# Hyperspectral Image Super-Resolution Based on Spatial Group Sparsity Regularization Unmixing
## 1 Introduction  
Hyperspectral imagery (HSI) can profile materials and organisms due to hundreds of narrow spectral bands that correspond to different wavelengths, which is not provided by traditional images. Hyperspectral imaging has been significantly developed and utilized in many computer vision tasks in recent years, including remote sensing [1], medical imaging [2], object classification [3, 4], and target tracking [5, 6]. To obtain an HSI with a good signal-to-noise ratio, the hyperspectral camera must use a long exposure time to ensure that enough photons are captured. Constrained by sensor technology, the current HSIs usually have low spatial resolution [7] and suffer from the spectral mixing effect, which leads to every single pixel consisting of spectra from several materials. This makes the exploitation of hyperspectral images a big challenge, which is difficult to overcome due to physical limits. _HSI super-resolution_ is a fundamental and essential problem in the field of hyperspectral imaging, based on the simple idea of fusing the spectral information of hyperspectral images with the spatial information of _conventional images_. In order to avoid misunderstandings, the term \"conventional images\" is used in this paper to represent RGB images and multispectral images (MSIs) which have at least an order of magnitude less spectra than HSIs. Thus, HSI super-resolution can also be referred to as hyperspectral and multispectral image fusion (HS-MS image fusion). There have been many HSI super-resolution approaches in recent years. The spatial and spectral regularization is an essential idea that widely used [8]. Most of the HSI super-resolution methods intend to seek breakthroughs in making full use of spatial and spectral information. From the point of view of image processing, the problem is a special case of image fusion. In particular, the HSI super-resolution problem can be seen as a generalisation of pan-sharpening [9; 10; 11]. However, this technique does improve the spatial resolution to some extent. To further improve the quality of super-resolution, other fusion methods such as matrix factorization-based methods [12; 13] maintain spatial structures of the HSI. Unmixing-based [12; 13] unmix the latent HSI into endmember and abundance matrices and then constrain the appropriate structures for those factored matrices to regularize the super-resolution problem. Tensor factorization-based [14; 15] and Deep Learning-based methods have also been developed in the literature [16; 17]. A brief review on the related works is given in Section 2. An equivalent definition of an HSI is given by the acquisition of a stack of images representing the radiance in the respective spectral band. Pixels in an HSI often include a mixture of multiple materials. Hence, the HSIs also seem as three-dimensional data cubes, which contain the spatial dimension and the spectral dimension. It leads to an observation that hyperspectral super-resolution is linked to the well-known problem of _hyperspectral unmixing_. Hyperspectral unmixing characterizes the complex mixture in a spectral signal as a combination of pure spectral compositions (which are called _endmembers_) and their corresponding weights (which are called _abundances_) [18]. There are two types of mixing models to describe HSIs: the linear mixing model and various nonlinear mixing models. The simplest model that meets the data are usually the most effective. The linear mixing model has been widely used, as it is simple and has clear physical meaning [19]. It assumes that each pixel of the HSI is a linear combination of endmembers [19], which leads to computationally efficient solutions. In the HSI super-resolution method proposed in this paper, we unmix the low spatial resolution HSI and high spatial resolution conventional image separately, following which the high spatial resolution HSI is estimated. The hyperspectral unmixing method introduced in this paper is based on spatial group sparsity regularization and the \(\ell_{2,1}\) norm.