0
$\begingroup$

I´m currently reading the paper (to be more precise: it´s a chapter from the book "Shearlets, Multiscale Analysis of Multivariate Data" by Kutyniok and Labate) "Image Processing Using Shearlets" by G.R. Easley and D. (Labate https://www.math.uh.edu/~dlabate/Chap_ImageAppl.pdf). I´m interested in the second part, the "Image Denoising".

I will explain the problem setting in the paper. Then i will share my question.

So the task is, to recover the function $f\in L^2(\mathbb{R}^2)$ from noisy data $y$: \begin{align} y=f+n, \end{align} where n is Gaussian white noise, with standard deviation $\sigma$. Hence we want to optimize the estimation $\tilde{f}$ of $f$. This is done by mimimizing the Mean Square Error (MSE), given by \begin{align} E[\vert\vert f-\tilde{f}\vert\vert^2], \end{align} where $E[.]$ is the expexted value, which is calculated with respect to the probability distribution of the noise $n$. The idea is to find and estimation $\tilde{f}$, satisfiying the minimax MSE, defined as: \begin{align} \text{inf}_{\tilde{f}}\text{sup}_{f\in F} E[\vert\vert f-\tilde{f}\vert\vert^2, \end{align} where F are the cartoonlike function (a model class of images) and we allow all measurable estimations in the infimum. When using Wavelet or Shearlet denoising the procedure is done as follows:

Let $W, W^{-1}$ denote the wavelet and inverse wavelet transform and $T_{N_\sigma}$ be the threshholding operator depending on $\sigma$, then the denoising process is done by: \begin{align} \tilde{f}_N=W^{-1}(T_{N_\sigma}(W(y))). \end{align} The threshholding operator only keeps the $N_\sigma$ coefficients with the highest absolute value. When doing denoising with Shearlets, the Wavelet transform is replaced by the Shearlet transform.

It is known, that the wavelet estimator $\tilde{f}_N$ satisfies \begin{align} \vert\vert f-\tilde{f}_N\vert\vert^2\leq CN^{-1},\quad as\quad N\to\infty \end{align} $C>0$ is a constant independent on $f$ and $f_N$. Now there is written, that this implies, that the Mean Square Error (MSE) of the wavelet estimator satisfies \begin{align} \text{sup}_{f\in F}E[\vert\vert f-\tilde{f}_N\vert\vert^2]=\Theta(\sigma),\quad as\quad \sigma\to 0. \end{align}

This is what I don´t understand. How does this follow? I think my problem is, that I don´t exactly know how to compute the expected value in this case. I hope someone can help me.

Thank you in advance,

Chris

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.