  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.
