Reflective tomography is an efficient method for optical imaging in the visible and near infrared ranges. It computes empirical reconstructions based on algorithms from X-ray tomography. This subject introduces mathematical gaps to be filled, about the meaning of the reconstructions, and about their artifacts. To tackle these questions, we study more generally the filtered backprojection on projections outside the range of the Radon transform. We consider generic projections that can involve any kind of physical and geometric parameters. We claim that the backprojection contains partially the geometry of the original scene. More precisely, we compare the singularities of the backprojection with the singularities of a representation of the scene. This comparison of wavefront sets, inspired by studies of the artifacts in X-ray tomography, is based on microlocal analysis. It gives a precise meaning to the well-reconstructed geometry, describes the invisible parts, and the artifacts. We illustrate the heuristic and the analysis principle on canonical cases that belong to various fields: shape from silhouettes, constructible tomography, cloaking, reconstruction from cartoon images, imaging of occluded lambertian objects. Numerical results show the relevance of the heuristic and its analysis. In a word, this study provides a mathematical framework that covers the solver of reflective tomography, and exhibits an imaging method whose range of application is wide.

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Keywords: 3D imaging, computational optics, reconstruction, Radon transform, geometric tomography

@article{MSIA_2020__9_1_1_0, author = {Jean-Baptiste Bellet and G\'erard Berginc}, title = {Heuristic imaging from generic projections: backprojection outside the range of the {Radon} transform}, journal = {MathematicS In Action}, pages = {1--16}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {9}, number = {1}, year = {2020}, doi = {10.5802/msia.12}, language = {en}, url = {https://msia.centre-mersenne.org/articles/10.5802/msia.12/} }

TY - JOUR TI - Heuristic imaging from generic projections: backprojection outside the range of the Radon transform JO - MathematicS In Action PY - 2020 DA - 2020/// SP - 1 EP - 16 VL - 9 IS - 1 PB - Société de Mathématiques Appliquées et Industrielles UR - https://msia.centre-mersenne.org/articles/10.5802/msia.12/ UR - https://doi.org/10.5802/msia.12 DO - 10.5802/msia.12 LA - en ID - MSIA_2020__9_1_1_0 ER -

%0 Journal Article %T Heuristic imaging from generic projections: backprojection outside the range of the Radon transform %J MathematicS In Action %D 2020 %P 1-16 %V 9 %N 1 %I Société de Mathématiques Appliquées et Industrielles %U https://doi.org/10.5802/msia.12 %R 10.5802/msia.12 %G en %F MSIA_2020__9_1_1_0

Jean-Baptiste Bellet; Gérard Berginc. Heuristic imaging from generic projections: backprojection outside the range of the Radon transform. MathematicS In Action, Volume 9 (2020) no. 1, pp. 1-16. doi : 10.5802/msia.12. https://msia.centre-mersenne.org/articles/10.5802/msia.12/

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