1 Aug 2011 09:03

## Re: deconvolution of 1-D signals

Hi Ralf,

5-15 times smaller would probably be fine, although you might want to watch the edges in the reconstruction - if they're at different dc levels you'll get edge artifacts (within ~ 1 kernel width of the edges). I'd tend to avoid spline filtering (or any form of noise reduction) before deconvolution, as this will also transform the data in a way not explained by the model you're using to deconvolve with.

Weiner filtering is a 2 liner ->

H = fft(kernel)
deconvolved = ifftshift(ifft(fft(signal)*np.conj(H)/(H*np.conj(H) + lambda**2)))

where lambda is your regularisation parameter, and white noise is assumed. There are various methods for choosing lambda optimally, but most people tend to use trial and error.

Iterative methods are typically ~1-2 orders of magnitude slower than a Weiner filter, but with fast fft libraries and modern computers still quite reasonable for modest data sizes (a 3D image stack of ~ 512x512x50 pixels will tend to be done in a bit under a minute, can't really comment on 1D data, but unless your signal is very long I'd expect it to be significantly quicker). Ffts scale with O(nlogn) so will generally dramatically outperform things based on a simple convolution or filtering approaches (O(n**2)) for large n. This might make an iterative approach using ffts faster than something like scipy.signal.deconvolve if your kernel is large.

cheers,
David

From: Ralf Gommers <ralf.gommers <at> googlemail.com>
To: David Baddeley <david_baddeley <at> yahoo.com.au>; SciPy Users List <scipy-user <at> scipy.org>
Sent: Mon, 1 August, 2011 6:03:10 PM
Subject: Re: [SciPy-User] deconvolution of 1-D signals

On Mon, Aug 1, 2011 at 2:51 AM, David Baddeley wrote:
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.

The kernel is typically about 5 to 15 times smaller than the signal extent, so I guess that may be problematic.

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 (

I'm aware of the problems with high frequency noise. This is why I tried the spline fitting - I figured that on a spline the deconvolution would be okay because the spline is very smooth. This should be fine for my data because the noise is much higher-frequency than the underlying signal, and the SNR is high to start with. But maybe there are better ways. I looked for a Python implementation of Wiener deconvolution but couldn't find one so quickly. Is there a package out there that has it?

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.

By constrained to be positive I'm guessing you mean monotonic? Otherwise I could just add a constant offset, but that's probably not what you mean. What's typically the speed penalty for an iterative method?

Ralf

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.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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Gmane