  Light rays incident on a transparent object of uniform refractive index
undergo deflections, which uniquely characterize the surface geometry of the
object. Associated with each point on the surface is a deflection map (or
spectrum) which describes the pattern of deflections in various directions.
This article presents a novel method to efficiently acquire and reconstruct
sparse deflection spectra induced by smooth object surfaces. To this end, we
leverage the framework of Compressed Sensing (CS) in a particular
implementation of a schlieren deflectometer, i.e., an optical system providing
linear measurements of deflection spectra with programmable spatial light
modulation patterns. We design those modulation patterns on the principle of
spread spectrum CS for reducing the number of observations. The ability of our
device to simultaneously observe the deflection spectra on a dense
discretization of the object surface is related to a Multiple Measurement
Vector (MMV) model. This scheme allows us to estimate both the noise power and
the instrumental point spread function.
  We formulate the spectrum reconstruction task as the solving of a linear
inverse problem regularized by an analysis sparsity prior using a translation
invariant wavelet frame. Our results demonstrate the capability and advantages
of using a CS based approach for deflectometric imaging both on simulated data
and experimental deflectometric data.
  Finally, the paper presents an extension of our method showing how we can
extract the main deflection direction in each point of the object surface from
a few compressive measurements, without needing any costly reconstruction
procedures. This compressive characterization is then confirmed with
experimental results on simple plano-convex and multifocal intra-ocular lenses
studying the evolution of the main deflection as a function of the object point
location.
