Mathematical Problems of Computerized Tomography

Proceedings of the IEEE

Volumn 71, Issue 3, 1983, Pages 379-389

  Louis, Alfred K ^a   Natterer, Frank ^a  

^a UNIVERSITY OF MÜNSTER   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

DIAGNOSTIC EVALUATION;

BIOMEDICAL ENGINEERING - COMPUTERIZED TOMOGRAPHY; IMAGE PROCESSING - RECONSTRUCTION; X-RAYS - ATTENUATION;

IMAGING TECHNIQUES;

COMPUTER ANALYSIS; COMPUTER ASSISTED TOMOGRAPHY; DIAGNOSIS; IMAGE PROCESSING; METHODOLOGY; NONBIOLOGICAL MODEL;

EID: 0020720276     PISSN: 00189219     EISSN: 15582256     Source Type: Journal    
DOI: 10.1109/PROC.1983.12596     Document Type: Article

References (71)

1. M. D. Altschuler and G. T. Herman, “Fully-three-dimensional image reconstruction using series expansion methods,” in A Review of Information Processing in Medical Imaging, A. B. Brill, Ed., Oak Ridge National Laboratory, Oak Ridge, TN, pp. 124-142.
2. M. Bertero, C. De Mol, and G. A. Viano, “The stability of inverse problems,” in H. P. Baltes, Ed., Inverse Scattering Problems. Berlin: Springer, 1980.
3. R. R. Bitmead and B.D.O. Anderson, “Asymptotically fast solutions of Toeplitz and related systems of linear equations,” Tech. Rep. EE 7915, Dep. Elec. Electron. Eng., James Cook Univ., Queensland, Australia, 4811.
4. T. F. Budinger, G. T. Gullberg, and R. H. Huesman, “Emission computed tomography,” in G. T. Herman, Ed., Image Reconstruction from Projections. Berlin: Springer, 1979.
5. A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications I,” J. Appl. Phys., vol. 34, pp. 2722-2727, 1963.
6. “Representation of a function by its line integrals, with some radiological applications II,” J. Appl. Phys., vol. 35, pp. 2908-2913, 1964.
7. R. A. Crowther, D. J. De Rosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. Roy. Soc. London, vol. A 317, pp. 319-340, 1970.
8. M. E. Davison and F. A. Grünbaum, “Tomographic reconstruction with arbitrary directions,” Commun. Pure Appl. Math., vol. 34, pp. 77-119, 1981.
9. M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” to appear in SIAM J. Appl. Math.
10. P.P.B. Eggermont, “Special discretization methods for the integral equations of image reconstruction and for Abel-type integral equations,” Tech. Rep. MIP G 50, Medical Image Processing Group, SUNY, 1980.
11. F. A. Grünbaum, “A study of Fourier space methods for limited angle image reconstruction,” Numer. Func. Anal. Opt., vol. 2, pp. 32-42, 1980.
12. “Reconstruction with arbitrary directions: Dimensions two and three,” in G. T. Herman and F. Natterer, Eds., Mathematical Aspects of Computerized Tomography. Berlin: Springer, 1981, pp.112-126.
13. F. S. Ham, “Theory of tomographic image reconstruction: Proof that the reconstruction matrix reduces to block form in rotation-reflection symmetry,” Gen. Elec, Tech. Rep.
14. C. Hamaker, K. T. Smith, D. C. Solmon, and S. L. Wagner, “The divergent beam X-ray transform,” Rocky Mountain J. Math., vol. 10, pp. 253-283, 1980.
15. C. Hamaker and D. C. Solmon, “The angles between the null spaces of X-rays,” J. Math. Anal. Appl., vol. 62, pp. 1-23, 1978.
16. S. Helgason, “Differential operators on homogeneous spaces,” Acta Math., vol. 102, pp. 239-299, 1959.
17. “The Radon transform on Euclidian spaces, compact two-point homogeneous spaces, and Grassmann manifolds,” Acta Mart., vol. 113, pp. 153-180, 1965.
18. The Radon Transform. Boston, MA: Birkhauser, 1980.
19. G. T. Herman, Ed., Image Reconstruction from Projections: Implementation and Applications. Berlin: Springer, 1979.
20. G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography. New York: Academic Press, 1980.
