  Wavelets have been used extensively for several years now in astronomy for
many purposes, ranging from data filtering and deconvolution, to star and
galaxy detection or cosmic ray removal. More recent sparse representations such
ridgelets or curvelets have also been proposed for the detection of anisotropic
features such cosmic strings in the cosmic microwave background.
  We review in this paper a range of methods based on sparsity that have been
proposed for astronomical data analysis. We also discuss what is the impact of
Compressed Sensing, the new sampling theory, in astronomy for collecting the
data, transferring them to the earth or reconstructing an image from incomplete
measurements.
