Abstract

In the local basis-function approach, a reconstruction is represented as a linear expansion of basis functions, which are arranged on a rectangular grid and possess a local region of support. The basis functions considered here are positive and may overlap. It is found that basis functions based on cubic B-splines offer significant improvements in the calculational accuracy that can be achieved with iterative tomographic reconstruction algorithms. By employing repetitive basis functions, the computational effort involved in these algorithms can be minimized through the use of tabulated values for the line or strip integrals over a single-basis function. The local nature of the basis functions reduces the difficulties associated with applying local constraints on reconstruction values, such as upper and lower limits. Since a reconstruction is specified everywhere by a set of coefficients, display of a coarsely represented image does not require an arbitrary choice of an interpolation function.

© 1985 Optical Society of America

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References

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  1. H. C. Andrews, C. L. Patterson, “Digital Interpolation of Discrete Images,” IEEE Trans. Comput. C-25, 196 (1976).
    [CrossRef]
  2. R. Gordon, G. T. Herman, “Three-Dimensional Reconstruction from Projections: a Review of Algorithms,” Cytol. 38, 111 (1974).
  3. G. T. Herman, A. Lent, “Iterative Reconstruction Algorithms,” Comput. Biol. Med. 6, 273 (1976).
    [CrossRef] [PubMed]
  4. R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GRS-18, 234 (1980).
    [CrossRef]
  5. G. T. Herman, A. Lent, “A Computer Implementation of a Bayesian Analysis of Image Reconstruction,” Inf. Control 31, 364 (1976).
    [CrossRef]
  6. B. R. Hunt, “Bayesian Methods in Nonlinear Digital Image Restoration,” IEEE Trans. Comput. C-26, 219 (1977).
    [CrossRef]
  7. K. M. Hanson, “Limited Angle CT Reconstruction Using a priori Information,” Proceedings First International Symposium on Medical Imaging and Image Interpretation, Berlin (IEEE Computer Society, Silver Spring, Md., 1982), pp. 527–533.
  8. K. M. Hanson, G. W. Wecksung, “Bayesian Approach to Limited-Angle Reconstruction in Computed Tomography,” J. Opt. Soc. Am. 73, 1501 (1983).
    [CrossRef]
  9. R. A. Brooks, G. DiChiro, “Theory of Image Reconstruction in Computed Tomography,” Radiology 117, 561 (1975).
    [PubMed]
  10. R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-Dimensional Electron Microscopy and X-Ray Photography,” J. Theoret. Biol. 29, 471 (1970).
    [CrossRef]
  11. R. A. Crowther, A. Klug, “ART and Science, or Conditions for 3-D Reconstruction from Electron Microscope Images,” J. Theoret. Biol. 32, 199 (1971).
    [CrossRef]
  12. P. Gilbert, “Iterative Methods for the Three-Dimensional Reconstruction of an Object from Projections,” J. Theoret. Biol. 36, 105 (1972).
    [CrossRef]
  13. S. H. Bellman, R. Bender, R. Gordon, J. E. Rowe, “ART is Science, Being a Defense of Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy,” J. Theoret. Biol. 32, 205 (1971).
    [CrossRef]
  14. G. T. Herman, A. Lent, S. W. Rowland, “ART: Mathematics and Applications,” J. Theoret. Biol. 42, 1 (1973).
    [CrossRef]
  15. R. Gordon, “A Tutorial on ART (Algebraic Reconstruction Techniques),” IEEE Trans. Nucl. Sci. NS-21, 78 (1974).
  16. H. S. Hou, H. C. Andrews, “Least Squares Image Restoration Using Spline Basis Functions,” IEEE Trans. Comput. C-26, 856 (1977).
    [CrossRef]
  17. H. S. Hou, H. C. Andrews, “Cubic Splines for Image Interpolation and Digital Filtering,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 508 (1978).
  18. K. M. Hanson, “Detectability in Computed Tomographic Images,” Med. Phys. 6, 441 (1979).
    [CrossRef] [PubMed]
  19. K. M. Hanson, “On the Optimality of the Filtered Backprojection Algorithm,” J. Comput. Assist. Tomogr. 4, 361 (1980).
    [CrossRef] [PubMed]
  20. P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
    [CrossRef] [PubMed]
  21. R. C. Allen, W. R. Boland, G. M. Wing, “Numerical Experiments Involving Galerkin and Collocation Methods for Linear Integral Equations of the First Kind,” J. Comput. Phys. 49, 465 (1983).
    [CrossRef]
  22. A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974).
  23. H. B. Buonocore, W. R. Brody, A. Macovski, “Natural Pixel Decomposition for Two-Dimensional Image Reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69 (1981).
    [CrossRef]
  24. D. G. McCaughey, H. C. Andrews, “Degrees of Freedom for Projection Imaging,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 63 (1977).
    [CrossRef]
  25. B. R. Hunt, “A Theorem on the Difficulty of Numerical Deconvolution,” IEEE Trans. Acoust. Ultrasound, AU-20, 94 (1972).
  26. A. Lent, “A Convergent Algorithm for Maximum Entropy Image Reconstruction, with a Medical X-Ray Application,” Proceedings, SPSE International Conference on Image Analysis and Evaluation, R. Shaw, Ed., Toronto, 19–23 July (Society of Photographic Scientists Engineers, Washington, D.C., 1976), pp. 249–257.
  27. G. Minerbo, “MENT: A Maximum-Entropy Algorithm for Reconstructing a Source from Projection Data,” Comput. Graphics Image Process. 10, 48 (1979).
    [CrossRef]
  28. A. M. Cormack, “Representation of a Function by its Line Integrals, with Some Radiological Applications,” J. Appl. Phys. 45, 2722 (1963).
    [CrossRef]
  29. A. M. Cormack, “Representation of a Function by its Line Integrals, with Some Radiological Applications. II,” J. Appl. Phys. 35, 2908 (1964).
    [CrossRef]
  30. A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
    [CrossRef]
  31. S. L. Wood, A. Macovski, M. Morf, “Reconstructions with Limited Data Using Estimation Theory,” Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv, J. F. Greenleaf, G. T. Herman, Eds. (North-Holland, Amsterdam, 1979), pp. 219–233.
  32. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  33. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), pp. 319–324.
  34. K. M. Hanson, “Variations in Task and the Ideal Observer,” Proc. Soc. Photo. Opt. Instrum. Eng. 419, 60 (1983).
  35. K. M. Hanson, “Optical Object and Edge Localization in the Presence of Correlated Noise,” Proc. Soc. Photo-Opt. Instrum. Eng. 454, 9 (1984).
  36. H. B. Kekre, S. C. Sahasrabudhe, N. C. Goyal, “Restoration of Noisy Images Using a Raised Cosine Function Approximation,” Comput. Vision Graphics Image Process. 26, 17 (1984).
    [CrossRef]
  37. G. T. Herman, “On Modifications to the Algebraic Reconstruction Techniques,” Comput. Biol. Med. 9, 271 (1979).
    [CrossRef] [PubMed]
  38. J. M. Hyman, “Accurate Monotonicity Preserving Cubic Interpolation,” SIAM J. Sci. Stat. Comput. 4, 645 (1983).
    [CrossRef]
  39. K. M. Hanson, “Image Processing: Science, Engineering, or Art?,” Proc. Soc. Photo-Opt. Instrum. Eng. 535, 70 (1985).
  40. K. Aki, A. Christoffersson, E. S. Husebye, “Determination of the Three-Dimensional Seismic Structure of the Lithosphere,” J. Geophys. Res. 82, 277 (1977).
    [CrossRef]
  41. K. Tanabe, “Projection Method for Solving a Singular System of Linear Equations and its Applications,” Numer. Math. 17, 203 (1971).
    [CrossRef]

