# A Learned Reduced-Rank Sharpening Method for Multiresolution Satellite Imagery
## 3 Results
### Ablation Study
#### 3.2.1 Regularization  
In [28; 29], a diagonal weighting matrix, \(\mathbf{W}\), is used in the regularization function to preserve edge sharpness. It is chosen to downweight the roughness penalty on each pixel of the subspace based on the maximum image gradient magnitude \(z_{\text{max}}\) of the corresponding pixel across all the high-resolution bands. Here, we show that adding this weighting matrix does not improve the results of our method. To test this weighted roughness penalty, the regularization function \(\phi_{\text{G}}(\mathbf{g})\) in (3) is replaced by  
\[\phi_{W}(\mathbf{g}_{j})=\mathbf{g}_{j}^{\top}(\mathbf{H}_{h}^{\top}\mathbf{W} \mathbf{H}_{h}+\mathbf{H}_{v}^{\top}\mathbf{W}\mathbf{H}_{v})\mathbf{g}_{j}, \tag{5}\] \begin{table}
\begin{tabular}{l r r r r r r} \hline \hline
**Time (s)** & **Proposed** & **ATPRK** & **S2 Sharp** & **SSSS** & **SupReME** & **SUSS** \\ \hline Mean & 238.77 & 17.20 & 29.40 & 66.00 & 8.65 & 204.00 \\ Std & 3.60 & 0.04 & 0.41 & 0.33 & 0.10 & 2.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computation time comparison between methods. where  
\[\text{diag}(\textbf{W})=\text{max}\Bigg{(}0.5,\exp\left(-\frac{z_{\text{max},k}^{2} }{2\sigma_{s}^{2}}\right)\Bigg{)},\qquad k=1,\ldots,n, \tag{6}\]  
where \(\sigma_{s}=1.0\) is a scaling factor, the gradient magnitude of each of the 10 m bands is found using the Prewitt discrete differentiation operator, and \(\textbf{z}_{\text{max}}=[z_{\text{max},k}]_{k=1}^{n}\) contains the maximum gradient magnitude for each pixel, just as in [28; 29]. Using (6) effectively fixes the values of the penalty weighting matrix within the value range \([0.5,1.0]\), where the maximum penalty is applied to pixels that are smooth in the high-resolution bands but weaker penalties are used where edges are apparent. When trained with (5) as the regularization term of the training loss, the results are slightly worse than using (4), as shown by the bar labeled \"Weighted regularization\" in Figure 6. Removing the regularization term completely makes the method perform significantly worse. This is equivalent to setting \(\lambda=0\) and removes any ability to tune the subspace through the hyperparameters \(\lambda\) and **q**. The bar labeled \"No regularization\" in Figure 6 shows the results. Additionally, removing **q** and **W**, leaving a version of (3) with a single \(\lambda\) parameter that is identical for all subspace bands, also gives worse results, as seen from the bar labeled \"No **q** parameter\" in Figure 6. As previously described, the **q** parameter was adapted from [29]. It compensates for the PCA dimensions being progressively more affected by noise and requiring more robust regularization [29; 43; 45]. We performed a grid search over the regularization strength \(\lambda\) and subspace dimension \(r\) with other parameters fixed to evaluate the method's sensitivity to these key parameters. The regularization parameter \(\lambda\) was varied from 0.0005 to 0.004 in steps of 0.0005, while the size of the low-rank subspace \(r\) was tested from 5 to 12 in steps of 1. Each parameter combination was evaluated using the average signal-to-reconstruction error (SRE) across the four synthetic datasets. The results, shown in Figure 7, reveal a broad region of stable  
Figure 6: A comparison of Root-Mean-Square-Error [63] for the proposed method and its variants; lower values are better. The bars represent the method as proposed, with and without weighted regularization, without orthogonal initialization, with parametrized orthogonal weights in the final layer, and with LeakyReLU instead of PReLU. While this figure only shows a single image quality metric, the proposed method performs best by all the metrics used in Section 3. The error bars show the standard error of the mean for ten runs with each of the four different images. performance with SRE values above 33.0 dB. The highest performance is observed in the region where \(\lambda\) ranges from 0.0015 to 0.0035 and \(r\) from 9 to 11, with peak SRE values around 33.3 dB.