# A Learned Reduced-Rank Sharpening Method for Multiresolution Satellite Imagery
## 2 Materials and Methods  
### Model  
The image acquisition of each S2 band can be described using an observational model,  
\[\mathbf{y}_{i}=\mathbf{M}_{i}\mathbf{B}_{i}\mathbf{x}_{i}+\mathbf{e}_{i},\qquad i =1,\dots,L, \tag{1}\]  
where \(\mathbf{x}_{i}\) is the vectorized \(n\) pixel target image at band \(i\), \(\mathbf{y}_{i}\) is the vectorized captured \(n/s^{2}\) pixel image of the same band, \(\mathbf{M}_{i}\) is a downsampling operator matching the band's resolution relative to the target resolution by a factor \(s\), \(\mathbf{B}_{i}\) is a circulant blurring matrix representative of the band's point spread function (PSF), \(L\) is the number of bands, and \(\mathbf{e}_{i}\) is noise [28]. To simplify the computation of the high-resolution target image \(\mathbf{x}=[\mathbf{x}_{1}^{\top}\mathbf{x}_{2}^{\top}\dots\mathbf{x}_{L}^{ \top}]^{\top}\) from the low-resolution observed image \(\mathbf{y}=[\mathbf{y}_{1}^{\top}\mathbf{y}_{2}^{\top}\dots\mathbf{y}_{L}^{ \top}]^{\top}\), a subspace representation of the target image defined by \(\mathbf{X}=\mathbf{GF}^{\top}\) is used. The spectral vectors of \(\mathbf{X}\in\mathbb{R}^{n\times L}\) can be reconstructed from \(\mathbf{F}\in\mathbb{R}^{L\times r}\) and \(\mathbf{G}\in\mathbb{R}^{n\times r}\), where \(r<L\). Since \(\mathbf{x}_{i}=\mathbf{G}\mathbf{f}_{(i)}\), where \(\mathbf{f}_{(i)}\) is row vector \(i\) from \(\mathbf{F}\), the observational model becomes  
\[\mathbf{y}_{i}=\mathbf{M}_{i}\mathbf{B}_{i}\mathbf{G}\mathbf{f}_{(i)}+\mathbf{e}_ {i},\qquad i=1,\dots,L, \tag{2}\]  
illustrated in Figure 1.  
Building upon the work of [28; 29], we define our cost function in the same manner,  
\[J(\mathbf{F},\mathbf{G})=\sum_{i=1}^{L}\frac{1}{2}\|\mathbf{y}_{i}-\mathbf{M}_{i }\mathbf{B}_{i}\mathbf{G}\mathbf{f}_{(i)}\|_{2}^{2}+\sum_{j=1}^{r}\lambda_{j} \phi_{G}(\mathbf{g}_{j}). \tag{3}\]  
It is minimized with respect to both \(\mathbf{F}\) and \(\mathbf{G}\) and is composed of an \(\ell_{2}\) reconstruction fidelity term and a regularization function \(\phi_{G}(\mathbf{g})\). \(\lambda_{j}\) are tuning parameters that control the strength of regularization, and \(r\) denotes the dimensionality of the low-rank subspace representation. The regularization term \(\phi_{G}(\mathbf{g})\) differs slightly from [28] and is defined as  
\[\phi_{G}(\mathbf{g}_{j})=\mathbf{g}_{j}^{-}(\mathbf{H}_{h}^{\top}\mathbf{H}_{ h}+\mathbf{H}_{v}^{\top}\mathbf{H}_{v})\mathbf{g}_{j},\qquad j=1,\dots,r, \tag{4}\]  
a roughness penalty using the horizontal and vertical \(n\times n\) finite difference matrices, \(\mathbf{H}_{h}\) and \(\mathbf{H}_{v}\), respectively. While this regularization shares similarities with total variation regularization [49; 50], it differs in that while the fidelity term is applied to the final full-resolution image, the regularization is applied within the low-rank subspace. Figure 2 shows a flow diagram of the method. Images are preprocessed by normalizing each band to zero mean and unit variance before training. Nearest-neighbor upsampling of the low-resolution bands is generated as \(\mathbf{Y}_{NN}=[\uparrow_{NN}(\mathbf{y}_{i})]_{i=1}^{L}\) so that all bands of the network input are at the intended final resolution. Nearest-neighbor upsampling was selected to preserve the original pixel values of low-resolution bands, ensuring precise compatibility with the observational model and loss function. This approach maintains the raw spectral characteristics of the original bands, enables exact retrieval of low-resolution values during downsampling, and provides a computationally efficient method that allows the network to focus on learning the transformation between low- and high-resolution bands without introducing interpolation artifacts. Symmetrical padding is also applied before feeding the inputs through the network to reduce boundary effects, and the network outputs are cropped back to the input size. The proposed method learns the weights to generate \(\mathbf{G}\) and \(\mathbf{F}\) during training.