A solution to Hammer's X-ray reconstruction problem

Advances in Mathematics

Volumn 214, Issue 1, 2007, Pages 323-343

  Gardner, Richard J ^a   Kiderlen, Markus ^b  

^a WESTERN WASHINGTON UNIVERSITY   (United States)

^b AARHUS UNIVERSITY   (Denmark)

Author keywords

Algorithm; Convex body; Convex polytope; Geometric tomography; Least squares; X ray

Indexed keywords

EID: 34250218200     PISSN: 00018708     EISSN: 10902082     Source Type: Journal    
DOI: 10.1016/j.aim.2007.02.005     Document Type: Article

References (22)

1. n. Elem. Math. 28 (1973) 43-44
2. Brunetti S., and Daurat A. Stability in discrete tomography: Some positive results. Discrete Appl. Math. 147 (2005) 207-226
3. Brunetti S., and Daurat A. Determination of Q-convex bodies by X-rays. Electron. Notes Discrete Math. vol. 20 (2005) 67-81
4. Falconer K.J. X-ray problems for point sources. Proc. London Math. Soc. (3) 46 (1983) 241-262
5. Falconer K.J. Hammer's X-ray problem and the stable manifold theorem. J. London Math. Soc. (2) 28 (1983) 149-160
6. Gardner R.J. Symmetrals and X-rays of planar convex bodies. Arch. Math. (Basel) 41 (1983) 183-189
7. Gardner R.J. Chord functions of convex bodies. J. London Math. Soc. (2) 36 (1987) 314-326
8. Gardner R.J. Geometric Tomography (1995), Cambridge Univ. Press, New York
9. Gardner R.J. Geometric Tomography. second ed. (2006)
10. Gardner R.J., and Gritzmann P. Discrete tomography: Determination of finite sets by X-rays. Trans. Amer. Math. Soc. 349 (1997) 2271-2295
11. Gardner R.J., and McMullen P. On Hammer's X-ray problem. J. London Math. Soc. (2) 21 (1980) 171-175
12. Gardner R.J., Kiderlen M., and Milanfar P. Convergence of algorithms for reconstructing convex bodies and directional measures. Ann. Statist. 34 (2006) 1331-1374
13. Gustafsson B., He C., Milanfar P., and Putinar M. Reconstructing planar domains from their moments. Inverse Problems 16 (2000) 1053-1070
14. Hadwiger H. Kurzer Beweis der isoperimetrischen Ungleichung für konvexe Bereiche. Elem. Math. 3 (1948) 111-112
15. Hammer P.C. Problem 2. Proc. Sympos. Pure Math., vol. VII: Convexity. Providence, RI (1963), Amer. Math. Soc., Providence, RI 498-499
16. Kölzow D., Kuba A., and Volčič A. An algorithm for reconstructing convex bodies from their projections. Discrete Comput. Geom. 4 (1989) 205-237
17. Lam D., and Solmon D.C. Reconstructing convex polygons in the plane from one directed X-ray. Discrete Comput. Geom. 26 (2001) 105-146
18. Milanfar P., Karl W.C., and Willsky A.S. Reconstructing binary polygonal objects from projections: A statistical view. Graph. Models Image Process. 56 (1994) 371-391
19. Schneider R. Convex Bodies: The Brunn-Minkowski Theory (1993), Cambridge Univ. Press, Cambridge
20. Shiryaev A.N. Probability. second ed. (1996), Springer, New York
21. van de Geer S. Applications of Empirical Process Theory (2000), Cambridge Univ. Press, New York
22. Volčič A. A three-point solution to Hammer's X-ray problem. J. London Math. Soc. (2) 34 (1986) 349-359