# A Learned Reduced-Rank Sharpening Method for Multiresolution Satellite Imagery
## 2 Materials and Methods
### Model
The network is trained end-to-end using only the input image \(\mathbf{y}\), with the loss function incorporating the observational model’s blurring and downsampling operations to maintain fidelity without requiring ground truth data. The minimization problem for \(\mathbf{G}\) and \(\mathbf{F}\) is solved using the NAdam optimizer [59], which performs stochastic gradient descent based on the loss function given in (3). The regularization parameters are chosen as \(\boldsymbol{\lambda}=\lambda\boldsymbol{q}=[\lambda q]_{j=1}^{r}=[\lambda_{j}]_{j =1}^{r}\), where \(\lambda\) is a small constant. The regularization parameters \(\boldsymbol{q}\) and \(r\) are adaptively determined based on the singular value spectrum of the input data during the initialization of \(\mathbf{F}\). Specifically, principle component analysis (PCA) is performed on the input image using singular value decomposition (SVD) [28; 29; 43]. By interpolating and blurring the input image \(\boldsymbol{\tilde{y}}\) after it is normalized to zero mean and unit variance, we generate a decomposition \(\boldsymbol{\tilde{Y}}=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{\top}\), where \(\boldsymbol{\tilde{Y}}\) is the interpolated and blurred normalized input image, \(\boldsymbol{\Sigma}\) is a diagonal matrix of the singular values of \(\boldsymbol{\tilde{Y}}\), and both \(\mathbf{U}\) and \(\mathbf{V}\) are unitary matrices composed of orthonormal basis vectors [45; 60]. We then use the values of \(\mathbf{s}=\text{diag}(\boldsymbol{\Sigma})\) to define the following:  
\[\mathbf{q}=\Bigg{[}1+\log(s_{1}/s_{1}),\sum_{i=1}^{2}1+\log(s_{1}/s_{i}), \ldots,\sum_{i=1}^{L}1+\log(s_{1}/s_{i})\Bigg{]}.\]  
For each coefficient in \(\mathbf{q}\), we compute \(q_{j}=\sum_{i=1}^{j}1+log(s_{1}/s_{i})\), where \(s_{1}\) is the largest singular value and \(s_{i}\) is the \(i\)th largest singular value from \(\mathbf{s}\). Since \(s_{1}\) is the largest singular value and \(\mathbf{s}\) is monotonically decreasing, \(q_{1}=1\) and \(\mathbf{q}\) is monotonically incresing. We also initialize \(\mathbf{F}\) from the orthogonal basis vectors of \(\mathbf{V}\) associated with the \(r\) largest singular values from \(\mathbf{s}\). Since the singular values are ordered by size, lower values provide less information to the final image, and below a certain threshold, their associated basis vectors add mostly noise. Therefore, components with smaller singular values, which are more likely to be dominated by noise, should receive stronger regularization, and those close to zero are often better discarded [45]. The logarithmic scaling ensures a smooth progression of regularization strength, while the cumulative sum creates monotonically increasing  
Figure 3: Diagram of the customized U-Net architecture [2; 58]. Network inputs, \(\mathbf{Y}_{\text{NN}}\), are padded with symmetrical mirror padding and the network output, and \(\mathbf{G}\) is cropped back to input size using dedicated layers (light orange). Each U-Net block consists of 2D convolutional layers followed by instance normalization and PReLU activation (yellow with dark orange bands). As an example, the U-Net blocks from the bottleneck are shown below the network diagram. In the contracting half, max pooling follows each pair of convolutions (red). In the expanding half, pixel shuffle and 2D convolutions (pink, yellow) precede each concatenation with the feature outputs from the contracting half (gray), followed by U-Net blocks. Skip connections scale features inversely in proportion to their length, i.e., full-resolution features are not scaled, the second-highest-resolution features are halved, and so on. A final convolutional layer outputs the estimated low-rank representation, also using PReLU activation (dark orange). weights across components. This approach automatically adapts to the data's inherent dimensionality structure, providing stronger regularization for components with lower signal-to-noise ratios while preserving significant features captured by the larger singular values.