# A Learned Reduced-Rank Sharpening Method for Multiresolution Satellite Imagery
###### Abstract
They can be further categorized as methods that require supervised training and methods that use unsupervised training. Supervised training requires large datasets with labeled training data where each input is paired with its corresponding target output, allowing the model to learn direct mappings through explicit feedback. This includes methods such as Deep Sentinel-2 (DSen2) [31] and Sentinel-2 Generative Adversarial Network (S2 GAN) [32]. Unsupervised training, on the other hand, works with unlabeled data, requiring the model to discover underlying patterns and structure without explicit ground truths, often by learning representations that capture the inherent relationships and variations within limited data. Examples of such methods include Self-Supervised Transfer Learning for Sentinel-2 (SelfS2) [33], Deep SURE for S2 Sharpening [34], Zero-Shot Sentinel-2 Sharpening Using a Symmetric Skipped Connection Convolutional Neural Network (SSC-CNN) [35], and the Simple Unrolled Sentinel-2 Sharpening algorithm (SUSS) [36], also giving good results. Applications that combine data from multiple satellite sources through data fusion are also increasingly popular [37]. Several established methods fuse Landsat 8 and Sentinel-2 data to leverage their complementary characteristics, typically harmonizing their spectral responses and spatial resolutions. While the widely-used HLS30 products combine observations from both satellites to achieve higher temporal frequency, it does so by re-sampling all bands to 30 m resolution [38; 39]. Other fusion approaches aim to preserve Sentinel-2's native 10 m resolution while incorporating Landsat 8 data, enabling applications that require both high spatial and temporal resolution [40; 41; 42]. This advanced fusion presents additional challenges in ensuring radiometric consistency and accurate spatial alignment between sensors but offers potential advantages over harmonization approaches that sacrifice spatial detail. The key innovation of our method lies in simultaneously optimizing a low-rank approximation and a linear transformation for Sentinel-2 image sharpening by leveraging neural network optimization techniques through a customized U-Net architecture. This results in a compact implementation while maintaining the interpretability of model-based approaches. Our novel design is tailored explicitly to multiresolution satellite imagery and incorporates instance normalization, PReLU activation, and resolution-matched pooling with scaled skip connections. Our simplified regularization approach operates directly in the low-rank subspace without requiring weighting matrices, and by initializing using principal component analysis (PCA) through singular value decomposition [43; 44; 45; 46] but removing the constraint of continued orthogonality during training, we achieve better convergence while maintaining image fidelity. We also use the singular value decomposition to determine regularization parameters that automatically adapt to the data's inherent dimensionality structure. This unique combination results in an end-to-end trainable framework for S2 sharpening that requires no external training data, adapts to different input images with minimal parameter tuning, and demonstrates consistent performance across both 20 m and 60 m bands. This also makes the method adaptable for other tasks, as we demonstrate by using it to fuse Sentinel-2 and Landsat 8 images. Future work could explore adapting our method for pansharpening and hypersharpening [19; 47] applications, where our learned reduced-rank approach might offer unique advantages in preserving spectral fidelity while enhancing spatial resolution. While the method's training time is longer than traditional methods due to the neural network optimization, this represents a run-time cost that eliminates the need for extensive parameter tuning across different image types--a significant advantage for practical applications where consistent performance is prioritized over raw execution speed.