21. G. T. Herman and F. Natterer, Eds., Mathematical Aspects of Computerized Tomography. Berlin: Springer, LNMI 8, 1981.
22. T. Inouye, “Image reconstruction with limited angle projection data,” IEEE Trans. Nucl. Sci., vol. NS-26, pp. 2666-2669, 1979.
23. F. John, “Bestimmung einer Funktion aus ihren Integralen über gewisse Mannigfaltigkeiten,” Math. Ann., vol. 109, pp. 488-520, 1934.
24. M. B. Katz, Questions of Uniqueness and Resolution in Reconstruction from Projections. Berlin: Springer, LNB 26, 1978.
25. G. Kowalski, “Multislice reconstruction from beam scanning,” IEEE Trans. Nucl. Sci., vol. NS-26, no. 2, Apr. 1979.
26. J. K. Leahy, K. T. Smith, and D. C. Solmon, “Uniqueness, non-uniqueness and inversion in the X-ray and Radon problems,” to appear in Proc. Int. Symp. on Ill-Posed Problems: Theory and Practice (Univ. Delaware, Newark, Oct. 2-6, 1979).
27. A. Lent, Seminar talk at the Biodynamic Research Unit, Mayo Clinic, Rochester, MN, 1975.
28. A. Lent and H. Tuy, “An iterative method for the extrapolation of band limited functions,” J. Math. Anal Appl., vol. 83, pp. 554-565,1981.
29. I. Lerche and E. Zeitler, “Projections, reconstructions and orthogonal functions,” J. Math. Anal Appl., vol. 56, pp. 634-649, 1976.
30. R. M. Lewitt, R.H.T. Bates, and T. M. Peters, “Image reconstruction from projections, II: Modified backprojection methods,” Optik, vol. 50, pp. 85-109, 1978.
31. R. M. Lewitt and R.H.T. Bates, “Image reconstruction from projections, III: Projection completion methods (theory),” Optik, vol. 50, pp. 180-205, 1978.
32. B. G. Logan, “The uncertainty principle in reconstructing functions from projections,” Duke Math. J., vol. 42, pp. 661-706, 1975.
33. A. K. Louis, “Picture reconstruction from projections in restricted range,” Math. Meth. in Appl Sci., vol. 2, pp. 209-220, 1980.
34. “Approximation of the Radon transform from samples in limited range,” in G. T. Herman and F. Natterer, Eds., Mathematical Aspects of Computized Tomography. Berlin: Springer, LNMI 8, 1981, pp. 127-139.
35. “Ghosts in tomography-The null space of the Radon transform,” Mart. Mert. in Appl Sci., vol. 3, pp. 1-10, 1981.
36. “Orthogonal function series expansions and the null space of the Radon transform,” SIAM J. Math. Anal., in press.
37. sampling in nuclear magnetic resonance tomography,” Comput. Assist. Tomogr., vol. 6, pp. 334-340, 1982.
38. “Analytische Methoden in der Computer Tomographic,” Habilitationsschrift, Münster Univ., 1981.
39. K. H. Low and F. Natterer, “An ultrafast algorithm in tomography,” Fachbereich der Universität des Saarlandes, 6600 Saarbriicken, Germany, Tech. Rep. A81/03.
40. D. Ludwig, “The Radon transform on Euclidean spaces,” Commun. Pure Appl Math., vol. 19, pp. 49-81, 1966.
41. W. R. Madych, “Degree of approximation in computerized tomography,” in E. W. Cheney, Ed., Approximation Theory III. New York: Academic Press, 1980.
42. R. B. Marr, “On the reconstruction of a function on a circular domain from a sampling of its line integrals,” J. Mart. Anal Appl., vol. 19, pp. 357-374, 1974.
43. “An overview of image reconstruction,” to appear in Proc. Int. Symp. on Ill-Posed Problems: Theory and Practice (Univ. of Delaware, Newark, Oct. 2-6, 1979).
44. R. B. Marr, C. N. Chen, and P. C. Lauterbur, “On two approaches to 3D reconstruction in NMR zeugmatography,” in G. T. Herman and F. Natterer, Eds., Mathematical Aspects of Computerized Tomography. Berlin: Springer, LNMI 8, 1981.
45. G. N. Minerbo, “Convolutional reconstruction from cone-beam projection data,” IEEE Trans. Nucl Sci., vol. NS-26, pp. 2682-2684, 1979.
