# Denoising and filtering

Universal schemes have been developed for discrete denoising and filtering problems.

## Discrete Denoising

In discrete denoising problem, a clean source, $x^n$, gets corrupted by a discrete memoryless channel (DMC) and results in an noisy sequence $Y^n$. A denoiser is an estimator that observes the entire noisy sequence $Y^n$ and estimates the original underlying clean source $x^n$ as good as possible with respect to a single letter loss function $\Lambda(x,\hat{x})$. The channel transition matrix $\mathbf{\Pi}$ is assumed to be invertible and known to the denoiser. The alphabet size of the source, noisy, and reconstruction symbols, $|\mathcal{X}|$, $|\mathcal{Y}|$, and $|\hat{\mathcal{X}}|$ are assumed to be finite.

\begin{eqnarray} L_{\hat{\mathbf{X}}^n}(x^n,y^n)= \frac{1}{n}\sum_{t=1}^n\Lambda(x_t,\hat{X}_t(y^n)) \end{eqnarray}

### Theorem

For any source sequence $x^n$, $$\lim_{n\rightarrow\infty}\Big[L_{\hat{\mathbf{X}}^n_{\text{DUDE}}}(x^n,Y^n) - D_k(x^n,Y^n)\Big] = 0$$ almost surely.