  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.
