We present a variational approach to reconstruct a multispectral image from an array of diffraction patterns captured in one monochromatic image. The mathematical model relies on the superposition of wavelength dependent calibration diffraction patterns and a spectral regularization with second order differences. The necessary preprocessing steps which influence the mathematical setting are discussed in detail. For computing a minimizer of our model we propose an active set method with model specific modifications. We validate our approach by experimental results for spectra within the range of 400 nm to 700 nm.