1985 (1)

K. M. Hanson, “Image Processing: Science, Engineering, or Art?,” Proc. Soc. Photo-Opt. Instrum. Eng. 535, 70 (1985).

1984 (2)

K. M. Hanson, “Optical Object and Edge Localization in the Presence of Correlated Noise,” Proc. Soc. Photo-Opt. Instrum. Eng. 454, 9 (1984).

H. B. Kekre, S. C. Sahasrabudhe, N. C. Goyal, “Restoration of Noisy Images Using a Raised Cosine Function Approximation,” Comput. Vision Graphics Image Process. 26, 17 (1984).
[CrossRef]

1983 (4)

K. M. Hanson, “Variations in Task and the Ideal Observer,” Proc. Soc. Photo. Opt. Instrum. Eng. 419, 60 (1983).

J. M. Hyman, “Accurate Monotonicity Preserving Cubic Interpolation,” SIAM J. Sci. Stat. Comput. 4, 645 (1983).
[CrossRef]

R. C. Allen, W. R. Boland, G. M. Wing, “Numerical Experiments Involving Galerkin and Collocation Methods for Linear Integral Equations of the First Kind,” J. Comput. Phys. 49, 465 (1983).
[CrossRef]

K. M. Hanson, G. W. Wecksung, “Bayesian Approach to Limited-Angle Reconstruction in Computed Tomography,” J. Opt. Soc. Am. 73, 1501 (1983).
[CrossRef]

1981 (1)

H. B. Buonocore, W. R. Brody, A. Macovski, “Natural Pixel Decomposition for Two-Dimensional Image Reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69 (1981).
[CrossRef]

1980 (3)

