[0, :] 38 Z[:half_size, half_size:-half_size] = band - np.abs( np.flipud( z[1:half_size+1, :] ) - band ) 39 # bottom band 40 band = z[-1, :] 41 Z[-half_size:, half_size:-half_size] = band + np.abs( np.flipud( z[-half_size-1:-1, :] ) -band ) 42 # left band 43 band = np.tile( z[:,0].reshape(-1,1), [1,half_size]) 44 Z[half_size:-half_size, :half_size] = band - np.abs( np.fliplr( z[:, 1:half_size+1] ) - band ) 45 # right band 46 band = np.tile( z[:,-1].reshape(-1,1), [1,half_size] ) 47 Z[half_size:-half_size, -half_size:] = band + np.abs( np.fliplr( z[:, -half_size-1:-1] ) - band ) 48 # central band 49 Z[half_size:-half_size, half_size:-half_size] = z 50 51 # top left corner 52 band = z[0,0] 53 Z[:half_size,:half_size] = band - np.abs( np.flipud(np.fliplr(z[1:half_size+1,1:half_size+1]) ) - band ) 54 # bottom right corner 55 band = z[-1,-1] 56 Z[-half_size:,-half_size:] = band + np.abs( np.flipud(np.fliplr(z[-half_size-1:-1,-half_size-1:-1]) ) - band ) 57 58 # top right corner 59 band = Z[half_size,-half_size:] 60 Z[:half_size,-half_size:] = band - np.abs( np.flipud(Z[half_size+1:2*half_size+1,-half_size:]) - band ) 61 # bottom left corner 62 band = Z[-half_size:,half_size].reshape(-1,1) 63 Z[-half_size:,:half_size] = band - np.abs( np.fliplr(Z[-half_size:, half_size+1:2*half_size+1]) - band ) 64 65 # solve system and convolve 66 if derivative == None: 67 m = np.linalg.pinv(A).reshape((window_size, -1)) 68 return scipy.signal.fftconvolve(Z, m, mode='valid') 69 elif derivative == 'col': 70 c = np.linalg.pinv(A).reshape((window_size, -1)) 71 return scipy.signal.fftconvolve(Z, -c, mode='valid') 72 elif derivative == 'row': 73 r = np.linalg.pinv(A).reshape((window_size, -1)) 74 return scipy.signal.fftconvolve(Z, -r, mode='valid') 75 elif derivative == 'both': 76 c = np.linalg.pinv(A).reshape((window_size, -1)) 77 r = np.linalg.pinv(A).reshape((window_size, -1)) 78 return scipy.signal.fftconvolve(Z, -r, mode='valid'), scipy.signal.fftconvolve(Z, -c, mode='valid')
Here is a demo
1 2 # create some sample twoD data 3 x = np.linspace(-3,3,100) 4 y = np.linspace(-3,3,100) 5 X, Y = np.meshgrid(x,y) 6 Z = np.exp( -(X**2+Y**2)) 7 8 # add noise 9 Zn = Z + np.random.normal( 0, 0.2, Z.shape ) 10 11 # filter it 12 Zf = sgolay2d( Zn, window_size=29, order=4) 13 14 # do some plotting 15 matshow(Z) 16 matshow(Zn) 17 matshow(Zf)
Gradient of a two-dimensional function
Since we have computed the best fitting interpolating polynomial surface it is easy to compute its gradient. This method of computing the gradient of a two dimensional function is quite robust, and partially hides the noise in the data, which strongly affects the differentiation operation. The maximum order of the derivative that can be computed obviously depends on the order of the polynomial used in the fitting.
The code provided above have an option derivative, which as of now allows to compute the first derivative of the 2D data. It can be "row"or "column", indicating the direction of the derivative, or "both", which returns the gradient.