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.
Differential phase-contrast interior tomography allows reconstruction of a refractive index distribution over a region of interest (ROI) for visualization and analysis of structures inside a large biological specimen. In the imaging mode, x-ray scanning only targets an ROI in an object and a narrow beam passes through the object, allowing a significant reduction of both radiation dose and system cost. Inspired by recently developed compressive sensing theory, in a numerical analysis framework we show that accurate interior reconstruction can be achieved on an ROI from truncated differential projection data through the ROI via the total variation minimization, assuming a piecewise constant distribution of the refractive indices in the ROI. Then, we develop a practical iterative algorithm for such an interior reconstruction and perform numerical experiments to demonstrate the feasibility of the proposed approach. We report on a method for tomographic phase contrast imaging of centimeter sized objects. As opposed to existing techniques, our approach can be used with low-brilliance, lab based x-ray sources and thus is of interest for a wide range of applications in medicine, biology, and nondestructive testing. The work is based on the recent development of a hard x-ray grating interferometer, which has been demonstrated to yield differential phase contrast projection images. Here we particularly focus on how this method can be used for tomographic reconstructions using filtered back projection algorithms to yield quantitative volumetric information of both the real and imaginary part of the samples's refractive index A novel quantitative method, based on the moire effect, for mapping ray deflections of a collimated light beam is described and. demonstrated. This method, which does not require coherent light, can replace interferometric techniques in many cases. The proposed setup is simple, and the interpretation is straightforward. We demonstrate deflection mapping of a candle’s flame with a resolution of 5 × 10−5 rad and lens mapping with a resolution of 10−2 rad. An analysis of the ray deflection for index-of-refraction mapping is provided.