Abstract

A convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections. The formula is approximate but has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation. It reduces to the standard fan-beam formula in the plane that is perpendicular to the axis of rotation and contains the point source. The algorithm is applied to a mathematical phantom as an example of its performance.

© 1984 Optical Society of America

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References

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  1. See, for example, references contained in G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 14.
  2. R. A. Robb, A. H. Lent, B. K. Gilbert, A. Chu, “The dynamic spatial reconstructor,”J. Med. Syst. 4, 253–288 (1980).
    [CrossRef]
  3. M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).
  4. J. G. Colsher, “Iterative three-dimensional image reconstruction, from tomographic projections,” Comput. Graphics Image Processing, 6, 513–537 (1977).
    [CrossRef]
  5. M. D. Altschuler, G. T. Herman, A. Lent, “Fully three dimensional reconstruction from cone-beam sources,” in Proceedings of the Conference on Pattern Recognition and Image Processing (IEEE Computer Society, Long Beach, Calif., 1978), pp. 194–199.
  6. M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).
  7. M. Schlindwein, “Iterative three-dimensional reconstruction from twin-cone beam projections,”IEEE Trans. Nucl. Sci. NS-25, 1135–1143 (1978).
    [CrossRef]
  8. G. Kowalski, “Multislice reconstruction from twin-cone beam scanning,”IEEE Trans. Nucl. Sci. NS-26, 2895–2903 (1979).
    [CrossRef]
  9. H. E. Knutsson, P. Edholm, G. H. Granlund, C. Petersson, “Ectomography—a new radiographic reconstruction method, I., II.,”IEEE Trans. Biomed. Eng. BME-27, 640–655 (1980).
    [CrossRef]
  10. O. Nalcioglu, Z. H. Cho, “Reconstruction of 3-d objects from cone-beam projections,” Proc. IEEE 66, 1584–1585 (1978).
    [CrossRef]
  11. R. V. Denton, B. Friedlander, A. J. Rockmore, “Direct three-dimensional image reconstruction from divergent rays,”IEEE Trans. Nucl. Sci. NS-26, 4695–4703 (1979).
    [CrossRef]
  12. H. K. Tuy, “An inversion formula for cone-beam reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., June, 1981).
  13. G. N. Minerbo, “Convolutional Reconstruction from cone-beam projection data,”IEEE Trans. Nucl. Sci. NS-26, 2682–2684 (1979).
    [CrossRef]
  14. R. M. Lewitt, M. R. McKay, “Description of a software package for computing cone-beam x-ray projections of time-varying structures, and for dynamic three-dimensional image reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1980).
  15. G. T. Herman, A. V. Lakshminarayanan, A. Naparstek, “Convolution reconstruction techniques for divergent beams,” Comput. Biol. Med. 6, 259–271 (1976).
    [CrossRef] [PubMed]
  16. L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

1980

R. A. Robb, A. H. Lent, B. K. Gilbert, A. Chu, “The dynamic spatial reconstructor,”J. Med. Syst. 4, 253–288 (1980).
[CrossRef]

H. E. Knutsson, P. Edholm, G. H. Granlund, C. Petersson, “Ectomography—a new radiographic reconstruction method, I., II.,”IEEE Trans. Biomed. Eng. BME-27, 640–655 (1980).
[CrossRef]

1979

G. Kowalski, “Multislice reconstruction from twin-cone beam scanning,”IEEE Trans. Nucl. Sci. NS-26, 2895–2903 (1979).
[CrossRef]

R. V. Denton, B. Friedlander, A. J. Rockmore, “Direct three-dimensional image reconstruction from divergent rays,”IEEE Trans. Nucl. Sci. NS-26, 4695–4703 (1979).
[CrossRef]

G. N. Minerbo, “Convolutional Reconstruction from cone-beam projection data,”IEEE Trans. Nucl. Sci. NS-26, 2682–2684 (1979).
[CrossRef]

