Abstract

A novel algebraic iterative algorithm based on deflection tomography is presented. This algorithm is derived from the essentials of deflection tomography with a linear expansion of the local basis functions. By use of this algorithm the tomographic problem is finally reduced to the solution of a set of linear equations. The algorithm is demonstrated by mapping a three-peak Gaussian simulative temperature field. Compared with reconstruction results obtained by other traditional deflection algorithms, its reconstruction results provide a significant improvement in reconstruction accuracy, especially in cases with noisy data added. In the density diagnosis of a hypersonic wind tunnel, this algorithm is adopted to reconstruct density distributions of an axial symmetry flow field. One cross section of the reconstruction results is selected to be compared with the inverse Abel transform algorithm. Results show that the novel algorithm can achieve an accuracy equivalent to the inverse Abel transform algorithm. However, the novel algorithm is more versatile because it is applicable to arbitrary kinds of distribution.

© 2006 Optical Society of America

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References

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  1. G. P. Montgomery, Jr., and D. L. Reuss, "Effects of refraction on axisymmetric flame temperatures measured by holographic interferometry," Appl. Opt. 21, 1373-1380 (1982).
    [CrossRef] [PubMed]
  2. J. Stricker, E. Keren, and O. Kafri, "Axisymmetric density field measurements by moire deflectometry," AIAA J. 21, 1767-1769 (1983).
    [CrossRef]
  3. E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, "Temperature mapping in flames by moiré deflectometry," Appl. Opt. 22, 689-705 (1983).
    [CrossRef]
  4. G. W. Faris and R. L. Byer, "Three-dimensional beam-deflection optical tomography of a supersonic jet," Appl. Opt. 27, 5202-5212 (1988).
    [CrossRef] [PubMed]
  5. D. Wu and A. He, "Measurement of three-dimensional temperature fields with interferometric tomography," Appl. Opt. 38, 3468-3473 (1999).
    [CrossRef]
  6. J. D. Posner and D. Dunn-Rankin, "Temperature field measurements of small, nonpremixed flames with use of an Abel inversion of holographic interferograms," Appl. Opt. 42, 952-959 (2003).
    [CrossRef] [PubMed]
  7. O. Sasaki and T. Kobayashi, "Beam-deflection optical tomography of the refractive-index distribution based on the Rytov approximation," Appl. Opt. 32, 746-751 (1993).
    [CrossRef] [PubMed]
  8. D. W. Sweeny and C. M. Vest, "Reconstruction of three-dimensional refractive index field from multi-direction interferometric data," Appl. Opt. 12, 2649-2664 (1973).
    [CrossRef]
  9. O. Kafri and I. Glatt, "Moiré deflectometry: a ray deflection approach to optical testing," Opt. Eng. 24, 944-960 (1985).
  10. Y. Song, B. Zhang, and A. He, "Bayesian approach to limited-projection reconstruction in moiré tomography," in ICO20: Optical Information Processing, Y. Sheng, S. Zhuang, and Y. Zhang, eds., Proc. SPIE 6027, 6027U (2006).
  11. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).
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    [CrossRef]
  13. R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography," J. Theor. Biol. 29, 471-481 (1970).
    [CrossRef] [PubMed]
  14. A. He, D. Yan, X. Ni, and H. Wang, "Design and application of real-time holographic large-aperture moire deflector with high sensitivity and high precision," Chin. J. Lasers 18, 827-831 (1991).
  15. D. Yan, A. He, and X. Ni, "New method of asymmetric flow field measurement in hypersonic shock tunnel," Appl. Opt. 30, 770-774 (1991).
    [CrossRef] [PubMed]
  16. J. Stricker, "Analysis of 3-D phase objects by moiré deflectometry," Appl. Opt. 23, 3657-3659 (1984).
    [CrossRef] [PubMed]
  17. Y.Song, B. Zhang, and A. He, "The filtered back-projection algorithm of deflection tomography and error analysis," Acta Opt. Sin. (to be published).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  26. C. Soller, R. Wenskus, P. Middendorf, G. Meier, and F. Obermeier, "Interferometric tomography for flow visualization of density fields in supersonic jets and convective flow," Appl. Opt. 33, 2921-2932 (1994).
    [CrossRef] [PubMed]
  27. J. Stricker and O. Kafri, "A New Method for Density Gradient Measurements in Compressible Flows," AIAA J. 20, 820-823 (1982).
    [CrossRef]
  28. M. Anastasio and X. Pan, "Full- and minimal-scan reconstruction algorithms for fan-beam diffraction tomography," Appl. Opt. 40, 3334-3345 (2001).
    [CrossRef]
  29. A. H. Andersen and A. C. Kak, "Digital ray tracing in two-dimensional refractive fields," J. Acoust. Soc. Am. 72, 1593-1606 (1982).
    [CrossRef]

