Reconstruction of Two-Valued Functions and Matrices
Part of the
Applied and Numerical Harmonic Analysis
book series (ANHA)
The reconstruction of a two-valued function from its two projections is considered. It is shown that this problem can be transformed into the solved problem of the reconstruction of characteristic functions (i.e., functions with values 0 and 1). Necessary and sufficient conditions are given to decide the existence and the uniqueness of a two-valued function if its projections and the two values are known. These conditions can also be applied if the two values are not given in advance. It is proved that merely the knowledge of the projections (i.e., without the two values) is not enough for the unique reconstruction of a two-valued function. However, on the basis of two given functions it is possible to decide whether they are the projections of a uniquely reconstructible two-valued function. Also the corresponding values can be determined in this way. The reconstruction of two-valued matrices from their row and column sums (projections) is also considered. It is shown that this problem can be transformed into the solved problem of the reconstruction of (0,1)-matrices. In the same way as in the case of two-valued functions, necessary and sufficient conditions are given to decide the existence and the uniqueness of a two-valued matrix if its projections and the two values are known. It is proved that, generally, there is only a finite number of solutions even if the two values are not fixed. Finally, an algorithm is given to reconstruct two-valued matrices from two projections.
G. G. Lorentz, “A problem of plane measure,” Amer. J. Math
, 417–426 (1949).CrossRefGoogle Scholar
H. G. Kellerer, “Masstheoretische Marginalprobleme,” Math. Annalen
, 168–198 (1964).CrossRefGoogle Scholar
A. Kuba and A. Volcic, “Characterization of measurable plane sets which are reconstructable from their two projections,” Inverse Problems
, 513–527 (1988).CrossRefGoogle Scholar
P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, “Sets uniquely determined by projections on axes,” Discrete Mathematics
, 149–159 (1991).CrossRefGoogle Scholar
A. Kuba, “Reconstruction of measurable plane sets from their two projections taken in arbitrary directions,” Inverse Problems
, 101–107 (1991).CrossRefGoogle Scholar
D. Gale, “Theorem on flows in networks,” Pacific. J. Math.
, 1073–1082 (1957).CrossRefGoogle Scholar
H. J. Ryser, “Combinatorial properties of matrices of zeros and ones,” Ganad. J. Math.
, 371–377 (1957).CrossRefGoogle Scholar
R. A. Brualdi, “Matrices of zeros and ones with fixed row and column sum vectors,” Linear Algebra and Its Applications
, 159–231 (1980).CrossRefGoogle Scholar
L. Mirsky, “Combinatorial theorems and integral matrices,” J. Comb. Theory
, 30–44 (1968).CrossRefGoogle Scholar
R. Aharoni, G. T. Herman, and A. Kuba, “Binary vectors partially determined by linear equation system,” Discrete Math
, 1–16 (1997).CrossRefGoogle Scholar
P. C. Fishburn, P. Schwander, L. A. Shepp, and J. Vanderbei, “The discrete Radon transform and its approximate inversion via linear programming,” Discrete Appl. Math.
, 39–61 (1997).CrossRefGoogle Scholar
S. K. Chang and C. K. Chow, “The reconstruction of three-dimensional objects from two orthogonal projections and its application to cardiac cineangiography,” IEEE Trans. Comp.
18–28 (1973).CrossRefGoogle Scholar
G. P. M. Prause and D. G. W. Onnasch, “Binary reconstruction of the heart chambers from biplane angiographic image sequence,” IEEE Trans. Medical Imaging
, 532–559 (1996).CrossRefGoogle Scholar
J. H. B. Kemperman and A. Kuba, “Reconstruction of two-valued matrices from their two projections,” Int. J. Imaging Systems and Technology
, 110–117 (1998).CrossRefGoogle Scholar
A. Kuba and A. Volcic, “The structure of the class of non-uniquely reconstructible sets,” Acta Sci. Szeged
, 363–388 (1993).Google Scholar
Y. R. Wang, “Characterization of binary patterns and their projections,” IEEE Trans. Comput.
, 1032–1035 (1995).CrossRefGoogle Scholar
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