Incomplete data problems in X-ray computerized tomography

SummaryIn the present paper 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.

[1]  Victor J. Sank,et al.  IMAGE RECONSTRUCTION FROM PROJECTIONS: ***I , 1978 .

[2]  M. E. Davison,et al.  Tomographic reconstruction with arbitrary directions , 1981 .

[3]  F. Natterer,et al.  Mathematical problems of computerized tomography , 1983, Proceedings of the IEEE.

[4]  Gabor T. Herman,et al.  Image reconstruction from projections : the fundamentals of computerized tomography , 1980 .

[5]  Alfred K. Louis,et al.  Tikhonov-Phillips Regularization of the Radon Transform , 1985 .

[6]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[7]  R. Marr,et al.  On the reconstruction of a function on a circular domain from a sampling of its line integrals , 1974 .

[8]  Kennan T. Smith,et al.  Practical and mathematical aspects of the problem of reconstructing objects from radiographs , 1977 .

[9]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[10]  A K Louis,et al.  Incomplete data problems in x-ray computerized tomography , 1986 .

[11]  A. Louis.  Inverse und schlecht gestellte Probleme , 1989 .

[12]  Kennan T. Smith,et al.  The divergent beam x-ray transform , 1980 .

[13]  Emil Wolf,et al.  Principles of Optics: Contents , 1999 .

[14]  W. Madych,et al.  Reconstruction from restricted Radon transform data: resolution and ill-conditionedness , 1986 .

[15]  Frank Natterer,et al.  A Sobolev Space Analysis of Picture Reconstruction , 1980 .

[16]  A. Louis,et al.  Picture reconstruction from projections in restricted range , 1980 .

[17]  Alfred K. Louis.  Orthogonal Function Series Expansions and the Null Space of the Radon Transform , 1984 .

[18]  M. E. Davison,et al.  The Ill-Conditioned Nature of the Limited Angle Tomography Problem , 1983 .

[19]  D. Slepian.  Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.