How to pan-sharpen images using the Gram-Schmidt pan-sharpen method - A recipe

Abstract

Since its publication in 1998 (Laben and Brower, 2000), the Gram-Schmidt pan-sharpen method has become one of the most popular algorithms to pan-sharpen multispectral (MS) imagery. It outperforms most other pan-sharpen methods in both maximizing image sharpness and minimizing color distortion. It is, on the other hand, also more complex and computationally expensive than most other methods, as it requires forward and backward transforming the entire image. Another complication is the lack of a clear recipe of how to compute the sensor dependent MS to Pan weights that are needed to compute the simulated low resolution pan band. Estimating them from the sensor's spectral sensitivity curves (in different ways), or using linear regression or least square methods are typical candidates which can include other degrees of freedom such as adding a constant offset or not. As a result, most companies and data providers do it somewhat differently. Here we present a solution to both problems. The transform coefficients can be computed directly and in advance from the MS covariance matrix and the MS to Pan weights. Once the MS covariance matrix is computed and stored with the image statistics, any small section of the image can be pan-sharpened on the fly, without having to compute anything else over the entire image. Similarly, optimal MS to Pan weights can be computed directly from the full MS-Pan covariance matrix, guaranteeing optimal image quality and consistency.