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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

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