  The direct measurement of a complex wavefunction has been recently realized
by using weak-values. In this paper, we introduce a method that exploits
sparsity for compressive measurement of the transverse spatial wavefunction of
photons. The procedure involves a weak measurement in random projection
operators in the spatial domain followed by a post-selection in the momentum
basis. Using this method, we experimentally measure a 192-dimensional state
with a fidelity of $90\%$ using only $25$ percent of the total required
measurements. Furthermore, we demonstrate measurement of a 19200 dimensional
state; a task that would require an unfeasibly large acquiring time with the
conventional direct measurement technique.