K. M. Hanson, “On the Optimality of the Filtered Backprojection Algorithm,” J. Comput. Assist. Tomogr. 4, 361 (1980).
[CrossRef] [PubMed]

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GRS-18, 234 (1980).
[CrossRef]

1979 (3)

K. M. Hanson, “Detectability in Computed Tomographic Images,” Med. Phys. 6, 441 (1979).
[CrossRef] [PubMed]

G. Minerbo, “MENT: A Maximum-Entropy Algorithm for Reconstructing a Source from Projection Data,” Comput. Graphics Image Process. 10, 48 (1979).
[CrossRef]

G. T. Herman, “On Modifications to the Algebraic Reconstruction Techniques,” Comput. Biol. Med. 9, 271 (1979).
[CrossRef] [PubMed]

1978 (1)

H. S. Hou, H. C. Andrews, “Cubic Splines for Image Interpolation and Digital Filtering,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 508 (1978).

1977 (4)

H. S. Hou, H. C. Andrews, “Least Squares Image Restoration Using Spline Basis Functions,” IEEE Trans. Comput. C-26, 856 (1977).
[CrossRef]

B. R. Hunt, “Bayesian Methods in Nonlinear Digital Image Restoration,” IEEE Trans. Comput. C-26, 219 (1977).
[CrossRef]

K. Aki, A. Christoffersson, E. S. Husebye, “Determination of the Three-Dimensional Seismic Structure of the Lithosphere,” J. Geophys. Res. 82, 277 (1977).
[CrossRef]

D. G. McCaughey, H. C. Andrews, “Degrees of Freedom for Projection Imaging,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 63 (1977).
[CrossRef]

1976 (3)

G. T. Herman, A. Lent, “Iterative Reconstruction Algorithms,” Comput. Biol. Med. 6, 273 (1976).
[CrossRef] [PubMed]

G. T. Herman, A. Lent, “A Computer Implementation of a Bayesian Analysis of Image Reconstruction,” Inf. Control 31, 364 (1976).
[CrossRef]

H. C. Andrews, C. L. Patterson, “Digital Interpolation of Discrete Images,” IEEE Trans. Comput. C-25, 196 (1976).
[CrossRef]

1975 (1)

R. A. Brooks, G. DiChiro, “Theory of Image Reconstruction in Computed Tomography,” Radiology 117, 561 (1975).
[PubMed]

1974 (2)

R. Gordon, G. T. Herman, “Three-Dimensional Reconstruction from Projections: a Review of Algorithms,” Cytol. 38, 111 (1974).

R. Gordon, “A Tutorial on ART (Algebraic Reconstruction Techniques),” IEEE Trans. Nucl. Sci. NS-21, 78 (1974).

1973 (1)

G. T. Herman, A. Lent, S. W. Rowland, “ART: Mathematics and Applications,” J. Theoret. Biol. 42, 1 (1973).
[CrossRef]

1972 (2)

P. Gilbert, “Iterative Methods for the Three-Dimensional Reconstruction of an Object from Projections,” J. Theoret. Biol. 36, 105 (1972).
[CrossRef]

B. R. Hunt, “A Theorem on the Difficulty of Numerical Deconvolution,” IEEE Trans. Acoust. Ultrasound, AU-20, 94 (1972).

1971 (3)

K. Tanabe, “Projection Method for Solving a Singular System of Linear Equations and its Applications,” Numer. Math. 17, 203 (1971).
[CrossRef]

S. H. Bellman, R. Bender, R. Gordon, J. E. Rowe, “ART is Science, Being a Defense of Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy,” J. Theoret. Biol. 32, 205 (1971).
[CrossRef]

R. A. Crowther, A. Klug, “ART and Science, or Conditions for 3-D Reconstruction from Electron Microscope Images,” J. Theoret. Biol. 32, 199 (1971).
[CrossRef]

1970 (1)

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-Dimensional Electron Microscopy and X-Ray Photography,” J. Theoret. Biol. 29, 471 (1970).
[CrossRef]

1964 (1)

A. M. Cormack, “Representation of a Function by its Line Integrals, with Some Radiological Applications. II,” J. Appl. Phys. 35, 2908 (1964).
[CrossRef]

1963 (1)

A. M. Cormack, “Representation of a Function by its Line Integrals, with Some Radiological Applications,” J. Appl. Phys. 45, 2722 (1963).
[CrossRef]

Aki, K.

K. Aki, A. Christoffersson, E. S. Husebye, “Determination of the Three-Dimensional Seismic Structure of the Lithosphere,” J. Geophys. Res. 82, 277 (1977).
[CrossRef]

Allen, R. C.

R. C. Allen, W. R. Boland, G. M. Wing, “Numerical Experiments Involving Galerkin and Collocation Methods for Linear Integral Equations of the First Kind,” J. Comput. Phys. 49, 465 (1983).
[CrossRef]

Andrews, H. C.

