## 1 Introduction  
Hyperspectral images with rich spectral information play an important role in agriculture [1], medicine [2], geological surveying [3], remote sensing [4], computer vision [5, 6], and other fields. Generally, the resolution of hyperspectral images directly affects the application effect of hyperspectral images. However, the spatial resolution of hyperspectral images directly obtained with hyperspectral cameras is relatively low. On the contrary, the multispectral image directly obtained by the multispectral camera has a higher spatial resolution. Obtaining high-resolution hyperspectral images by improving hardware processes is usually costly, so fusing high-resolution multispectral images with low-resolution hyperspectral images is one of the most economical ways to obtain high-resolution hyperspectral images. In the practical application of the fusion method, two images of the same area obtained by the hyperspectral camera and the multispectral camera can be pre-processed by image alignment [7] and enhancement. Then, the feature points in the two images can be extracted and matched separately, and then the two images can be geometrically aligned. After alignment, the two images can be used as input images for the fusion algorithm. The application flow chart is shown in Figure 1a. However, in this paper, in order to compare the fusion ability of various algorithms more effectively, we use the original image in the dataset as the target HR-HSI in the experimental part, simulate the LR-HSI obtained by the hyperspectral camera by spatially blurred downsampling of the target HR-HSI, and simulate the HR-MSI obtained by the multispectral camera by spectral downsampling of the target HR-HSI. The LR-HSI and HR-MSI involved in the fusion are already aligned images, so that we can easily check the fusion performance of the fusion algorithm by comparing the similarity between the fused images and the target image (the better the similarity, the better the fusion effect). The application flow chart is shown in Figure 1b. The earliest methods of spatial-spectral image fusion can be traced back to pan-sharpening techniques, which fuse low spatial resolution RGB images with high resolution panchromatic images to obtain high-resolution RGB images. Representative pan-sharpening techniques include component substitution [8], methods based on principal component analysis [9], and methods based on compressed sensing [10]. Since panchromatic images contain only one piece of spectral information, pan-sharpening techniques usually lead to spectral distortion. To enhance the fusion quality, HR-MSI is usually used instead of panchromatic images to fuse with LR-HSI. Typical fusion methods include Bayesian-based fusion methods, spectral unmixing-based fusion methods, matrix decomposition-based fusion methods, and tensor decomposition-based fusion methods. [PERSON] et al. first used the Bayesian method based on maximum posterior probability (MAP) estimation to realize fusion. This method uses a stochastic mixed model (SMM) to estimate potential spectral scene features and formulates a cost function to optimize the corresponding hyperspectral image data estimated by LR-HSI and HR-MSI. The optimization of MAP-SMM is carried out in the principal component subspace. The idea of using two input images to merge spectral information in a subspace is the primary source of inspiration for many later fusion methods [11; 12; 13]. Compared with pan-sharpening technology, this fusion method has made a significant breakthrough in improving the spatial resolution of hyperspectral images [14; 15]. With the development of fusion technology, fusion methods based on spectral unmixing [16; 17; 18] have been proposed, based on the fact that hyperspectral images often have a large number of so-called \"mixed pixels\", i.e., overlapping spectral responses of several different surface materials, due to the much broader pixel coverage.