Abstract
We give some uniqueness results for the problem of determining a finite set in the plane knowing its projections alongm directions. We apply the results to the problem of the reconstruction of a homogeneous convex body with a finite set of spherical disjoint holes. Ifm X-ray pictures with directions in some plane are given, then the problem is well posed provided the number of the holes is less than or equal tom and the set of the directions satisfies a suitable condition.
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Y. Das and W. M. Boerner, On radar target shape estimation using algorithms for reconstruction from projections,IEEE Trans. Antennas and Propagation,26 (1978), 274–279.
R. J. Gardner and P. McMullen, On Hammer's X-ray problem,J. London Math.-Soc. 21(2) (1980), 171–175.
M. Longinetti, An isoperimetric inequality for convex polygons and convex sets with the same symmetrals,Geom. Dedicata 20 (1986), 27–41.
G. G. Lorentz, A problem of plane measure,Amer. J. Math. 71 (1949), 417–426.
C. Mägerl and A. Volčič, On the well-posedness of the Hammer X-ray problem,Ann. Mat. Pura. Appl. 144(4) (1986), 173–182.
M. E. Mermikides, Data analysis for bubble chamber and hybrid systems,Proc. 1980 CERN School of Computing, CERN, Geneva, 1981, pp. 106–135.
A. Renyi, On projections of probability distributions,Acta Math. Hungar.,3 (1952), 131–142.
K. T. Smith, D. C. Solmon, and S. L Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs,Bul. Amer. Math. Soc. 83 (1977), 1227–1270.
A. Volčič, Well-posedness of the Gardner-McMullen reconstruction problem,Proc. Conf. Measure Theory, Oberwolfach, 1983, Lecture Notes in Mathematics, Vol. 1089, Springer-Verlag, Berlin, 1984.
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Bianchi, G., Longinetti, M. Reconstructing plane sets from projections. Discrete Comput Geom 5, 223–242 (1990). https://doi.org/10.1007/BF02187787
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DOI: https://doi.org/10.1007/BF02187787