1978

M. Schlindwein, “Iterative three-dimensional reconstruction from twin-cone beam projections,”IEEE Trans. Nucl. Sci. NS-25, 1135–1143 (1978).
[CrossRef]

O. Nalcioglu, Z. H. Cho, “Reconstruction of 3-d objects from cone-beam projections,” Proc. IEEE 66, 1584–1585 (1978).
[CrossRef]

1977

J. G. Colsher, “Iterative three-dimensional image reconstruction, from tomographic projections,” Comput. Graphics Image Processing, 6, 513–537 (1977).
[CrossRef]

1976

G. T. Herman, A. V. Lakshminarayanan, A. Naparstek, “Convolution reconstruction techniques for divergent beams,” Comput. Biol. Med. 6, 259–271 (1976).
[CrossRef] [PubMed]

1974

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

Alschuler, M. D.

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

Altschuler, M. D.

M. D. Altschuler, G. T. Herman, A. Lent, “Fully three dimensional reconstruction from cone-beam sources,” in Proceedings of the Conference on Pattern Recognition and Image Processing (IEEE Computer Society, Long Beach, Calif., 1978), pp. 194–199.

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

Censor, Y.

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

Cho, Z. H.

O. Nalcioglu, Z. H. Cho, “Reconstruction of 3-d objects from cone-beam projections,” Proc. IEEE 66, 1584–1585 (1978).
[CrossRef]

Chu, A.

R. A. Robb, A. H. Lent, B. K. Gilbert, A. Chu, “The dynamic spatial reconstructor,”J. Med. Syst. 4, 253–288 (1980).
[CrossRef]

Colsher, J. G.

J. G. Colsher, “Iterative three-dimensional image reconstruction, from tomographic projections,” Comput. Graphics Image Processing, 6, 513–537 (1977).
[CrossRef]

Denton, R. V.

R. V. Denton, B. Friedlander, A. J. Rockmore, “Direct three-dimensional image reconstruction from divergent rays,”IEEE Trans. Nucl. Sci. NS-26, 4695–4703 (1979).
[CrossRef]

Edholm, P.

H. E. Knutsson, P. Edholm, G. H. Granlund, C. Petersson, “Ectomography—a new radiographic reconstruction method, I., II.,”IEEE Trans. Biomed. Eng. BME-27, 640–655 (1980).
[CrossRef]

Eggermont, P. P. B.

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

Friedlander, B.

R. V. Denton, B. Friedlander, A. J. Rockmore, “Direct three-dimensional image reconstruction from divergent rays,”IEEE Trans. Nucl. Sci. NS-26, 4695–4703 (1979).
[CrossRef]

Gilbert, B. K.

R. A. Robb, A. H. Lent, B. K. Gilbert, A. Chu, “The dynamic spatial reconstructor,”J. Med. Syst. 4, 253–288 (1980).
[CrossRef]

Granlund, G. H.

H. E. Knutsson, P. Edholm, G. H. Granlund, C. Petersson, “Ectomography—a new radiographic reconstruction method, I., II.,”IEEE Trans. Biomed. Eng. BME-27, 640–655 (1980).
[CrossRef]

Herman, G. T.

G. T. Herman, A. V. Lakshminarayanan, A. Naparstek, “Convolution reconstruction techniques for divergent beams,” Comput. Biol. Med. 6, 259–271 (1976).
[CrossRef] [PubMed]

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

See, for example, references contained in G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 14.

M. D. Altschuler, G. T. Herman, A. Lent, “Fully three dimensional reconstruction from cone-beam sources,” in Proceedings of the Conference on Pattern Recognition and Image Processing (IEEE Computer Society, Long Beach, Calif., 1978), pp. 194–199.

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

Knutsson, H. E.

H. E. Knutsson, P. Edholm, G. H. Granlund, C. Petersson, “Ectomography—a new radiographic reconstruction method, I., II.,”IEEE Trans. Biomed. Eng. BME-27, 640–655 (1980).
[CrossRef]

Kowalski, G.

G. Kowalski, “Multislice reconstruction from twin-cone beam scanning,”IEEE Trans. Nucl. Sci. NS-26, 2895–2903 (1979).
[CrossRef]

Kuo, Y. H.

