Reconstruction of convex bodies from brightness functions

Discrete and Computational Geometry

Volumn 29, Issue 2, 2003, Pages 279-303

  Gardner, R J ^a   Milanfar, Peyman ^b  

^a WESTERN WASHINGTON UNIVERSITY   (United States)

^b University of California Santa Cruz   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0012933716     PISSN: 01795376     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00454-002-0759-2     Document Type: Article

References (26)

1. C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Software 22 (1996), 469-483.
2. J. Bourgain and J. Lindenstrauss. Projection bodies, in: Geometric Aspects of Functional Analysis (1986/7), Lecture Notes in Mathematics 1317, Springer-Verlag, Berlin, 1988, pp. 250-270.
3. B. Büeler, A. Enge, and K. Fukuda, Exact volume computation for polytopes: a practical study, in: Polytopes - Combinatorics and Compulation (Oberwolfach. 1997), Birkhäuser, Basel, 2000. pp. 131-154.
4. S. Campi, Recovering a centred convex body from the areas of its shadows: a stability estimate. Ann. Mat. Pura Appl. (4) 151 (1988), 289-302.
5. S. Campi, A. Colesanti, and P. Gronchi, Convex bodies with extremal volumes having prescribed brightness in finitely many directions, Geom. Dedicata 57 (1995), 121-133.
6. S. Campi, D, Haas, and W. Weil, Approximation of zonoids by zonotopes in fixed directions, Discrete Comput. Geom. 11 (1994), 419-431.
7. J. H. Conway, R. H. Hardin, and N. J. A. Sloane, Packing lines, planes, etc.: packings in Grassmannian spaces, Experiment. Math. 5 (1996), 139-159.
8. R. J. Gardner, Geometric Tomography, Cambridge University Press, New York, 1995.
9. R. J. Gardner and A. Volčič. Determination of convex bodies by their brightness functions, Mathematika 40(1993), 161-168.
10. G. H. Golub and V. Pereyra, The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate, SIAM J. Numer. Anal. 10 (1973), 413-432.
11. P. Gritzmann and A. Hufnagel, On the algorithmic complexity of Minkowski's reconstruction problem, J. London Math. Soc. (2) 59 (1999), 1081-1100.
12. P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. I. Containment problems, Discrete Math. 13(1994), 129-174.
13. P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. II. Volume and mixed volumes, in: Polytopes: Abstract, Convex and Computational (Scarborough, ON, 1993). Kluwer, Dordrecht, 1994, pp. 373-466.
14. H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press. New York, 1996.
15. M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988.
16. W. C. Karl, Reconstructing Objects from Projections, Ph.D. dissertation, Department of Electrical Engineering and Computer Science. Massachusetts Institute of Technology, Cambridge. MA. 1991.
17. M. Kiderlen, Non-parametric estimation of the directional distribution of stationary line and fibre processes, Adv. in Appl. Probab. 33 (2001). 6-24.
18. L. Lamberg, On the Minkowski problem and the lightcurve operator, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 87 (1993).
19. n", J. Optim. Theory Appl. 39 (1983), 363-377.
20. A. S. Lele, Convex Set Reconstruction from Support Line Measurements and Its Application to Laser Radar Data, M.Sc. dissertation, Department of Electrical Engineering and Computer Science. Massachusetts Institute of Technology, Cambridge, MA, 1991.
21. A. S. Lele, S. R. Kulkarni. and A. S. Willsky, Convex polygon estimation from support line measurements and applications to target reconstruction from laser radar data, J. Opt. Soc. Amer. A 9 (1992), 1693-1714.
22. J. Lemordant, P. D. Tao, and H. Zouaki, Modélisation et optimisation numérique pour la reconstruction d'un polyèdre à partir de son image gaussienne généralisée, RAIRO Modél. Math. Anal. Numér. 27 (1993), 349-374.
23. J. J. Little. An iterative method for reconstructing convex polyhedra from extended Gaussian images, in: Proc. AAAI, National Conf. Artificial Intelligence (Washington, D.C.), 1983, pp. 247-250.
24. C. A. Rogers, Covering a sphere with spheres, Mathematika 10 (1963). 157-164.
25. R. Schneider. Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge. 1993.
26. M. A. Sumbatyan and E. A. Troyan, Reconstruction of the shape of a convex defect from a scattered wave field in the ray approximation, J. Appl. Math. Mech. 56 (1992), 464-468.