Nano-Fourier-transform infrared spectroscopy (nano-FTIR) combines infrared spectroscopy with scanning probe microscopy (SPM) techniques and enables spectroscopic imaging of molecular and electronic properties of matter at nanometer spatial resolution. The spectroscopic imaging can be used to derive chemical mappings, i.e. the spatial distribution of concentrations of the species contained in a given sample. However, due to the sequential scanning principle underlying SPM, recording the complete spectrum over a large spatial area leads to long measurement times. Furthermore, the acquired spectrum often contains additional signals from species and lineshape effects that are not explicitly accounted for. A compressive chemical mapping approach is proposed for undersampled nano-FTIR data that utilizes sparsity of these additional signals in the spectral domain. The approach combines a projection technique with standard compressed sensing, followed by a spatially regularized regression. Using real nano-FTIR measurements superimposed by simulated interferograms representing the chemical mapping of the contained species, it is demonstrated that the proposed procedure performs well even in cases in which the simulated interferograms and the sparse additional signals exhibit a strong spectral overlap.