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

Lakshminarayanan, A. V.

G. T. Herman, A. V. Lakshminarayanan, A. Naparstek, “Convolution reconstruction techniques for divergent beams,” Comput. Biol. Med. 6, 259–271 (1976).
[CrossRef] [PubMed]

Lent, A.

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

M. D. Altschuler, G. T. Herman, A. Lent, “Fully three dimensional reconstruction from cone-beam sources,” in Proceedings of the Conference on Pattern Recognition and Image Processing (IEEE Computer Society, Long Beach, Calif., 1978), pp. 194–199.

Lent, A. H.

R. A. Robb, A. H. Lent, B. K. Gilbert, A. Chu, “The dynamic spatial reconstructor,”J. Med. Syst. 4, 253–288 (1980).
[CrossRef]

Lewitt, R. M.

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

R. M. Lewitt, M. R. McKay, “Description of a software package for computing cone-beam x-ray projections of time-varying structures, and for dynamic three-dimensional image reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1980).

Logan, B. F.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

McKay, M.

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

McKay, M. R.

R. M. Lewitt, M. R. McKay, “Description of a software package for computing cone-beam x-ray projections of time-varying structures, and for dynamic three-dimensional image reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1980).

Minerbo, G. N.

G. N. Minerbo, “Convolutional Reconstruction from cone-beam projection data,”IEEE Trans. Nucl. Sci. NS-26, 2682–2684 (1979).
[CrossRef]

Nalcioglu, O.

O. Nalcioglu, Z. H. Cho, “Reconstruction of 3-d objects from cone-beam projections,” Proc. IEEE 66, 1584–1585 (1978).
[CrossRef]

Naparstek, A.

G. T. Herman, A. V. Lakshminarayanan, A. Naparstek, “Convolution reconstruction techniques for divergent beams,” Comput. Biol. Med. 6, 259–271 (1976).
[CrossRef] [PubMed]

Petersson, C.

H. E. Knutsson, P. Edholm, G. H. Granlund, C. Petersson, “Ectomography—a new radiographic reconstruction method, I., II.,”IEEE Trans. Biomed. Eng. BME-27, 640–655 (1980).
[CrossRef]

Robb, R. A.

R. A. Robb, A. H. Lent, B. K. Gilbert, A. Chu, “The dynamic spatial reconstructor,”J. Med. Syst. 4, 253–288 (1980).
[CrossRef]

Rockmore, A. J.

R. V. Denton, B. Friedlander, A. J. Rockmore, “Direct three-dimensional image reconstruction from divergent rays,”IEEE Trans. Nucl. Sci. NS-26, 4695–4703 (1979).
[CrossRef]

Schlindwein, M.

M. Schlindwein, “Iterative three-dimensional reconstruction from twin-cone beam projections,”IEEE Trans. Nucl. Sci. NS-25, 1135–1143 (1978).
[CrossRef]

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

Srihari, S. N.

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

Tuy, H.

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

Tuy, H. K.

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., June, 1981).

Udupa, J.

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

Udupa, J. K.

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

Yau, M. M.

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

Comput. Biol. Med.

G. T. Herman, A. V. Lakshminarayanan, A. Naparstek, “Convolution reconstruction techniques for divergent beams,” Comput. Biol. Med. 6, 259–271 (1976).
[CrossRef] [PubMed]

Comput. Graphics Image Processing

J. G. Colsher, “Iterative three-dimensional image reconstruction, from tomographic projections,” Comput. Graphics Image Processing, 6, 513–537 (1977).
[CrossRef]

IEEE Trans. Biomed. Eng.

H. E. Knutsson, P. Edholm, G. H. Granlund, C. Petersson, “Ectomography—a new radiographic reconstruction method, I., II.,”IEEE Trans. Biomed. Eng. BME-27, 640–655 (1980).
[CrossRef]

IEEE Trans. Nucl. Sci.