H. S. Hou, H. C. Andrews, “Cubic Splines for Image Interpolation and Digital Filtering,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 508 (1978).

D. G. McCaughey, H. C. Andrews, “Degrees of Freedom for Projection Imaging,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 63 (1977).
[CrossRef]

H. S. Hou, H. C. Andrews, “Least Squares Image Restoration Using Spline Basis Functions,” IEEE Trans. Comput. C-26, 856 (1977).
[CrossRef]

H. C. Andrews, C. L. Patterson, “Digital Interpolation of Discrete Images,” IEEE Trans. Comput. C-25, 196 (1976).
[CrossRef]

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Bellman, S. H.

S. H. Bellman, R. Bender, R. Gordon, J. E. Rowe, “ART is Science, Being a Defense of Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy,” J. Theoret. Biol. 32, 205 (1971).
[CrossRef]

Bender, R.

S. H. Bellman, R. Bender, R. Gordon, J. E. Rowe, “ART is Science, Being a Defense of Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy,” J. Theoret. Biol. 32, 205 (1971).
[CrossRef]

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-Dimensional Electron Microscopy and X-Ray Photography,” J. Theoret. Biol. 29, 471 (1970).
[CrossRef]

Ben-Israel, A.

A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974).

Boland, W. R.

R. C. Allen, W. R. Boland, G. M. Wing, “Numerical Experiments Involving Galerkin and Collocation Methods for Linear Integral Equations of the First Kind,” J. Comput. Phys. 49, 465 (1983).
[CrossRef]

Brody, W. R.

H. B. Buonocore, W. R. Brody, A. Macovski, “Natural Pixel Decomposition for Two-Dimensional Image Reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69 (1981).
[CrossRef]

Brooks, R. A.

R. A. Brooks, G. DiChiro, “Theory of Image Reconstruction in Computed Tomography,” Radiology 117, 561 (1975).
[PubMed]

Buonocore, H. B.

H. B. Buonocore, W. R. Brody, A. Macovski, “Natural Pixel Decomposition for Two-Dimensional Image Reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69 (1981).
[CrossRef]

Christoffersson, A.

K. Aki, A. Christoffersson, E. S. Husebye, “Determination of the Three-Dimensional Seismic Structure of the Lithosphere,” J. Geophys. Res. 82, 277 (1977).
[CrossRef]

Cormack, A. M.

A. M. Cormack, “Representation of a Function by its Line Integrals, with Some Radiological Applications. II,” J. Appl. Phys. 35, 2908 (1964).
[CrossRef]

A. M. Cormack, “Representation of a Function by its Line Integrals, with Some Radiological Applications,” J. Appl. Phys. 45, 2722 (1963).
[CrossRef]

Crowther, R. A.

R. A. Crowther, A. Klug, “ART and Science, or Conditions for 3-D Reconstruction from Electron Microscope Images,” J. Theoret. Biol. 32, 199 (1971).
[CrossRef]

DiChiro, G.

R. A. Brooks, G. DiChiro, “Theory of Image Reconstruction in Computed Tomography,” Radiology 117, 561 (1975).
[PubMed]

Dines, K. A.

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GRS-18, 234 (1980).
[CrossRef]

Gilbert, P.

P. Gilbert, “Iterative Methods for the Three-Dimensional Reconstruction of an Object from Projections,” J. Theoret. Biol. 36, 105 (1972).
[CrossRef]

Gordon, R.

R. Gordon, G. T. Herman, “Three-Dimensional Reconstruction from Projections: a Review of Algorithms,” Cytol. 38, 111 (1974).

R. Gordon, “A Tutorial on ART (Algebraic Reconstruction Techniques),” IEEE Trans. Nucl. Sci. NS-21, 78 (1974).

S. H. Bellman, R. Bender, R. Gordon, J. E. Rowe, “ART is Science, Being a Defense of Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy,” J. Theoret. Biol. 32, 205 (1971).
[CrossRef]

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-Dimensional Electron Microscopy and X-Ray Photography,” J. Theoret. Biol. 29, 471 (1970).
[CrossRef]

Goyal, N. C.

H. B. Kekre, S. C. Sahasrabudhe, N. C. Goyal, “Restoration of Noisy Images Using a Raised Cosine Function Approximation,” Comput. Vision Graphics Image Process. 26, 17 (1984).
[CrossRef]

Greville, T. N. E.

A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974).

Hanson, K. M.

K. M. Hanson, “Image Processing: Science, Engineering, or Art?,” Proc. Soc. Photo-Opt. Instrum. Eng. 535, 70 (1985).

K. M. Hanson, “Optical Object and Edge Localization in the Presence of Correlated Noise,” Proc. Soc. Photo-Opt. Instrum. Eng. 454, 9 (1984).

