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We propose a new efficient region-based scheme for the compression of multispectral remote-sensing images. The region-based description of an image comprises a segmentation map, which singles out the relevant regions and provides their main features, followed by the detailed (possibly lossless) description of each region. The map conveys information on the image structure and could even be the only item of interest for the user; moreover, it enables the user to perform a selective download of the regions of interest, or can be used for high-level data mining and retrieval applications. This approach, with the multiple pieces of information required, may seem inherently inefficient. The goal of this research is to show that, by carefully selecting the appropriate segmentation and coding tools, region-based compression of multispectral images can be also effective in a rate-distortion sense, thus providing an image description that is both insightful and efficient. To this end, we define a generic coding scheme, based on Bayesian image segmentation and on transform coding, where several key design choices, however, are left open for optimization, from the type of transform, to the rate allocation procedure, and so on. Then, through an extensive experimental phase on real-world multispectral images, we gain insight on such key choices, and finally single out an efficient and robust coding scheme, with Bayesian segmentation, class-adaptive Karhunen-Loeve spectral transform, and shape-adaptive wavelet spatial transform, which outperforms state-of-the-art and carefully tuned conventional techniques, such as JPEG-2000 multicomponent or SPIHT-based coders.