46. F. Natterer, “Computerized tomography with unknown sources,” University Saarbriicken, Tech. Rep. TR A81/06, FB 10, 1981.
47. Genauigkeitsfragen bei der numerischen Rekonstruktion von Bildem. Basel, Switzerland: Birkhäuser, ISNM 49, 1979, pp. 131-146.
48. “A Sobolev space analysis of picture reconstruction,” SIAM J. Appl Math., vol. 39, pp. 402-411, 1980.
49. “The identification problem in emission computed tomography,” in: G. T. Herman and F. Natterer, Eds., Mathematical Aspects of Computerized Tomography. Berlin: Springer, LNMI 8, 1981, pp. 45-56.
50. “Efficient implementation of 'optimal' algorithms in computerized tomography,” Math. Meth. in Appl Sci., vol. 2, pp. 545-555, 1980.
51. A. Peres, “Tomographic reconstruction from limited angular data,” J. Comput. Assist. Tomogr., vol. 3, pp. 800-803, 1979.
52. R. M. Perry, “Reconstructing a function by circular harmonic analysis of its line integrals,” in Image Processing for 2D and 3D Reconstruction from Projections, Stanford, CA, 1975.
53. E. T. Quinto, “Null spaces and ranges for the classical and spherical Radon transforms,” J. Math. Anal Appl, vol. 90, pp. 408-429, 1982.
54. J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Sächs. Adad. Wiss. Leipzig, vol. 69, pp. 262-277, 1917.
55. S. W. Rowland, “Computer implementation of image reconstruction formulas,” in G. T. Herman, Ed., Image Reconstruction from Projections: Implementation and Applications. Berlin: Springer, 1979, pp. 9-79.
56. L. A. Shepp, “Computerized tomography and nuclear magnetic resonance,” J. Comput. Assist. Tomogr., vol. 4, pp. 94-107, 1980.
57. L. A. Shepp and J. B. Kruskal, “Computerized tomography: The new medical X-ray technology,” Amer. Math. Monthly, vol.85, pp. 420-439, 1978.
58. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case,” Bell. Syst. Tech. J., vol. 57, pp. 1371-1430, 1978.
59. K. T. Smith, D. C. Solmon, and S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bul AMS 83, pp. 1227-1270, 1977.
60. D. C. Solmon, “The X-ray transform,” J. Mart. Anal Appl., vol. 56, pp. 61-83, 1976.
61. H. Stark, J. W. Woods, I. Paul, and R. Hingorani, “An investigation of computerized tomography by direct Fourier inversion and optimum interpolation,” IEEE Trans. Biomed. Eng., vol. BME-28, pp. 496-505, 1981.
62. K. C. Tarn, V. Perez-Mendez, and B. MacDonald, “Limited angle 3-D reconstruction from continuous and pinhole projection,” IEEE Trans. Nucl. Sci., vol. NS-27, pp. 445-458, 1980.
63. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems. New York: Wiley, 1977.
64. O. J. Tretiak, “The point-spread function for the convolution algorithm,” in R. Gordon, Ed., Image Processing for 2-D and 3-D Reconstruction from Projections. Stanford, 1975.
65. O. I. Tretiak and P. Delaney, “The exponential convolution algorithm for emission computed axial tomography,” in Proc. 5th Int. Conf. on Information Processing in Medical Imaging, 1978.
66. O. J. Tretiak and C. Metz, “The exponential Radon transform,” SIAM J. Appl Math., vol. 39, pp. 341-354, 1980.
67. H. Triebel, Interpolation Theory, Function Spaces, Differential Operation. Amsterdam: North-Holland, 1978.
68. H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SUNY Buffalo, Tech. Rep. TR MlPG 57, 1981.
69. “Reconstruction of a three-dimensional object from a limited range of views,” J. Mart. Anal. Appl., vol. 80, pp. 598-616, 1981.
70. G. Wahba, “A new approach to the numerical evaluation of the inverse Radon transform with discrete, noisy data,” in G. T. Herman and F. Natterer, Eds., Mathematical Aspects of Computerized Tomography. Berlin: Springer, 1981.
71. K. Yosida, Functional Analysis, 6th ed. Berlin: Springer, 1980.