# Calculating the Mean Square Error (MSE) in Wavelet Denoising

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.