Multiband image fusion via linear mappings

Despite the development of more sophisticated airborne systems that are equipped with hy-perspectral sensors, the more cost effective commercially sponsored multispectral satellite sys-tems are still in use today. Consequently, the research topic of multispectral panchromatic image sharpening is still active, whose sole purpose is to produce high spatial resolution imagery while preserving the spectral integrity of the original spectral image. The component substitution injection scheme is still a foundation for many of today’s techniques. Working under the as-sumption that a low-pass filtered panchromatic image can be constructed from a linear mapping of the spectral bands, an unsharp mask can be created and fused into the original multispectral image. The challenge in deciding how the band weights are computed is typically solved using linear regression, however, this produces a greyscale with different global statistics to the stand-ard intensity mapping thus altering the spectral properties of the pan-sharpened image. We tackle this problem with a different approach; that is to produce a method of mapping a four band image into a real greyscale that preserves the global statistics of the standard intensity mapping all the while increasing the correlation with the panchromatic. Starting with colour to greyscale mappings, we produce a closed form solution to a maximum variance greyscale subject to preserving image mean. By exploiting the cubic constraints on the band weights, we reduce the time complexity from that of quadratic programming to one which is limited by sorting three numbers. We further expand on this optimisation to produce greyscales with maximum image variance which has application to multi-banded images for dimensions limited by the ability to compute a convex-hull. Lastly we adapt our solution by introducing a quadratic constraint on image variance. This method is based on a novel solution to the geometric intersection of a hyper-ellipse with hyper-planes. Lastly we further this solution to finding the band weights that will enhance the correlation of our mapping to the panchromatic image. Post image evaluation our technique was seen to improve on classical component substitution methods