[SciPy-User] deconvolution of 1-D signals

[David Baddeley]

Hi Ralf,

I do a reasonable amount of (2 & 3D) deconvolution of microscopy images and the 
method I use depends quite a lot on the exact properties of the signal. You can 
usually get away with fft based convolutions even if your signal is not periodic 
as long as your kernel is significantly smaller than the signal extent.

As Joe mentioned, for a noisy signal convolving with the inverse or performing 
fourier domain division doesn't work as you end up amplifying high frequency 
noise components. You thus need some form of regularisation. The thresholding of 
fourier components that Joe suggests does this, but you might also want to 
explore more sophisticated options, the simplest of which is probably Wiener 
filtering (http://en.wikipedia.org/wiki/Wiener_deconvolution). 

If you've got a signal which is constrained to be positive, it's often useful to 
introduce a positivity constraint on the deconvolution result which generally 
means you need an iterative algorithm. The choice of algorithm should also 
depend on the type of noise that is present in your signal -  my image data is 
constrained to be +ve and typically has either Poisson or a mixture of Poisson 
and Gaussian noise and I use either the Richardson-Lucy or a weighted version of 
ICTM (Iterative Constrained Tikhonov-Miller) algorithm. I can provide more 
details of these if required.

cheers,
David






________________________________
From: Ralf Gommers <ralf.gommers at googlemail.com>
To: SciPy Users List <scipy-user at scipy.org>
Sent: Mon, 1 August, 2011 5:56:49 AM
Subject: [SciPy-User] deconvolution of 1-D signals

Hi,

For a measured signal that is the convolution of a real signal with a response 
function, plus measurement noise on top, I want to recover the real signal. 
Since I know what the response function is and the noise is high-frequency 
compared to the real signal, a straightforward approach is to smooth the 
measured signal (or fit a spline to it), then remove the response function by 
deconvolution. See example code below.

Can anyone point me towards code that does the deconvolution efficiently? 
Perhaps signal.deconvolve would do the trick, but I can't seem to make it work 
(except for directly on the output of np.convolve(y, window, mode='valid')).

Thanks,
Ralf


import numpy as np
from scipy import interpolate, signal
import matplotlib.pyplot as plt

# Real signal
x = np.linspace(0, 10, num=201)
y = np.sin(x + np.pi/5)

# Noisy signal
mode = 'valid'
window_len = 11.
window = np.ones(window_len) / window_len
y_meas = np.convolve(y, window, mode=mode)
y_meas += 0.2 * np.random.rand(y_meas.size) - 0.1
if mode == 'full':
    xstep = x[1] - x[0]
    x_meas = np.concatenate([ \
        np.linspace(x[0] - window_len//2 * xstep, x[0] - xstep, 
num=window_len//2),
        x,
        np.linspace(x[-1] + xstep, x[-1] + window_len//2 * xstep, 
num=window_len//2)])
elif mode == 'valid':
    x_meas = x[window_len//2:-window_len//2+1]
elif mode == 'same':
    x_meas = x

# Approximating spline
xs = np.linspace(0, 10, num=500)
knots = np.array([1, 3, 5, 7, 9])
tck = interpolate.splrep(x_meas, y_meas, s=0, k=3, t=knots, task=-1)
ys = interpolate.splev(xs, tck, der=0)

# Find (low-frequency part of) original signal by deconvolution of smoothed
# measured signal and known window.
y_deconv = signal.deconvolve(ys, window)[0]  #FIXME

# Plot all signals
fig = plt.figure()
ax = fig.add_subplot(111)

ax.plot(x, y, 'b-', label="Original signal")
ax.plot(x_meas, y_meas, 'r-', label="Measured, noisy signal")
ax.plot(xs, ys, 'g-', label="Approximating spline")
ax.plot(xs[window.size//2-1:-window.size//2], y_deconv, 'k-',
        label="signal.deconvolve result")
ax.set_ylim([-1.2, 2])
ax.legend()

plt.show()
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