K. M. Hanson, G. W. Wecksung, “Bayesian Approach to Limited-Angle Reconstruction in Computed Tomography,” J. Opt. Soc. Am. 73, 1501 (1983).
[CrossRef]

K. M. Hanson, “Variations in Task and the Ideal Observer,” Proc. Soc. Photo. Opt. Instrum. Eng. 419, 60 (1983).

K. M. Hanson, “On the Optimality of the Filtered Backprojection Algorithm,” J. Comput. Assist. Tomogr. 4, 361 (1980).
[CrossRef] [PubMed]

K. M. Hanson, “Detectability in Computed Tomographic Images,” Med. Phys. 6, 441 (1979).
[CrossRef] [PubMed]

K. M. Hanson, “Limited Angle CT Reconstruction Using a priori Information,” Proceedings First International Symposium on Medical Imaging and Image Interpretation, Berlin (IEEE Computer Society, Silver Spring, Md., 1982), pp. 527–533.

Herman, G. T.

G. T. Herman, “On Modifications to the Algebraic Reconstruction Techniques,” Comput. Biol. Med. 9, 271 (1979).
[CrossRef] [PubMed]

G. T. Herman, A. Lent, “A Computer Implementation of a Bayesian Analysis of Image Reconstruction,” Inf. Control 31, 364 (1976).
[CrossRef]

G. T. Herman, A. Lent, “Iterative Reconstruction Algorithms,” Comput. Biol. Med. 6, 273 (1976).
[CrossRef] [PubMed]

R. Gordon, G. T. Herman, “Three-Dimensional Reconstruction from Projections: a Review of Algorithms,” Cytol. 38, 111 (1974).

G. T. Herman, A. Lent, S. W. Rowland, “ART: Mathematics and Applications,” J. Theoret. Biol. 42, 1 (1973).
[CrossRef]

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-Dimensional Electron Microscopy and X-Ray Photography,” J. Theoret. Biol. 29, 471 (1970).
[CrossRef]

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), pp. 319–324.

Hou, H. S.

H. S. Hou, H. C. Andrews, “Cubic Splines for Image Interpolation and Digital Filtering,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 508 (1978).

H. S. Hou, H. C. Andrews, “Least Squares Image Restoration Using Spline Basis Functions,” IEEE Trans. Comput. C-26, 856 (1977).
[CrossRef]

Hunt, B. R.

B. R. Hunt, “Bayesian Methods in Nonlinear Digital Image Restoration,” IEEE Trans. Comput. C-26, 219 (1977).
[CrossRef]

B. R. Hunt, “A Theorem on the Difficulty of Numerical Deconvolution,” IEEE Trans. Acoust. Ultrasound, AU-20, 94 (1972).

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Husebye, E. S.

K. Aki, A. Christoffersson, E. S. Husebye, “Determination of the Three-Dimensional Seismic Structure of the Lithosphere,” J. Geophys. Res. 82, 277 (1977).
[CrossRef]

Hyman, J. M.

J. M. Hyman, “Accurate Monotonicity Preserving Cubic Interpolation,” SIAM J. Sci. Stat. Comput. 4, 645 (1983).
[CrossRef]

Joseph, P. M.

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

Kekre, H. B.

H. B. Kekre, S. C. Sahasrabudhe, N. C. Goyal, “Restoration of Noisy Images Using a Raised Cosine Function Approximation,” Comput. Vision Graphics Image Process. 26, 17 (1984).
[CrossRef]

Klug, A.

R. A. Crowther, A. Klug, “ART and Science, or Conditions for 3-D Reconstruction from Electron Microscope Images,” J. Theoret. Biol. 32, 199 (1971).
[CrossRef]

Lent, A.

G. T. Herman, A. Lent, “Iterative Reconstruction Algorithms,” Comput. Biol. Med. 6, 273 (1976).
[CrossRef] [PubMed]

G. T. Herman, A. Lent, “A Computer Implementation of a Bayesian Analysis of Image Reconstruction,” Inf. Control 31, 364 (1976).
[CrossRef]

G. T. Herman, A. Lent, S. W. Rowland, “ART: Mathematics and Applications,” J. Theoret. Biol. 42, 1 (1973).
[CrossRef]

A. Lent, “A Convergent Algorithm for Maximum Entropy Image Reconstruction, with a Medical X-Ray Application,” Proceedings, SPSE International Conference on Image Analysis and Evaluation, R. Shaw, Ed., Toronto, 19–23 July (Society of Photographic Scientists Engineers, Washington, D.C., 1976), pp. 249–257.

Lytle, R. J.

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GRS-18, 234 (1980).
[CrossRef]

Macovski, A.

