Pan-sharpening algorithm utilizes panchromatic image and multispectral image to obtain a high spatial and high spectral image. However, the optimizations of the algorithms are designed with different standards. We adopt the simple matrix equation to describe the Pan-sharpening problem. The solution existence condition and the acquirement of spectral and spatial resolution are discussed. A down-sampling enhancement method was introduced for better acquiring the spatial and spectral down-sample matrices. By the generalized inverse theory, we derived two forms of general inverse matrix formulations that can correspond to the two prominent classes of Pan-sharpening methods, that is, component substitution and multi-resolution analysis methods. Specifically, the Gram Schmidt Adaptive(GSA) was proved to follow the general inverse matrix formulation of component substitution. A model prior to the general inverse matrix of the spectral function was rendered. The theoretical errors are analyzed. Synthetic experiments and real data experiments are implemented. The proposed methods are better and sharper than other methods qualitatively in both synthetic and real experiments. The down-sample enhancement effect is shown of better results both quantitatively and qualitatively in real experiments. The generalized inverse matrix theory help us better understand the Pan-sharpening.

翻译：全色锐化算法利用全色图像和多光谱图像获取高空间分辨率和高光谱分辨率的图像。然而，不同算法的优化标准各异。我们采用简单矩阵方程描述全色锐化问题，讨论了解的存在条件以及光谱与空间分辨率的获取方式。引入了一种降采样增强方法，以更好地获取空间和光谱降采样矩阵。基于广义逆理论，推导出了两种通用逆矩阵形式，分别对应全色锐化方法的两大主流类别：分量替换法和多分辨率分析法。具体而言，证明了自适应格拉姆-施密特方法（GSA）符合分量替换法的通用逆矩阵形式，并提出了光谱函数广义逆矩阵的先验模型。理论误差被系统分析，同时实施了合成实验与真实数据实验。在合成与真实实验中，所提方法在定性比较中优于其他方法，表现更清晰锐利；真实实验中降采样增强效果在定性与定量指标上均更优。广义逆矩阵理论有助于深入理解全色锐化。