Linear Denoising

Section: Random Signal

Let $y$ be a noisy measure of “clean” signal $x$ corrupted by some additive noise $n$: $$ y = x + n $$

Signal to Noise Ratio (SNR)

The SNR measures the quality of a signal, or its estimation. Knowing $x$, the SNR of $y$ is given by:

$$ SNR(y|x) = 20 \log\left(\frac{\Vert x \Vert }{\Vert x - y \Vert}\right) $$

Denoising by filtering

  • Observation: $y = x + n$ where $x$ is a clean signal and $n$ some noise
  • Goal: find a filter $h$ such that $x_{est} = h*y$ is a denoised estimation of $x$.

The best filter, which maximise the SNR, is the Wiener filter given in the frequency domain: $$ \hat h[k] = \frac{E(|\hat x[k]|^2)}{E(|\hat x[k]|^2) + E(|\hat n[k]|^2)} $$

More on the numerical tours !

Take a look on the [Simple Denoising Methods tour] (https://www.numerical-tours.com/matlab/#denoisingsimp), and in particular:

  • Linear Signal Denoising
  • Linear Image Denoising