Electron microscopy has shown to be a very powerful tool to deeply analyze the chemical composition at various scales. However, many samples can not be analyzed with an acceptable signal-to-noise ratio because of the radiation damage induced by the electron beam. Particularly, electron energy loss spectroscopy (EELS) which acquires a spectrum for each spatial position requires high beam intensity. Scanning transmission electron microscopes (STEM) sequentially acquire data cubes by scanning the electron probe over the sample and record a spectrum for each spatial position. Recent works developed new acquisition procedures, which allow for partial acquisition schemes following a predetermined scan pattern. A reconstruction of the full data cube is conducted as a post-processing step. A multi-band image reconstruction procedure which exploits the spectral structure and the spatial smoothness of STEM-EELS images is explained here. The performance of the proposed scheme is illustrated thanks to experiments conducted on a realistic phantom dataset as well as real EELS spectrum-image.