Summary: Earth-observing satellites usually not only take ordinary red-green-blue images but also provide several images including the near-infrared and infrared spectrum. These images are called multispectral, for about four to seven different bands, or hyperspectral, for higher-dimensional images of up to 210 bands. The drawback of the additional spectral information is that each spectral band has rather low spatial resolution. In this paper we propose a new variational method for sharpening high-dimensional spectral images with the help of a high resolution gray-scale image while preserving the spectral characteristics used for classification and identification tasks. We describe the application of split Bregman minimization to our energy, prove convergence speed, and compare the split Bregman method to a descent method based on the ideas of alternating directions minimization. Finally, we show results on Quickbird multispectral as well as on AVIRIS hyperspectral data.