# A Learned Reduced-Rank Sharpening Method for Multiresolution Satellite Imagery
## 3 Results
### Fusing Landsat Data  
It is straightforward to extend the proposed method for other applications. As a demonstration, an experiment was run that fuses Sentinel-2 and Landsat 8 images, bringing all bands to 10 m resolution. In [42], the S2 Sharp algorithm [28] was extended to handle Landsat 8 (L8) data through several key modifications. The original cost function was expanded to include L8 bands, increasing the number of processed bands from 12 to 20 while maintaining its mathematical structure. For handling L8's point spread function (PSF), the method adopted the same 2D Gaussian function approach used for Sentinel-2, where the average spatial response was calculated by combining along-track and across-track OLI spatial responses, with the Gaussian function variance determined through least squares estimation. A particular challenge was processing L8's 15 m panchromatic band, which is not an integer multiple of the target 10 m resolution. This was solved by implementing a dual-processing approach where the band was both downsampled to 20 m and upsampled to 10 m using bicubic interpolation, allowing the method to preserve information while maintaining compatibility with the algorithm's framework. The resulting method, termed SLSharp, successfully enabled joint sharpening of both Sentinel-2 and Landsat 8 bands to 10 m resolution while preserving the beneficial properties of the original S2 Sharp algorithm [42]. Adopting our proposed method in the same manner also enables it to fuse Sentinel-2 and Landsat 8 data. An image of Glen Canyon, Arizona, which was used in [42], was processed to demonstrate the feasibility of adapting our method to similar problems, and the results are presented in Section Fusion Evaluation.  
Several modifications were implemented to adapt our method for Sentinel-2 and Landsat 8 fusion. The kernel sizes of the pooling layers on the downward path and the corresponding pixel shuffle operations on the upward path were updated so that the first two contractions now match the resolutions of the 30 and 60 m bands. The regularization was changed to \(\lambda=0.001\), and the regularization strength was capped at 45\(\lambda\) to maintain similar maximum regularization with a larger subspace \(r\). Landsat 8 bands were appended to the Sentinel-2 input tensor, with the panchromatic band being both downsampled to 20 m resolution and upscaled using bicubic interpolation to 10 m to maintain compatibility with the network architecture. The learning rate was reduced slightly to ensure stable convergence with the expanded input set. Point spread functions for the Landsat 8 bands were estimated following the approach detailed in [42], where the average spatial response was calculated by combining along-track and across-track OLI spatial responses through least squares estimation.  
The extended method is evaluated using the same image quality metrics as in Section 3.1.1: ERGAS, RMSE, SAM, SRE, SSIM, and UIQI. A set of real S2 and L8 images of Glen Canyon, Arizona, was evaluated at reduced scales using [PERSON]'s protocol. The evaluation process began by reducing the image resolution by factors of 2, 3, and 6. These data were then processed to create a comprehensive test set by applying appropriate PSF filters and downsampling to generate multiple resolution versions of the 10 m, 15 m, 20 m, 30 m, and 60 m bands, providing a controlled environment for comparing our proposed method against SLSharp [42], a baseline method based on SupReME [29] and ATPRK [26].  
Figure 9 shows the RGB bands of the image used, and inset images show details used for qualitative evaluations at full scale.