This thesis contributes to the fields of data compression and compressed sensing and their application to imaging mass spectrometry and sporadic communication. Compressed sensing is mainly built on the knowledge that most data is compressible or sparse, meaning that most of its content is redundant and not worth being measured. As a main result in this work, a compressed sensing model for imaging mass spectrometry is introduced. It combines peak-picking of the spectra and denoising of the m/z-images A robustness result for the reconstruction of compressed measured data is presented which generalizes known reconstruction guarantees.