# Pansharpening by Complementing Compressed Sensing with Spectral Correction
## 1 Introduction  
The optical sensors installed in satellites acquire panchromatic (PAN) and multispectral (MS) region images. PAN sensors observe a wide range of visible and near-infrared (NIR) regions as one band with a high spatial resolution, and the MS sensor observes multiple bands. Pansharpening (PS) is a method used to generate a high-resolution MS images from these two types of data. Due to physical constraints [1], MS sensors are not designed to acquire high-resolution images. In general, high-resolution MS images are obtained via the pansharpening process. PS techniques are used for change detection, target recognition, classification, backgrounds for map application, visual image analysis, etc. PS has been studied for decades [2; 3], and its methodologies can roughly be classified into four types. The first is methods used to generate a pansharpened image by substituting the intensity component of the MS images for that of the PAN image via component substitution. The intensity-hue-saturation (IHS) transform [4], generalized IHS method (GIHS) [5], Brovey transform [6], principal component analysis [6], and Gram-Schmidt transform [7] belong to this group. The second group contains methods used to extract high-frequency components of PAN images via multiresolution analysis(MRA) and then add them to the MS images. To extract high-frequency components, high-pass filter, decimated wavelet transforms [8], a \"trous\" wavelet transform [9], Laplacian pyramid [10; 11], and non-subsampled contourlet transform [12] methods are used. The third group contains methods that use machine-learning techniques such as compressed sensing [13] and deep learning [14; 15]. The fourth group is the hybrid methods that combine multiple methods described above. The methods proposed by [PERSON] et al. [16], [PERSON] et al. [17], and [PERSON] [18] are the ones that combined the component substitution method and the MRA. The combination of the MRA, convolutional neural network (CNN), and sparse modeling was proposed by [PERSON] et al. [19], and the combination of the MRA and CNN was proposed by [PERSON] et al. [20]. Recently, proposals for hybrid methods using machine-learning techniques have been increasing. Many methods based on compressed sensing (CS) have been proposed based on the 2006 theory [21]. Since [PERSON] et al. [13] proposed a method for super-resolution, it has been frequently applied to PS. [PERSON] et al. proposed a method using a learning-free dictionary and CS [22], followed by the proposal of a method using dictionary learning and CS [23]. Although they showed the possibility of using a dictionary without learning, the feasibility of these methods was low because high-resolution MS images that were not realistically available were necessary for dictionary construction. [PERSON] et al. proposed a method using a high-resolution dictionary generated using a set of pairs of low-resolution MS images and high-resolution PAN images [24]. They then proposed a method for reconstruction by calculating sparse representations in two stages using a learning-free dictionary [25]. [PERSON] et al. constructed a pair of high-resolution and low-resolution learning-free dictionaries from PAN images [26; 27], [PERSON] et al. proposed a method called online coupled dictionary learning (OCDL) [28], which iteratively performs dictionary learning and reconstruction processes until the reconstructed image becomes stable, based on the sparse representation (SR) theory described by [PERSON] [29]. SR theory shows that better reconstruction results are obtained when the dictionary's atoms are highly related to the reconstructed image. [PERSON] et al. proposed the generation of a learning-free dictionary using the high-resolution and low-resolution dictionaries as the detailed image information [30]. [PERSON] et al. proposed a learning-free dictionary with low-resolution PAN images and high-resolution detailed information extracted from by the ripplet transform of the edges and textures of high-resolution PAN images [31]. [PERSON] et al.