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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).

Sample Code

   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     .. [1] 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     .. [2] 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[0] - np.abs( y[1:half_window+1][::-1] - y[0] )
  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')

Code explanation

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).

Figure

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[1]>xvals[0] : #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:

  1. 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".
  2. 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.

  3. 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[0]) * (dy**exp[1])
  32 
  33     # pad input array with appropriate values at the four borders
  34     new_shape = z.shape[0] + 2*half_size, z.shape[1] + 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)[0].reshape((window_size, -1))
  68         return scipy.signal.fftconvolve(Z, m, mode='valid')
  69     elif derivative == 'col':
  70         c = np.linalg.pinv(A)[1].reshape((window_size, -1))
  71         return scipy.signal.fftconvolve(Z, -c, mode='valid')
  72     elif derivative == 'row':
  73         r = np.linalg.pinv(A)[2].reshape((window_size, -1))
  74         return scipy.signal.fftconvolve(Z, -r, mode='valid')
  75     elif derivative == 'both':
  76         c = np.linalg.pinv(A)[1].reshape((window_size, -1))
  77         r = np.linalg.pinv(A)[2].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)

Original.pdf Original data Original+noise.pdf Original data + noise Original+noise+filtered.pdf (Original data + noise) filtered

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.



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