# A Learned Reduced-Rank Sharpening Method for Multiresolution Satellite Imagery
## 2 Materials and Methods
### Model
The former is generated using a U-Net neural network that learns a transformation from the input image \(\mathbf{y}\) to the low-rank representation \(\mathbf{G}\). The preprocessed inputs are fed through the neural network to generate the low-rank representation, \(\mathbf{G}=\text{UNet}(\mathbf{Y}_{NN})\). A trainable linear layer implements the mapping from \(\mathbf{G}\) to the final image \(\mathbf{X}\). The weights of this layer correspond to the transformation matrix \(\mathbf{F}\). This two-stage architecture enables end-to-end training while maintaining the computational efficiency of a low-rank representation. The custom U-Net consists of 2D convolutional blocks with instance normalization [51] and Parametric Rectified Linear Unit (PReLU) activation [52]. PReLU activation is similar to the well-known Rectified Linear Unit (ReLU) activation but adds a learnable parameter \(a\) that independently controls the leakiness of negative value activations for each band [52],  
Figure 1: Block diagram showing the relationship between the observed low-resolution image \(\mathbf{y}_{i}\), the high-resolution image \(\mathbf{x}_{i}\), and the reduced-rank representation \(\mathbf{G}\). The full-resolution band \(\mathbf{x}_{i}\) is represented as \(\mathbf{G}f_{(i)}\) in the reduced-rank subspace and then transformed through blurring and downsampling operations with additive noise to produce the observed data. \[\text{PReLU}(x)= \begin{cases}x,&\text{if }x\geq 0,\\ ax,&\text{otherwise}.\end{cases} \]  
PReLU was chosen over standard ReLU due to its ability to learn the leakiness of negative value activations independently and adaptively for each band. Unlike standard LeakyReLU with a fixed slope, PReLU automatically allows the network to determine the most appropriate negative slope during training. This is particularly beneficial for multispectral satellite imagery, where different bands may require different activation characteristics. Max pooling is used on the downward path, and pixel shuffle with 2D convolutions is used on the upward path [53]. The kernel sizes of each Max pooling layer are chosen so that the spatial resolution of the features in the first two contractions matches the resolutions of the 20 and 60 m bands, respectively. The skip connections are scaled proportionately with their length, which has been shown to improve training stability in other U-Net architectures [54]. Pixel shuffle offers a more computationally efficient learnable alternative to traditional upsampling methods like transposed convolutions and has been applied to varied super-resolution problems with good results [55; 56; 57]. By rearranging elements from a convolutional feature map, pixel shuffle allows the network to learn the upsampling transformation directly rather than using fixed interpolation techniques. This approach helps maintain spatial information while enabling the network to adaptively learn how to increase resolution, which is particularly valuable for multiresolution satellite imagery where precise upsampling can significantly impact image quality [53]. A single 2D convolution layer and PReLU activation generate the low-rank approximation that is cropped to the correct shape before the final linear layer. Figure 3 shows the layout of the U-net architecture. Figure 2: Flow diagram of the proposed sharpening method. Preprocessing begins with bandwise normalization to zero mean and unit variance, followed by nearest-neighbor upsampling of the low-resolution bands to align all inputs to 10 m resolution. Symmetric padding is applied to reduce boundary effects in the convolutional operations. The preprocessed input passes through a customized U-Net architecture, which generates the low-rank spatial representation \(\mathbf{G}\). A trainable linear layer implements the transformation \(\mathbf{F}\) to produce the sharpened output \(\mathbf{\hat{X}}\). After processing, outputs are cropped to remove padding and denormalized to restore the original data range.