deconvolution of 1-D signals

David Baddeley | 1 Aug 2011 02:51

Re: deconvolution of 1-D signals

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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