H. B. Buonocore, W. R. Brody, A. Macovski, “Natural Pixel Decomposition for Two-Dimensional Image Reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69 (1981).
[CrossRef]

S. L. Wood, A. Macovski, M. Morf, “Reconstructions with Limited Data Using Estimation Theory,” Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv, J. F. Greenleaf, G. T. Herman, Eds. (North-Holland, Amsterdam, 1979), pp. 219–233.

McCaughey, D. G.

D. G. McCaughey, H. C. Andrews, “Degrees of Freedom for Projection Imaging,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 63 (1977).
[CrossRef]

Minerbo, G.

G. Minerbo, “MENT: A Maximum-Entropy Algorithm for Reconstructing a Source from Projection Data,” Comput. Graphics Image Process. 10, 48 (1979).
[CrossRef]

Morf, M.

S. L. Wood, A. Macovski, M. Morf, “Reconstructions with Limited Data Using Estimation Theory,” Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv, J. F. Greenleaf, G. T. Herman, Eds. (North-Holland, Amsterdam, 1979), pp. 219–233.

Patterson, C. L.

H. C. Andrews, C. L. Patterson, “Digital Interpolation of Discrete Images,” IEEE Trans. Comput. C-25, 196 (1976).
[CrossRef]

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

Rowe, J. E.

S. H. Bellman, R. Bender, R. Gordon, J. E. Rowe, “ART is Science, Being a Defense of Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy,” J. Theoret. Biol. 32, 205 (1971).
[CrossRef]

Rowland, S. W.

G. T. Herman, A. Lent, S. W. Rowland, “ART: Mathematics and Applications,” J. Theoret. Biol. 42, 1 (1973).
[CrossRef]

Sahasrabudhe, S. C.

H. B. Kekre, S. C. Sahasrabudhe, N. C. Goyal, “Restoration of Noisy Images Using a Raised Cosine Function Approximation,” Comput. Vision Graphics Image Process. 26, 17 (1984).
[CrossRef]

Spital, R. D.

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

Stockham, C. D.

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

Tanabe, K.

K. Tanabe, “Projection Method for Solving a Singular System of Linear Equations and its Applications,” Numer. Math. 17, 203 (1971).
[CrossRef]

Wecksung, G. W.

Wing, G. M.

R. C. Allen, W. R. Boland, G. M. Wing, “Numerical Experiments Involving Galerkin and Collocation Methods for Linear Integral Equations of the First Kind,” J. Comput. Phys. 49, 465 (1983).
[CrossRef]

Wood, S. L.

S. L. Wood, A. Macovski, M. Morf, “Reconstructions with Limited Data Using Estimation Theory,” Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv, J. F. Greenleaf, G. T. Herman, Eds. (North-Holland, Amsterdam, 1979), pp. 219–233.

Comput. Biol. Med. (2)

G. T. Herman, A. Lent, “Iterative Reconstruction Algorithms,” Comput. Biol. Med. 6, 273 (1976).
[CrossRef] [PubMed]

G. T. Herman, “On Modifications to the Algebraic Reconstruction Techniques,” Comput. Biol. Med. 9, 271 (1979).
[CrossRef] [PubMed]

Comput. Graphics Image Process. (1)

G. Minerbo, “MENT: A Maximum-Entropy Algorithm for Reconstructing a Source from Projection Data,” Comput. Graphics Image Process. 10, 48 (1979).
[CrossRef]

Comput. Tomogr. (1)

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

Comput. Vision Graphics Image Process. (1)

H. B. Kekre, S. C. Sahasrabudhe, N. C. Goyal, “Restoration of Noisy Images Using a Raised Cosine Function Approximation,” Comput. Vision Graphics Image Process. 26, 17 (1984).
[CrossRef]

Cytol. (1)

R. Gordon, G. T. Herman, “Three-Dimensional Reconstruction from Projections: a Review of Algorithms,” Cytol. 38, 111 (1974).

IEEE Trans. Acoust. Speech Signal Process. (1)

D. G. McCaughey, H. C. Andrews, “Degrees of Freedom for Projection Imaging,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 63 (1977).
[CrossRef]

IEEE Trans. Acoust. Ultrasound (1)

B. R. Hunt, “A Theorem on the Difficulty of Numerical Deconvolution,” IEEE Trans. Acoust. Ultrasound, AU-20, 94 (1972).

IEEE Trans. Acoust., Speech, Signal Process. (1)

H. S. Hou, H. C. Andrews, “Cubic Splines for Image Interpolation and Digital Filtering,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 508 (1978).

