# Variational Multiscale Nonparametric Regression: Algorithms and Implementation
## 4 Numerical Study
### Unknown Noise Level  
We now investigate the issue of unknown noise level \(\sigma\). This is illustrated by implementing two different estimators for the noise level into the MIND estimator with total variation regularisation and shearlets dictionary. We stress, however, that the results are to be expected to be similar for other choices of the dictionary and regularisation functional. The MIND optimisation is solved by the Chambolle-Pock algorithm. However, in contrast to previous simulations the 50%-quantile of \(\|\sigma\epsilon\|_{\text{MS}}\) in (5) with known \(\sigma\) is now replaced by an estimated noise level \(\hat{\sigma}\) which then gives a new threshold \(q_{n}\) for MIND. We display all results for the well known test image \"butterfly\" (\(256\times 256\) pixels, see, e.g., [72]). The SNR is set to 20, or equivalently \(\sigma=12.6\). The choice of test image and SNR is purely arbitrary, and the results would remain quite the same for other choices. We investigate the influence on MIND for two different noise level estimators. One is a second-order difference-based estimator by [PERSON] et al. [35], which tends to overestimate the noise level, and the other one is a patch based estimator by [PERSON] et al. [71], which often underestimates the noise level (see Figure 12). The comparison results are summarised in Figure 10. Visually, there is little difference between the use of MIND with known noise level and that of the estimated ones. A slight improvement in terms of PSNR is even shown for the use of the estimator that tends to underestimate the noise level. However, a caution should be taken that an underestimated noise level leads to a sacrifice in statistical confidence. For instance, it is no longer possible to guarantee that the true image \(f\) lies in the constraint of (3) with given probability 50%. In contrast, when using the true noise level or an overestimated noise level, such a statistical confidence statement is valid. In short, there is nearly no loss of performance of MIND due to the unknown noise level in practice. A heuristic explanation of this is that the mean squared error of the estimated noise level \(\sigma\) scaled as \(O(n^{-1})\), whereas for estimating \(f\) this rate is slower. Hence, its effect on the overall performance is of minor order.  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & & **Dyadic Cubes** & **Small Cubes** & **Wavelets** & **Curvelets** & **Shearlets** \\ \hline \multirow{4}{*}{\
\begin{tabular}{} \end{tabular} } & MISE & 0.00176 (\(5.4\times 10^{-6}\)) & 0.00149 (\(1.9\times 10^{-6}\)) & 0.00147 (\(2.9\times 10^{-5}\)) & 0.0013 (\(2.5\times 10^{-5}\)) & **0.000871** (\(3\times 10^{-4}\)) \\  & PSNR & 27.5 (\(0.013\)) & 28.3 (\(0.005\)) & 28.3 (\(0.085\)) & 28.9 (\(0.086\)) & **30.6** (\(0.015\)) \\  & SSIM & 0.871 (\(3.1\times 10^{-6}\)) & 0.806 (\(0.002\)) & **0.872** (\(0.0023\)) & 0.852 (\(0.0091\)) & 0.715 (\(0.0043\)) \\  & VIF & 0.726 (\(0.00058\)) & 0.823 (\(0.00036\)) & 0.767 (\(0.0021\)) & 0.809 (\(0.0022\)) & **0.852** (\(0.00086\)) \\ \hline \multirow{4}{*}{
\begin{tabular}{} \end{tabular} } & MISE & 0.000671 (\(1.3\times 10^{-6}\)) & 0.000438 (\(8.5\times 10^{-7}\)) & 0.00059 (\(5\times 10^{-7}\)) & 0.000554 (\(1.3\times 10^{-5}\)) & **0.000412** (\(2.6\times 10^{-6}\)) \\  & PSNR & 31.37 (\(0.0082\))