# Convolutive noise removal

I have the time domain signal $$u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t)$$ where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is known, but only numerically. I also have prior knowledge that $u(t)$ is a sum of a small number of sinusoids. How can I recover $u(t)$ from $u_o(t)$?

In the case where $\eta$ is not present, I can Fourier transform to obtain: $$\hat{u}_o(\xi) = \hat{u}(\xi)*l(\xi) + \sigma(\xi)$$ where $l$ is a Lorentzian. The deconvolution is easy to solve with basis pursuit: $$argmin |u|_1 \; subject \; to \; \|l*u - \hat{u_o} \|^2 \leq \mu$$ This ignores $\eta$ as well as our statistical knowledge of $\eta$. Are there ideas on how I can incorporate $\eta$ into my denoising model? Is there a different model I should look into?

edit: looks like I need to set up a MAP estimate for $f = u*\eta + \sigma$. I think I can sort it out when it's just $\eta$ or just $\sigma$.

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How does the distribution of $\eta(t)$ look like? Can it be approximated by some well-known distribution? –  Stanislav Jun 3 '12 at 22:16
We can not approximate it with a well-known distribution. It needs to be recorded each time we run an experiment. –  dranxo Jun 4 '12 at 1:38

A lot depends on $\hat{\eta}(\xi)$. When you convolve this with $\hat{u}$ and $l(\xi)$, you will lose the sparsity if $\hat{\eta}(\xi)$ has broad support.

Just how much do you know about the spectrum of $\eta(t)$? If you know it well enough, you can deconvolve it before trying your basis pursuit approach. If you don't know it very well, then you're in deep trouble.

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Yes, the $l$ and the $\eta$ hurt the sparsity. The $l$ is no problem since I have an explicit form for it. As for $\eta$, it's noise, all I know is the distribution of $\eta$. I suppose the problem I'm trying to solve is: remove additive noise + remove convolutive noise + remove convolution with $l$. –  dranxo Jun 3 '12 at 5:32
Let's be clear about "all I know is the distribution of $\eta$." Are you saying that you know a probability distribution for $\eta(t_{i})$ at each point in time $t_{i}$, or that you also know the correlation between $\eta(t_{i})$ at different points in time? –  Brian Borchers Jun 3 '12 at 13:09
At each point in time we draw $\eta(t_i)$ from a known distribution. –  dranxo Jun 4 '12 at 1:37
Are these draws independent, or correlated in time? If they're correlated, do you know anything about the correlation? –  Brian Borchers Jun 4 '12 at 1:42
Other researchers have established, using a physical model, that this isn't white noise. A method to solve my problem is still interesting even if we have to ignore that. I'm not sure what I can learn about the correlations in time, the way we can get the density right now is to record only the noise and then use that data to generate noise statistics. We do this at t=0. There might be a way to obtain statistics at future times but I haven't figured out how to do it –  dranxo Jun 4 '12 at 6:08

You can try Bayesian approach:

solution $= x = \arg \max_x P(u = x | u_o = y) = \arg \max_x \frac{P(u_o = y|u = x)P(u=x)}{P(u_o = y)} =$ $= \arg \max_x P(u_o = y|u = x)P(u=x)$.

$P(u_o = y|u = x) = P(u e^{-t/\tau}\eta=y|u=x) = P(\eta=y e^{t/\tau}/x)$ - this probability can be computed, because you know the distribution of $\eta$. Gaussian noise $\sigma$ is not considered here (but it should have zero mean).

$P(u=x)$ is the prior information about the true signal. You can construct it as a decreasing function of the number of sinusoids.

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This is what I think I'll have to do (but I'm not good with Bayesian approaches). It looks similar to Aubert and Aujol's denoising method: www.math.u-bordeaux1.fr/~jaujol/PAPERS/uv.pdf . The sticking point is that I need to minimize something non-convex in $x$. Also, I think we may need to split the $P$ across discretization points and take logs. Otherwise, well need the distribution of $\eta$ over all time which I don't think I can get. –  dranxo Jun 5 '12 at 23:09
Also, shouldn't that be argmax? –  dranxo Jun 5 '12 at 23:12
I fixed $\arg \max$, sorry for that. Yes, after the computation of the Euler-Lagrange equation, you will need the discretization anyway. –  Stanislav Jun 6 '12 at 8:43