# Variational Multiscale Nonparametric Regression: Algorithms and Implementation
## 4 Numerical Study
### Comparison of Different Dictionaries  
We next investigate the choice of dictionary \(\{\phi_{\lambda}|\lambda\in\Lambda_{n}\}\) and its impact on the practical performance. Five different dictionaries are considered. The first consists of the indicator functions of dyadic cubes defined in (18), and the second is composed of the indicator functions of small cubes of edge length \(\leq 30\), i.e.,  
\[\frac{1}{n}\left[i,i+\ell\right]\times\left[j,j+\ell\right]\quad\subseteq\quad \left[0,1\right]^{2}\qquad\text{for $\ell=1,\ldots,30$, and all possible $i,j\in\mathbb{N}$}\,.\]  
The third is the system of tensor wavelets, in particular, the Daubechies' symlets with six vanishing moments. The fourth is the frame of curvelets and the last is that of shearlets, both of which are constructed using default parameters in packages CurveLab ([[http://www.curvelet.org](http://www.curvelet.org)]([http://www.curvelet.org](http://www.curvelet.org))) and ShearLab3D ([[http://shearlab.math.lmu.de](http://shearlab.math.lmu.de)]([http://shearlab.math.lmu.de](http://shearlab.math.lmu.de))), respectively. As shown in Figures 3 and 4, different algorithms lead to almost the same final result, we only report the Chambolle-Pock algorithm, due to its empirically fast convergence. Figure 5: Performance of Chambolle-Pock (line), ADMM (dash-dot) and semismooth Newton (dash) algorithms for MIND with \(H^{1}\) regularisation over time. (**a**) Objective value; (**b**) Constraint gap; (**c**) Distance to limit; (**d**) PSNR; (**e**) SSIM; (**f**) VIF. As a reflection of versatility of images, we consider three test images of different types. They are a magnetic resonance tomography image of mouse \"brain' (cf. Figure 7a) from Radiopaedia.org (rID: 6777), a \"cell\" image (cf. Figure 8a) taken from [70] and a mouse \"BIRN\" image (cf. Figure 9a) from cellimagelibrary.org (doi:10.7295/W9 CCDB17) (see also Figure 1 in the Introduction as well as Figure 10 in Section 4.3 for the case of natural images). All images are rescaled to \(256\times 256\) pixels via bicubic interpolation. The SNR is set to 20 over all cases (see Figure 11 for noisy image). The results on all test images are shown in Figures 7-9. In general, the dictionary of shearlets performed the best, which is followed by that of curvelets, while the rest (i.e., dyadic cubes, small cubes and wavelets) had similar performance. One exception is the \"cell\" image, where the dictionary of indicator functions of cubes, in particular of small cubes, was better at detecting tiny white balls (see Figure 8). This is because of the similarity of such features and the cubes at small scales in the dictionary. The average performance over 10 random repetitions measured by aforementioned image quality measures as well as mean integrated squared error  
Figure 6: Performance of Chambolle–Pock (line) and ADMM (dash-dot) algorithms for MIND with TV regularisation over time. (**a**) Objective value; (**b**) Constraint gap; (**c**) Distance to limit; (**d**) PSNR; (**e**) SSIM; (**f**) VIF. \[\text{MISE}=\frac{1}{n}\sum_{i=1}^{n}(\hat{f}(x_{i})-f(x_{i}))^{2}\]  
is reported in Table 2, which is consistent with the virtual inspections in Figures 7-9. We note that 10 repetitions are sufficient here, since the variation in each repetition is comparably small (cf. standard deviations reported in parenthesis in Table 2). As a remark, we emphasise that the difference in performance due to the choice of dictionaries is generally negligible, but the dictionary of shearlets slightly outperforms the other choices. Figure 8: Results on “cell” by MIND with TV regularisation and different dictionaries. (**a**) Truth; (**b**) Dyadic cubes, PSNR = 31.8; (**c**) Small cubes, PSNR = 33.6; (**d**) Wavelets, PSNR = 32.9; (**e**) Curvelets, PSNR = 32.9; (**f**) Shearlets, PSNR = 33.9. Figure 7: Results on “brain” by MIND with TV regularisation and different dictionaries.