M. Schlindwein, “Iterative three-dimensional reconstruction from twin-cone beam projections,”IEEE Trans. Nucl. Sci. NS-25, 1135–1143 (1978).
[CrossRef]

G. Kowalski, “Multislice reconstruction from twin-cone beam scanning,”IEEE Trans. Nucl. Sci. NS-26, 2895–2903 (1979).
[CrossRef]

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

R. V. Denton, B. Friedlander, A. J. Rockmore, “Direct three-dimensional image reconstruction from divergent rays,”IEEE Trans. Nucl. Sci. NS-26, 4695–4703 (1979).
[CrossRef]

G. N. Minerbo, “Convolutional Reconstruction from cone-beam projection data,”IEEE Trans. Nucl. Sci. NS-26, 2682–2684 (1979).
[CrossRef]

J. Med. Syst.

R. A. Robb, A. H. Lent, B. K. Gilbert, A. Chu, “The dynamic spatial reconstructor,”J. Med. Syst. 4, 253–288 (1980).
[CrossRef]

Proc. IEEE

O. Nalcioglu, Z. H. Cho, “Reconstruction of 3-d objects from cone-beam projections,” Proc. IEEE 66, 1584–1585 (1978).
[CrossRef]

Other

M. D. Alschuler, Y. Censor, G. T. Herman, A. Lent, R. M. Lewitt, S. N. Srihari, H. Tuy, J. K. Udupa, “Mathematical aspects of image reconstruction from projections,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1981).

M. D. Altschuler, G. T. Herman, A. Lent, “Fully three dimensional reconstruction from cone-beam sources,” in Proceedings of the Conference on Pattern Recognition and Image Processing (IEEE Computer Society, Long Beach, Calif., 1978), pp. 194–199.

M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman, Y. H. Kuo, R. M. Lewitt, M. McKay, H. Tuy, J. Udupa, M. M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone-beam x-ray projections,” (State University of New York at Buffalo, Buffalo, N.Y., August, 1979).

R. M. Lewitt, M. R. McKay, “Description of a software package for computing cone-beam x-ray projections of time-varying structures, and for dynamic three-dimensional image reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1980).

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., June, 1981).

See, for example, references contained in G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 14.

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Figures (8)

Fig. 1
Fig. 1

Schematic physical arrangement of the 3D tomographic system. The source-to-rotation axis distance is d; the source-to-detector plane distance is D = d + d′.

Fig. 2
Fig. 2

Geometry in the midplane for derivation of the fan-beam formula. The detector system is here represented by its projection on a line (aa′) through the origin and parallel to the actual detector line.

Fig. 3
Fig. 3

Coordinate system for describing projection and reconstruction in the midplane of a 3D system. The unit vectors m ^ , n ^, and k ^ form an orthonormal set. The axis of rotation is along k ^.

Fig. 4
Fig. 4

Coordinate system for projections above the midplane. The axis of rotation is along z. The vector n ^ is parallel to the midplane. The vector k ^ is inclined with respect to z and is given by k ^ = m ^ × n ^. ρ′ lies in the shaded plane.

Fig. 5
Fig. 5

Comparison of representative slices of a phantom with its reconstruction. The phantom and the detector array used are defined in Table 1. The lower row of slices is an exact digitized representation of the phantom, with the corresponding reconstruction just above. The horizontal line defines the position corresponding to the line drawing, in which the density of the phantom (solid line) is compared with that of the reconstruction (points). The scale is linear with a range of 0.0 to 1.0.

Fig. 6
Fig. 6

Same as Fig. 5 except that 128 angles and twice the linear detector density were employed.

Fig. 7
Fig. 7

Comparison of vertical (x = constant) slices of the exact phantom (middle row) with corresponding slices from the present method (row below middle) and the unmodified fan-beam method (row above middle). The display scale has been concentrated in the density range 0.64 to 1.44 for clarity. Absolute differences from exact are shown in the bottom row (present method) and the top row (unmodified fan-beam method). In order to highlight small differences, the scale is linear from 0.0 to 0.2. The phantom contains the two additional ellipsoids noted in Table 1.

