Understanding Pan-Sharpening via Generalized Inverse

Pan-sharpening algorithms utilize a panchromatic image and a multispectral image to generate a high spatial and high spectral image. However, the optimizations of the algorithms are designed with different standards. We employ a simple matrix equation to describe the Pan-sharpening problem. The conditions for the existence of a solution and the acquisition of spectral and spatial resolution are discussed. A down-sampling enhancement method is introduced to improve the estimation of spatial and spectral down-sample matrices. Using generalized inverse theory, we discovered two kinds of solution spaces of generalized inverse matrix formulations, which correspond to the two prominent classes of Pan-sharpening methods: component substitution and multi-resolution analysis. Specifically, the Gram-Schmidt adaptive method is demonstrated to align with the generalized inverse matrix formulation of component substitution. A model prior of the generalized inverse matrix of the spectral function is rendered. Theoretical errors are analyzed. The diffusion prior is naturally embedded with the help of general solution spaces of the generalized inverse form, enabling the acquisition of refined Pan-sharpening results. Extensive experiments, including comparative, synthetic, real-data ablation and diffusion-related tests are conducted. The proposed methods produce qualitatively sharper and superior results in both synthetic and real experiments. The down-sampling enhancement method demonstrates quantitatively and qualitatively better outcomes in real-data experiments. The diffusion prior can significantly improve the performance of our methods across almost all evaluation measures. The generalized inverse matrix theory helps deepen the understanding of Pan-sharpening mechanisms.