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In this paper, we propose an optimum algorithm, in the minimum mean-square-error (mmse) sense, for panchromatic (Pan) sharpening of very high resolution multispectral (MS) images. The solution minimizes the squared error between the original MS image and the fusion result obtained by spatially enhancing a degraded version of the MS image through a degraded version, by the same scale factor, of the Pan image. The fusion result is also optimal at full scale under the assumption of invariance of the fusion parameters across spatial scales. The following two versions of the algorithm are presented: a local mmse (lmmse) solution and a fast implementation which globally optimizes the fusion parameters with a moderate performance loss with respect to the lmmse version. We show that the proposed method is computationally practical, even in the case of local optimization, and it outperforms the best state-of-the-art Pan-sharpening algorithms, as resulted from the IEEE Data Fusion Contest 2006, on true Ikonos and QuickBird data and on simulated Pleiades data.