IEEE Trans. Biomed. Eng. (1)

H. B. Buonocore, W. R. Brody, A. Macovski, “Natural Pixel Decomposition for Two-Dimensional Image Reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69 (1981).
[CrossRef]

IEEE Trans. Comput. (3)

H. S. Hou, H. C. Andrews, “Least Squares Image Restoration Using Spline Basis Functions,” IEEE Trans. Comput. C-26, 856 (1977).
[CrossRef]

H. C. Andrews, C. L. Patterson, “Digital Interpolation of Discrete Images,” IEEE Trans. Comput. C-25, 196 (1976).
[CrossRef]

B. R. Hunt, “Bayesian Methods in Nonlinear Digital Image Restoration,” IEEE Trans. Comput. C-26, 219 (1977).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (1)

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GRS-18, 234 (1980).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

R. Gordon, “A Tutorial on ART (Algebraic Reconstruction Techniques),” IEEE Trans. Nucl. Sci. NS-21, 78 (1974).

Inf. Control (1)

G. T. Herman, A. Lent, “A Computer Implementation of a Bayesian Analysis of Image Reconstruction,” Inf. Control 31, 364 (1976).
[CrossRef]

J. Appl. Phys. (2)

A. M. Cormack, “Representation of a Function by its Line Integrals, with Some Radiological Applications,” J. Appl. Phys. 45, 2722 (1963).
[CrossRef]

A. M. Cormack, “Representation of a Function by its Line Integrals, with Some Radiological Applications. II,” J. Appl. Phys. 35, 2908 (1964).
[CrossRef]

J. Comput. Assist. Tomogr. (1)

K. M. Hanson, “On the Optimality of the Filtered Backprojection Algorithm,” J. Comput. Assist. Tomogr. 4, 361 (1980).
[CrossRef] [PubMed]

J. Comput. Phys. (1)

R. C. Allen, W. R. Boland, G. M. Wing, “Numerical Experiments Involving Galerkin and Collocation Methods for Linear Integral Equations of the First Kind,” J. Comput. Phys. 49, 465 (1983).
[CrossRef]

J. Geophys. Res. (1)

K. Aki, A. Christoffersson, E. S. Husebye, “Determination of the Three-Dimensional Seismic Structure of the Lithosphere,” J. Geophys. Res. 82, 277 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Theoret. Biol. (5)

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-Dimensional Electron Microscopy and X-Ray Photography,” J. Theoret. Biol. 29, 471 (1970).
[CrossRef]

R. A. Crowther, A. Klug, “ART and Science, or Conditions for 3-D Reconstruction from Electron Microscope Images,” J. Theoret. Biol. 32, 199 (1971).
[CrossRef]

P. Gilbert, “Iterative Methods for the Three-Dimensional Reconstruction of an Object from Projections,” J. Theoret. Biol. 36, 105 (1972).
[CrossRef]

S. H. Bellman, R. Bender, R. Gordon, J. E. Rowe, “ART is Science, Being a Defense of Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy,” J. Theoret. Biol. 32, 205 (1971).
[CrossRef]

G. T. Herman, A. Lent, S. W. Rowland, “ART: Mathematics and Applications,” J. Theoret. Biol. 42, 1 (1973).
[CrossRef]

Med. Phys. (1)

K. M. Hanson, “Detectability in Computed Tomographic Images,” Med. Phys. 6, 441 (1979).
[CrossRef] [PubMed]

Numer. Math. (1)

K. Tanabe, “Projection Method for Solving a Singular System of Linear Equations and its Applications,” Numer. Math. 17, 203 (1971).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

K. M. Hanson, “Optical Object and Edge Localization in the Presence of Correlated Noise,” Proc. Soc. Photo-Opt. Instrum. Eng. 454, 9 (1984).

K. M. Hanson, “Image Processing: Science, Engineering, or Art?,” Proc. Soc. Photo-Opt. Instrum. Eng. 535, 70 (1985).

Proc. Soc. Photo. Opt. Instrum. Eng. (1)

K. M. Hanson, “Variations in Task and the Ideal Observer,” Proc. Soc. Photo. Opt. Instrum. Eng. 419, 60 (1983).

Radiology (1)

R. A. Brooks, G. DiChiro, “Theory of Image Reconstruction in Computed Tomography,” Radiology 117, 561 (1975).
[PubMed]

SIAM J. Sci. Stat. Comput. (1)

J. M. Hyman, “Accurate Monotonicity Preserving Cubic Interpolation,” SIAM J. Sci. Stat. Comput. 4, 645 (1983).
[CrossRef]

Other (7)

A. Lent, “A Convergent Algorithm for Maximum Entropy Image Reconstruction, with a Medical X-Ray Application,” Proceedings, SPSE International Conference on Image Analysis and Evaluation, R. Shaw, Ed., Toronto, 19–23 July (Society of Photographic Scientists Engineers, Washington, D.C., 1976), pp. 249–257.

A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974).

