Heuristic imaging from generic projections: backprojection outside the range of the Radon transform
MathematicS In Action, Volume 9 (2020) no. 1, pp. 1-16.

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.

Received: 2018-10-03
Revised: 2019-11-08
Accepted: 2019-11-08
Published online: 2020-04-01
DOI: https://doi.org/10.5802/msia.12
Classification: 78A97,  94A12,  44A12
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},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {9},
     number = {1},
     year = {2020},
     pages = {1-16},
     doi = {10.5802/msia.12},
     language = {en},
     url = {msia.centre-mersenne.org/item/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/item/MSIA_2020__9_1_1_0/

[1] G. Berginc; M. Jouffroy Optronic system and method dedicated to identification for formulating three-dimensional images, US patent 20110254924 A1, European patent 2333481 A1, FR 09 05720 B1, 2009

[2] G. Berginc; M. Jouffroy 3D Laser Imaging, PIERS Online, Tome 7 (2011) no. 5, pp. 411-415 | Article

[3] L. Borg; J. Frikel; J. S. Jørgensen; E. T. Quinto Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data, SIAM J. Imaging Sci., Tome 11 (2018) no. 4, pp. 2786-2814 | Article | MR 3884226 | Zbl 07115015

[4] A. Faridani; E. L. Ritman; K. T. Smith Local tomography, SIAM J. Appl. Math., Tome 52 (1992) no. 2, pp. 459-484 | Article | MR 1154783 | Zbl 0758.65081

[5] J. Frikel; E. T. Quinto Characterization and reduction of artifacts in limited angle tomography, Inverse Probl., Tome 29 (2013) no. 12, 125007, 21 pages | Article | MR 3129117 | Zbl 1284.92044

[6] D. T. Gering; W. M. Wells Object modeling using tomography and photography, Multi-View Modeling and Analysis of Visual Scenes, 1999.(MVIEW’99) Proceedings. (1999), pp. 11-18

[7] A. Grigis; J. Sjöstrand Microlocal analysis for differential operators: an introduction Tome 196, Cambridge University Press, 1994 | Zbl 0804.35001

[8] F. K. Knight; S. R. Kulkarni; R. M. Marino; J. K. Parker Tomographic Techniques Applied to Laser Radar Reflective Measurements, Lincoln Laboratory Journal, Tome 2 (1989) no. 2, pp. 143-160

[9] A. Laurentini The visual hull concept for silhouette-based image understanding, IEEE Trans. Pattern Anal. Mach. Intell., Tome 16 (1994) no. 2, pp. 150-162 | Article

[10] F. Natterer; F. Wübbeling Mathematical methods in image reconstruction, SIAM Monographs on Mathematical Modeling and Computation, Tome 5, Society for Industrial and Applied Mathematics, 2001

[11] E. T. Quinto Singularities of the X-ray transform and limited data tomography in 2 and 3 , SIAM J. Math. Anal., Tome 24 (1993) no. 5, pp. 1215-1225

[12] A. G. Ramm; A. I. Katsevich The Radon transform and local tomography, CRC Press, 1996 | Zbl 0863.44001

[13] G. Rigaud; J.-B. Bellet; G. Berginc; I. Berechet; S. Berechet Reflective Imaging Solved by the Radon Transform, IEEE Geoscience and Remote Sensing Letters, Tome 13 (2016), pp. 936-938 | Article

[14] P. Schapira Tomography of constructible functions, International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes (1995), pp. 427-435 | Article | Zbl 0878.44002

[15] P. Stefanov Microlocal Analysis Methods, Encyclopedia of Applied and Computational Mathematics, Springer, 2015, pp. 914-920 | Article