Geometric problems from 3D reflective tomography
Résumé
Reflective tomography is an emerging method in three-dimensional optical imaging. It empirically extends the domain of validity of X-ray inversion to the visible and near-infrared spectra. In this paper, we show that this extension introduces mathematical challenges, and offers new opportunities in geometry. In the spirit of reflective tomography, we formulate properly new geometrical problems and we derive a heuristics that solves efficiently original problems of geometric tomography. We discuss this heuristics on a canonical case and on numerical results. The argumentation emphasizes the contribution of the singularities, and shows that the method reconstructs especially features, by backprojection of the discontinuities of the input projections. On one side, this shows to opticians that the scope of reflective tomography covers new possibilities, including imaging of active surfaces. On the other side, we address to mathematicians conjectures based on the previous observations, and we suggest approaches to be explored. In a word, this work lays the foundation for further mathematical studies that could upgrade optical applications.
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