Helium atom scattering (HAS) is a well established technique for examining the surface structure and dynamics of materials at atomic sized resolution. The HAS technique Helium spin-echo spectroscopy opens up the possibility of compressing the data acquisition process. Compressed sensing (CS) methods demonstrating the compressibility of spin-echo spectra are presented. In addition, wavelet based CS approximations, founded on a new continuous CS approach, are used to construct continuous spectra that are compatible with variable transformations to the energy/momentum transfer domain. Moreover, recent developments on structured multilevel sampling that are empirically and theoretically shown to substantially improve upon the state of the art CS techniques are implemented. These techniques are demonstrated on several examples including phonon spectra from a gold surface.
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