Fig. 8
Fig. 8

Same as Fig. 7 except that the source-axis distance d has been reduced to d = 40.0 with a corresponding increase in the detector coverage.

Tables (1)

Tables Icon

Table 1 Detector Array and Phantom Used in Algorithm Testa

Equations (45)

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

l = Y d / ( d 2 + Y 2 ) 1 / 2             or             Y = l d / ( d 2 - l 2 ) 1 / 2 .
θ = Φ + π / 2 + α ,
α = tan - 1 ( Y / d ) = tan - 1 [ l / ( d 2 - l 2 ) 1 / 2 ] .
p ( l , θ ) = r d r d ϕ f ( r , ϕ ) δ [ r cos ( θ - ϕ ) - l ] ,
p ( l , θ ) = P Φ ( Y ) , l < d = 0 , l > d .
f ( r , ϕ ) = 1 4 π 2 d θ - d l r cos ( θ - ϕ ) - l l p ( l , θ ) .
p ( l , θ ) = - d ω 2 π exp ( i ω l ) q ( ω , θ ) .
f ( r , ϕ ) = 1 8 π 2 d θ - ω d ω q ( ω , θ ) exp [ i ω r cos ( θ - ϕ ) ] .
f ( r , ϕ ) = 1 4 π 2 Re d θ - ω d ω - d l p ( l , θ ) × exp { i ω [ r cos ( θ - ϕ ) - l ] } ,
d θ d l = d Φ d Y d 3 / ( d 2 + Y 2 ) 3 / 2 .
ω = ω [ d + r cos ( ϕ - Φ ) ] / ( d 2 + Y 2 ) 1 / 2 ,
f ( r , ϕ ) = 1 4 π 2 Re d Φ d 2 [ d + r cos ( ϕ - Φ ) ] 2 × 0 ω d ω - d Y d ( d 2 + Y 2 ) 1 / 2 P Φ ( Y ) × exp [ i ω ( d r sin ( ϕ - Φ ) d + r cos ( ϕ - Φ ) - Y ) ] .
f ( r , ϕ ) = 1 4 π 2 d Φ d 2 [ d + r cos ( ϕ - Φ ) ] 2 P ˜ Φ [ Y ( r , ϕ ) ] ,
Y ( r , ϕ ) = d r sin ( ϕ - Φ ) / [ d + r cos ( ϕ - Φ ) ] ,
P ˜ Φ ( Y ) = - d Y d ( d 2 + Y 2 ) 1 / 2 P Φ ( Y ) g ( Y - Y ) ,
g ( Y ) = Re 0 ω y 0 exp ( i ω Y ) ω d ω .
P ˜ Φ ( i ) ( Y j ) = j P Φ ( i ) ( Y j ) cos θ j Y j - Δ Y / 2 Y j + Δ Y / 2 g ( Y j - Y ) d Y .
f ( ρ ) = 1 4 π 2 Re d Φ d 2 ( d + ρ · m ^ ) 2 0 ω d ω × - d Y d ( d 2 + Y 2 ) 1 / 2 P Φ ( Y , Z = 0 ) × exp [ i ω ( d ρ · n ^ d + ρ · m ^ - Y ) ] ,
δ m ^ = δ Φ z ^ × m ^
= δ Φ d ( d 2 + Z 2 ) 1 / 2 n ^
δ m ^ = δ Φ k ^ × m ^
= δ Φ n ^ .
δ Φ = δ Φ d / ( d 2 + Z 2 ) 1 / 2 .
r = ρ + Z z ^ ,
ρ · k ^ = 0.
δ f ( ρ + Z z ^ ) = 1 4 π 2 Re δ ϕ d 2 ( d + ρ · m ^ ) 2 0 ω d ω × - d Y d ( d 2 + Y 2 ) 1 / 2 P Φ ( Y , Z ) × exp [ i ω ( d ρ · n ^ d + ρ · m ^ - Y ) ] .
