## 2 Materials and Methods  
### Detail Injection Sharpening Framework  
Given the observable low-spatial-resolution HS image \(\mathbf{X}\in\mathbb{R}^{m\times n\times\Lambda}\) and the high-spatial-resolution MS image \(\mathbf{Y}\in\mathbb{R}^{M\times N\times\Lambda}\), where \(m,n,M,\) and \(N\) are the rows and columns of images respectively, and \(\Lambda\) and \(\lambda\) are the number of bands, (generally, \(M>m\), \(N>n\), and \(\Lambda>\lambda\)), \(s=M/m\) represents the spatial resolution ratio, and the goal of HS and MS image fusion is to estimate the unobservable HS image \(\mathbf{Z}\in\mathbb{R}^{M\times N\times\Lambda}\) with high spatial and spectral resolutions. Generally, the LR HS image can be spatially upscaled to the size of \(M\times N\times\Lambda\) by bilinear interpolation method [2; 38], which can be denoted as \(\mathbf{X}^{\text{up}}\in\mathbb{R}^{M\times N\times\Lambda}\). Therefore, the difference between the LR and HR HS images is the lack of HR spatial details, which makes the LR HS image have blurred edges and textures. The detail injection model aims to estimate the missing details of the HS image, and to inject into the LR HS image directly to obtain a new HS image that would be observed at the same resolution with the HR MS image. Technically, a detail-injection model can be described as follows:  
\[\widetilde{\mathbf{Z}}_{k}=\mathbf{X}_{k}^{\text{up}}+\mathbf{D}_{k},k=1,2, \ldots,\Lambda \tag{1}\]  
where \(\widetilde{\mathbf{Z}}\) is the spatially enhanced HS image, \(\mathbf{D}\) denotes the residual image, i.e., the missing detail component, which can be generally computed from the corresponding MS image bands, e.g.,  
\[\mathbf{D}_{k}=\sum_{i=1}^{\lambda}\alpha_{ik}\cdot\left(\mathbf{Y}_{i}- \mathbf{Y}_{i}^{\text{low}}\right),k=1,2,\ldots,\Lambda \tag{2}\]  
where \(\mathbf{Y}^{\text{low}}\) is the spatially degraded MS image that can be simulated by several approaches [39]. \(\alpha_{ik}\) is the injection gain associated with the corresponding \(i\)-th MS band. Thus, we have  
\[\widetilde{\mathbf{Z}}_{k}=\mathbf{X}_{k}^{\text{up}}+\sum_{i=1}^{\lambda} \alpha_{ik}\cdot\left(\mathbf{Y}_{i}-\mathbf{Y}_{i}^{\text{low}}\right),k=1,2, \ldots,\Lambda \tag{3}\]  
Conventional detail injection methods usually employ a multi-resolution analysis (MRA) model to obtain the high-frequency detail component in a band-by-band manner, which highly depends on the selection and performance of the MRA model [14]. In particular, for hyper-sharpening tasks, due to the large difference in spectral resolution between the HS and MS images, it is difficult to accurately predict the missing details by manually selecting the MRA model and computing the injection gains under various circumstances. By contrast, a convolutional neural network has the ability of learning and predicting the spatial structure features from the pending images in an automatic and supervised fashion. The details are driven from the context, which effectively reduces the model uncertainty and achieves higher image quality and adaptability. Hence, in this paper, we propose to utilize CNN to automatically learn the spatial details from HS and MS images themselves to promote the performance of image fusion.