My mathematical class task is to de-noise a function using wavelet transform. I am to select a function $f(x)$ and noise function with zero-mean $n(x)$. I am to add noise like this: $$f_{noise}(x) = f(x) + \epsilon n(x)$$ The thing is, I should perform de-noising process not just by using thresholding or a similar technique, but taking into account the available information about the noise function. If I were to do de-noising using Foruier transform, I would null frequencies corresponding to noise function, but wavelets are different since detailing coefficients aren't actually frequencies, so I fail to come up with a solution. So, my question is: how to denoise a function with wavelet transform relying on the fact that $n(x)$ is known?

  • $\begingroup$ Can you explain what information you know about the noise function? Surely you don't simply know $n(x)$ exactly, because then you could just subtract it from $f_\text{noise}(x)$ and be done... $\endgroup$ – Rahul Dec 24 '13 at 15:20
  • $\begingroup$ That's true! Let's say I know, that this is a Gaussian noise $\endgroup$ – Dmitry Dec 24 '13 at 15:25
  • $\begingroup$ I don't have an answer, but I'd like to point out another SE site, Signal Processing. $\endgroup$ – Post No Bulls Dec 25 '13 at 7:03

I assume the signal is being sampled on a discrete set $\{ x_i \}$, say evenly spaced on $[0,1]$. Fix a model space $X$ for the signal $f$, e.g. Sobolev, Besov, etc.

Typically, one would perform a discrete wavelet transform (where the filter is given by some chosen wavelet basis for $X$) and threshold the wavelet coefficients using some criterion involving $\epsilon$. For example, hard thresholding means setting any wavelet coefficients $ < \epsilon$ to $0$. Then you invert the thresholded coefficients. This gives the estimate for $f$.

The mathematical and statistical justification of this procedure lies in the fact that

  1. Wavelets can describe signals parsimoniously.
  2. Wavelets form an unconditional basis for $X$; so thresholding preserves smoothness.
  • $\begingroup$ What do you mean by 'invert thresholding coefficients'? If, say, detailing coefficients are [1 2 1 3 4] and the threshold is 2 then the resulting detailing coefficients would be [0 2 0 3 4] and isn't it enough? $\endgroup$ – Dmitry Dec 25 '13 at 17:21
  • $\begingroup$ By "invert", I mean applying the inverse DWT to the thresholded coefficients $[0\; 2\; 0\; 3\; 4]$. $\endgroup$ – Michael Dec 26 '13 at 22:42

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