# Pansharpening by Complementing Compressed Sensing with Spectral Correction
## 1 Introduction
introduced non-negative matrix factorization and proposed estimation of a high-resolution matrix by solving an optimization problem of decomposing the matrix into basis and coefficient matrices [32]. [PERSON] et al. created a dictionary of high-resolution MS image features by incorporating the tradeoff parameter [24; 26] and back-projection [13; 33]. It was shown that the spectral distortion was reduced by incorporating back-projection. [PERSON] proposed the cross-resolution projection and the offset [18]. The cross-resolution projection generates high-resolution MS images by assuming that the position estimated by CS is the same for high-resolution and low-resolution images. The offset is used to adjust the reconstructed image. In these studies, the results of PS depended on the model selection and dictionary selection. Various studies on the structure and construction of dictionaries, model construction, and optimization processes have been conducted. In addition, it was pointed out that CS-based reconstruction does not guarantee the reproduction of the original image [13; 29; 34], which means that the spatial characteristics may not be incorporated accurately. The fact that it is not an exact reconstruction should be considered when the method is used for spectral analysis. The process of back-projection was introduced by [PERSON] et al. [13] to improve the reconstructed image. This process was also incorporated by [PERSON] et al. [34] to reduce the spectral distortion. On the other hand, it is also known that ringing artefacts occur when back-projection is performed. In the PS process, low spectral distortion is important as well as high resolution. It is important to consider how to achieve the fidelity of the reproduced image to the ideal image in terms of the spectral and spatial resolution. In this paper, we focused on reducing the spectral distortion more effectively than the back-projection for resolution enhancement of a visible light image. To this end, we propose a method for pansharpening by combining the CS technique and a component substitution method that calculates the intensity with high spatial resolution and low spectral distortion to enhance the reproducibility. As the component substitution, spectrum correction using modeled panchromatic image (SCMP) [35] is introduced. The observation band of the PAN sensor of IKONOS, an earth observation satellite, was 526-929 nm. The red, green, blue, and NIR bands of the MS sensor of IKONOS were 632-698 nm, 505-595 nm, 445-516 nm, and 757-853 nm, respectively. Therefore, the PAN sensor covered the observation band of the MS sensor including NIR. The NIR information needs to be included when generating PS images using component substitution in order to avoid spectral distortion. The SCMP is a model that can correct this distortion. On the other hand, processing using the CS is expected to have high data reproducibility and it reflects the characteristics of the input image, while the resolution is lower than that generated by the SCMP. In order to improve the fidelity of the reproduced image to the original image, a tradeoff process was applied on the high-resolution intensity images obtained by the CS and the SCMP. It was found that a spatial resolution equivalent to that of PAN image was obtained and spectral distortion was reduced by the proposed method.