ρ · m ^ = d r · x ^ / d
d ( d 2 + Y 2 ) 1 / 2 = ( d 2 + Z 2 ) 1 / 2 ( d 2 + Y 2 + Z 2 ) 1 / 2 .
δ f ( r ) = 1 4 π 2 Re δ Φ d 2 ( d + r · x ^ ) 2 0 ω d ω × - d Y d ( d 2 + Y 2 + Z 2 ) 1 / 2 P Φ ( Y , Z ) × exp [ i ω ( d r · y ^ d + r · x ^ - Y ) ] ,
Z = z d / [ d + r · x ^ ] .
f ( r ) = 1 4 π 2 d Φ d 2 ( d + r · x ^ ) 2 P ˜ Φ [ Y ( r ) , Z ( r ) ] ,
Y ( r ) = r · y ^ d / ( d + r · x ^ ) ,
Z ( r ) = r · z ^ d / ( d + r · x ^ ) ,
P ˜ Φ ( Y , Z ) = - d Y - d Z g y ( Y - Y ) g z ( Z - Z ) × P Φ ( Y , Z ) d / ( d 2 + Y 2 + Z 2 ) 1 / 2 ,
g y ( Y ) = Re 0 ω y 0 ω d ω exp ( i ω Y ) ,
g z ( Z ) = sin ω z 0 Z / π Z .
P ˜ Φ ( i ) ( Y j , Z k ) = j , k P Φ ( i ) ( Y j , Z k ) × cos θ j k Y j - Δ Y / 2 Y j + Δ Y / 2 g y ( Y j - Y ) d Y × Z k - Δ Z / 2 Z k + Δ Z / 2 g z ( Z k - Z ) d Z ,
P Φ ( Y ) = ( d 2 + Y 2 ) 1 / 2 / ( d + ρ 0 · x ^ ) × δ [ Y - ( ρ 0 · y ^ d ) / ( d + ρ 0 · x ^ ) ] .
1 4 π 2 d Φ d 2 ( d + ρ · x ^ ) 2 g [ ( ρ · y ^ d ) / ( d + ρ · x ^ ) - ( ρ 0 · y ^ d ) / ( d + ρ 0 · x ^ ) ] d d + ρ 0 · x ^ δ ( ρ - ρ 0 ) ,
P Φ ( Y , Z ) = d 2 ρ 0 d z 0 f 0 ( r 0 ) d ( d 2 + Y 2 + Z 2 ) 1 / 2 ( d + ρ 0 · x ^ ) 2 × δ [ Y - ( ρ 0 · y ^ d ) / ( d + ρ 0 · x ^ ) ] δ [ Z - z 0 d / ( d + ρ 0 · x ^ ) ] ,
d z f ( r ) = 1 4 π 2 d Φ d 2 ( d + r · x ^ ) 2 d 2 ρ 0 d z 0 f 0 ( r 0 ) × g y [ Y ( r ) - ( ρ 0 · y ^ d ) / ( d + ρ 0 · x ^ ) ] d 2 ( d + ρ 0 · x ^ ) 2 × d z g z [ z d / ( d + r · x ^ ) - z 0 d / ( d + ρ 0 · x ^ ) ] .
d z f ( r ) = d 2 ρ 0 d z 0 f 0 ( r 0 ) 1 4 π 2 × d Φ d ( d + ρ · x ^ ) g y [ ( ρ · y ^ d ) / ( d + ρ · x ^ ) - ( ρ 0 · y ^ d ) / ( d + ρ 0 · x ^ ) ] d 2 ( d + ρ 0 · x ^ ) 2 .
d z f ( r ) d z 0 f 0 ( ρ , z 0 ) .
P Φ ( Y , Z ) = ( d 2 + Y 2 + Z 2 ) 1 / 2 ( d 2 + Y 2 ) 1 / 2 P Φ ( Y , Z = 0 )
P ˜ Φ ( Y , Z ) = - d Y g y ( Y - Y ) × P Φ ( Y , Z = 0 ) d / ( d 2 + Y 2 ) 1 / 2 ,

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