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

S. L. Wood, A. Macovski, M. Morf, “Reconstructions with Limited Data Using Estimation Theory,” Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv, J. F. Greenleaf, G. T. Herman, Eds. (North-Holland, Amsterdam, 1979), pp. 219–233.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), pp. 319–324.

K. M. Hanson, “Limited Angle CT Reconstruction Using a priori Information,” Proceedings First International Symposium on Medical Imaging and Image Interpretation, Berlin (IEEE Computer Society, Silver Spring, Md., 1982), pp. 527–533.

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Figures (9)

Fig. 1
Fig. 1

Projection measurement geometry of the tomographic problem. Each measurement is an integral over a thin strip of width w. The ith measurement is designated by the perpendicular distance of the strip from the origin ri and angle θi. The assumed region of support of the unknown function is a circle with radius R.

Fig. 2
Fig. 2

Geometry associated with the ith strip integral across the jth local basis function, which is centered on (xj,yj) and is assumed to have square support.

Fig. 3
Fig. 3

(a) One-dimensional profile functions ϕ(x) used to produce the 2-D separable basis functions studied. From bottom to top, they are the square, triangle, Hanning, Gaussian, and cubic B-spline profile functions. (b) Fourier transforms of the profile functions.

Fig. 4
Fig. 4

Image of the original object used in the following examples.

Fig. 5
Fig. 5

Reconstructions of Fig. 4 obtained using basis functions on a 32 × 32 grid and the ART reconstruction algorithm. The input data consist of 32 strip integrals (projections) at each of 60 angles, evenly spaced over π rad. The separable basis functions employed are based on the profile functions (a) square, (b) triangle, and (c) cubic B-spline. In the display of these results, the reconstruction values are obtained everywhere in the x-y plane by using the corresponding basis-function expansion Eq. (6).

Fig. 6
Fig. 6

Reconstructions on a 32 × 32 grid using (a) Gaussian and (b) Hanning profile functions.

Fig. 7
Fig. 7

Result of bilinear interpolation of the coefficients obtained in the reconstruction employing the square basis function on a 32 × 32 grid, that is, the same coefficients as in Fig. 5(a). Although improvement in the display over Fig. 5(a) results, there are more artifacts than in Fig. 5(b) in which the triangle profile function was consistently used throughout.

Fig. 8
Fig. 8

Reconstructions on a 128 × 128 grid using the B-spline profile function. In (a) projections are assumed to be line integrals and in (b), strip integrals. The improvement in the resolution of the image representation afforded by finer sampling does not necessarily improve the appearance of the final result. These reconstructions should approximate the measurement-space component of the original function. This demonstrates the unsuitability of the mathematically pure minimum-norm solution, which is restricted to the measurement space.

Fig. 9
Fig. 9

Geometry associated with separable basis functions. The limits of integration along the dashed line are s1 and s2.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

g i = h i ( x , y ) f ( x , y ) d x d y ,
L ( f ; r , θ ) = + f ( r cos θ s sin θ , r sin θ + cos θ ) d s ,
S ( f ; r , θ , w ) = r w 2 r + w / 2 L ( f ; t , θ ) d t .
S ( f ; r , θ , w ) w L ( f ; r , θ ) .
g i = S ( f ; r i , θ i , w ) i = 1 , , M .
f ˆ ( x , y ) = j = 1 N a j b j ( x , y ) .
j = 1 N a j S ( b j ; r i , θ i , w ) = g i , i = 1 , , M .
p i j = S ( b j ; r i , θ i , w ) ,
P a = g ,
b j ( x , y ) = b ( x x j , y y j ) ,
p i j = S ( b j ; r i , θ i , w ) = S ( b ; d i j , θ i , w ) ,
b ( x , y ) d x d y = Δ 2 .
F ( r , θ ) = 0 r L ( b ; t , θ ) d t .
F ( r , θ ) = F ( r , θ ) , r > 0 ,
S ( b ; r , θ , w ) = F ( r + w / 2 , θ ) F ( r w / 2 , θ )
b ( x , y ) = ϕ ( x ) ϕ ( y ) ,
ϕ ( x ) = ϕ ( x ) .
L ( b ; r , θ ) = s 1 s 2 ϕ ( r cos θ s sin θ ) ϕ ( r sin θ + s cos θ ) d s ,
L ( b ; r , π / 4 + θ ) = L ( b ; r , π / 4 θ ) .
s 1 = max ( a + r sin θ cos θ , r cos θ α sin θ ) θ 0 , = a , θ = 0 , s 2 = a r sin θ cos θ
L ( b ; r , θ ) = s 2 s 1 = 2 a / cos θ , r a ( cos θ sin θ ) , = ( R θ r ) / ( sin θ cos θ ) , a ( cos θ sin θ ) r R θ , = 0 , r > R θ .

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