Abstract
Reconstruction of an object from a series of silhouettes is obtained through a binary geometric tomography technique since both the objects and the projections, which are measured as a series of silhouettes, are binary. In this paper, we formulate the method Shape from Silhouettes in two- and in three-dimensional discrete space. This approach to the problem derives an ambiguity theorem for the reconstruction of objects in the discrete space. The theorem shows that Shape from Silhouettes in discrete space results in an object which is over-reconstructed, if the object is convex. Furthermore, we show that in three-dimensional space, it is possible to reconstruct a class of non-convex objects from a set of silhouettes although in the plane a non-convex object is not completely reconstructible from projections.
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Notes
- 1.
The cross-section of the cone l(s), s∈Ω(s) with the hyperplane s ⊤ x=d is geometrically defined as the silhouette generated by source s.
References
Aloimonos, J.: Visual shape computation. Proc. IEEE 76, 899–916 (1988)
Boltyanski, V., Martin, H., Soltan, P.S.: Excursions into Combinatorial Geometry. Springer, Berlin (1997)
Chollet, A., Wallet, G., Fuchs, L., Andres, A., Largeteau-Skapin, G.: Ω-Arithmetization: a discrete multi-resolution representation of real functions. In: IWCIA 2009. Lecture Notes in Computer Science, vol. 5852, pp. 316–329 (2009)
Dobkin, D.P., Edelsbrunner, H., Yap, C.K.: Probing convex polytopes. In: Proc. 18th ACM Symposium on Theory of Computing, pp. 424–432 (1986)
Frank, J. (ed.): Electron Tomography: Methods for Three-Dimensional Visualization of Structures in the Cell, 2nd edn. Springer, Berlin (2006)
Herman, G.T., Kuba, A. (eds.): Advances in Discrete Tomography and Its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2007)
Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms, and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (1999)
Imiya, A., Kawamoto, K.: Shape reconstruction from an image sequences. In: Lecture Notes in Computer Science, vol. 2059, pp. 677–686 (2001)
Imiya, A., Kawamoto, K.: Mathematical aspects of shape reconstruction from an image sequence. In: Proc. 1st Intl. Symp. 3D Data Processing Visualization and Transformations, pp. 632–635 (2002)
Jordan, E.K., Kirby, R.M., Silvia, C., Peters, T.: Through a new looking glass∗: mathematically precise visualization. SIAM News 43(5) (2010). Archive: http://www.siam.org/news/archives.php
Kawamoto, K., Imiya, A.: Detection of spatial points and lines by random sampling and voting process. Pattern Recognit. Lett. 22, 199–207 (2001)
Kutulakos, K., Seitz, S.M.: A theory of shape by space carving. In: Proceedings of 7th ICCV, vol. 1, pp. 307–314 (1999)
Laurentini, A.: The visual hull concept for silhouette-bases image understanding. IEEE Trans. Pattern Anal. Mach. Intell. 16, 150–163 (1994)
Laurentini, A.: How a 3D shape can be understood from 2D silhouettes. IEEE Trans. Pattern Anal. Mach. Intell. 17, 88–195 (1995)
Li, R.S.-Y.: Reconstruction of polygons from projections. Inf. Process. Lett. 28, 235–240 (1988)
Lindembaum, M., Bruckstein, A.: Reconstructing a convex polygon from binary perspective projections. Pattern Recognit. 23, 1343–1350 (1990)
Natterer, F.: The Mathematics of Computerized Tomography. Wiley, New York (1986)
Plitzko, J.M., Baumeister, W.: Cryoelectron tomography (CET). In: Hawkes, P.W., Spence, J.C.H. (eds.) Science of Microscopy, 2nd edn., pp. 535–604. Springer, Berlin (2006)
Prince, J.L., Willsky, A.S.: Reconstructing convex sets from support line measurements. IEEE Trans. Pattern Anal. Mach. Intell. 12, 377–389 (1990)
Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss. 69, 262–277 (1917)
Rao, A.S., Goldberg, Y.K.: Shape from diameter: recognizing polygonal parts with parallel-jaw gripper. Int. J. Robot. Res. 13, 16–37 (1994)
Said, M., Lachaud, J.-O., Feschet, F.: Multiscale discrete geometry. In: DGCI 2009. Lecture Notes in Computer Science, vol. 5810, pp. 118–131 (2009)
Said, M., Lachaud, J.-O., Feschet, F.: Multiscale analysis of digital segments by intersection of 2D digital lines. In: Proc. ICPR 2010, pp. 4097–4100 (2010)
Skiena, S.S.: Interactive reconstruction via geometric probing. Proc. IEEE 80, 1364–1383 (1992)
Skiena, S.S.: Probing convex polygon with half-planes. J. Algorithms 12, 359–374 (1991)
Tuy, H.K.: An inversion formula for cone-beam reconstruction. SIAM J. Appl. Math. 43, 546–552 (1983)
Acknowledgements
This research was supported by the grant “Computational anatomy for computer-aided diagnosis and therapy: Frontiers of medical image sciences” funded by the Grant-in-Aid for Scientific Research on Innovative Areas, MEXT, Japan, the Grants-in-Aid for Scientific Research funded by Japan Society of the Promotion of Sciences, Japan. A part of the research is based on the Master’s thesis of Kosuke Sato submitted to the School of Science and Technology, Chiba University 2007.
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Imiya, A., Sato, K. (2012). Shape from Silhouettes in Discrete Space. In: Brimkov, V., Barneva, R. (eds) Digital Geometry Algorithms. Lecture Notes in Computational Vision and Biomechanics, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4174-4_11
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DOI: https://doi.org/10.1007/978-94-007-4174-4_11
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