We demonstrate through numerical simulations with real data
the feasibility of using compressive sensing techniques for the acquisition of spectro-polarimetric
data. This allows us to combine the measurement and the compression process into one
consistent framework. Signals are recovered thanks to a sparse reconstruction
scheme from projections of the signal of interest onto appropriately chosen vectors, typically noise-like
vectors. The compressibility properties of spectral lines are analyzed in detail. The results shown
in this paper demonstrate that, thanks to the compressibility properties of spectral lines, it is
feasible to reconstruct the signals using only a small fraction of the information that is measured
nowadays. We investigate in depth the quality of the reconstruction as a function of the amount of
data measured and the influence of noise. This change of paradigm also allows us to define new instrumental
strategies and to propose modifications to existing instruments in order to take advantage of compressive sensing techniques.