  Optical Deflectometric Tomography (ODT) provides an accurate characterization
of transparent materials whose complex surfaces present a real challenge for
manufacture and control. In ODT, the refractive index map (RIM) of a
transparent object is reconstructed by measuring light deflection under
multiple orientations. We show that this imaging modality can be made
"compressive", i.e., a correct RIM reconstruction is achievable with far less
observations than required by traditional Filtered Back Projection (FBP)
methods. Assuming a cartoon-shape RIM model, this reconstruction is driven by
minimizing the map Total-Variation under a fidelity constraint with the
available observations. Moreover, two other realistic assumptions are added to
improve the stability of our approach: the map positivity and a frontier
condition. Numerically, our method relies on an accurate ODT sensing model and
on a primal-dual minimization scheme, including easily the sensing operator and
the proposed RIM constraints. We conclude this paper by demonstrating the power
of our method on synthetic and experimental data under various compressive
scenarios. In particular, the compressiveness of the stabilized ODT problem is
demonstrated by observing a typical gain of 20 dB compared to FBP at only 5% of
360 incident light angles for moderately noisy sensing.
