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Incomplete data problems in X-ray computerized tomography. II: Truncated projections and region-of-interest tomography. (English) Zbl 0675.65137

Summary: [For part I see ibid. 48, 251-262 (1986; Zbl 0578.65131).]
We study truncated projections for the fanbeam geometry in computerized tomography. First we derive consistency conditions for the divergent beam transform. Then we study a singular value decomposition for the case where only the interior rays in the fan are provided, as for example in region-of-interest tomography. We show that the high angular frequency components of the searched-for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition.

MSC:

65R10 Numerical methods for integral transforms
44A15 Special integral transforms (Legendre, Hilbert, etc.)
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
45H05 Integral equations with miscellaneous special kernels

Citations:

Zbl 0578.65131
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References:

[1] Bartmann, A.: Unvollständige Daten in der Computer-Tomographie. Dissertation LMU München 1986 · Zbl 0599.65087
[2] Hamaker, C., Smith, K.T., Solmon, D.C., Wagner, S.L.: The divergent beam X-ray transform. Rocky Mountain J. Math.10, 253-283 (1980) · Zbl 0443.44005 · doi:10.1216/RMJ-1980-10-1-253
[3] Lewitt, R.M.: Image reconstruction from projections. I: General theoretical considerations. Optik50, 19-33 (1978)
[4] Louis, A.K.: Fast scanning geometries in X-ray computerized tomography. In: Boffi, V., Neunzert, H. (eds.) Applications of mathematics in technology, pp. 324-329. Stuttgart: Teubner 1984
[5] Louis, A.K.: Tikhonov-Phillips regulatization of the radon transform. In: Hämmerlin, G., Hoffmann, K.H. (eds.) Constructive methods for the practical treatment of integral equations, pp. 211-223. Basel: Birkhäuser 1985
[6] Louis, A.K.: Incomplete data problems in X-ray computerized tomography. I: Singular value decomposition of the limited angle transform. Numer. Math.48, 251-262 (1986) · Zbl 0578.65131 · doi:10.1007/BF01389474
[7] Louis, A.K.: Inverse und schlecht gestellte Probleme, 1st Ed. Stuttgart: Teubner 1989 · Zbl 0667.65045
[8] Madych, W.R., Nelson, S.A.: Reconstruction from restricted Radon transform data: Resolution and ill-conditionedness. SIAM J. Math. Anal.17, 1447-1453 (1986) · Zbl 0627.44006 · doi:10.1137/0517102
[9] Natterer, F.: The mathematics of computerized tomography, 1st Ed. New York: Wiley-Teubner 1987 · Zbl 0638.65091
[10] Slepian, D.: Prolate spheroidal wave functions. Fourier analysis and uncertainty, V.: The Discrete Case. Bell Syst. Techn. J.57, 1371-1430 (1978) · Zbl 0378.33006
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