2006

Y. Song, B. Zhang, and A. He, "Bayesian approach to limited-projection reconstruction in moiré tomography," in ICO20: Optical Information Processing, Y. Sheng, S. Zhuang, and Y. Zhang, eds., Proc. SPIE 6027, 6027U (2006).

2004

2003

2001

1999

1998

1996

D. Yan, F. Liu, Z. Wang, and A. He, "Moiré tomography by ART," in Laser Interferometry: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 2861, 146-150 (1996).
[CrossRef]

1994

1993

1991

A. He, D. Yan, X. Ni, and H. Wang, "Design and application of real-time holographic large-aperture moire deflector with high sensitivity and high precision," Chin. J. Lasers 18, 827-831 (1991).

D. Yan, A. He, and X. Ni, "New method of asymmetric flow field measurement in hypersonic shock tunnel," Appl. Opt. 30, 770-774 (1991).
[CrossRef] [PubMed]

1988

1987

1985

1984

1983

R. M. Lewitt, "Reconstruction algorithms: transform methods," Proc. IEEE 71, 390-408 (1983).
[CrossRef]

J. Stricker, E. Keren, and O. Kafri, "Axisymmetric density field measurements by moire deflectometry," AIAA J. 21, 1767-1769 (1983).
[CrossRef]

E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, "Temperature mapping in flames by moiré deflectometry," Appl. Opt. 22, 689-705 (1983).
[CrossRef]

1982

G. P. Montgomery, Jr., and D. L. Reuss, "Effects of refraction on axisymmetric flame temperatures measured by holographic interferometry," Appl. Opt. 21, 1373-1380 (1982).
[CrossRef] [PubMed]

J. Stricker and O. Kafri, "A New Method for Density Gradient Measurements in Compressible Flows," AIAA J. 20, 820-823 (1982).
[CrossRef]

A. H. Andersen and A. C. Kak, "Digital ray tracing in two-dimensional refractive fields," J. Acoust. Soc. Am. 72, 1593-1606 (1982).
[CrossRef]

1973

1970

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[CrossRef] [PubMed]

Agrawal, A. K.

Anastasio, M.

Andersen, A. H.

A. H. Andersen and A. C. Kak, "Digital ray tracing in two-dimensional refractive fields," J. Acoust. Soc. Am. 72, 1593-1606 (1982).
[CrossRef]

Bar-Ziv, E.

E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, "Temperature mapping in flames by moiré deflectometry," Appl. Opt. 22, 689-705 (1983).
[CrossRef]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[CrossRef] [PubMed]

Butuk, N. K.

Byer, R. L.

Cai, G.

Dhawan, A. P.

Dunn-Rankin, D.

Faris, G. W.

Gao, Y.

Glatt, I.

O. Kafri and I. Glatt, "Moiré deflectometry: a ray deflection approach to optical testing," Opt. Eng. 24, 944-960 (1985).

Gollahalli, S. R.

Gordon, R.

R. Rangayyan, A. P. Dhawan, and R. Gordon, "Algorithms for limited-view computed tomography: an annotated bibliography and a challenge," Appl. Opt. 24, 4000-4012 (1985).
[CrossRef] [PubMed]

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[CrossRef] [PubMed]

Griffin, D.

He, A.

Y. Song, B. Zhang, and A. He, "Bayesian approach to limited-projection reconstruction in moiré tomography," in ICO20: Optical Information Processing, Y. Sheng, S. Zhuang, and Y. Zhang, eds., Proc. SPIE 6027, 6027U (2006).

D. Wu and A. He, "Measurement of three-dimensional temperature fields with interferometric tomography," Appl. Opt. 38, 3468-3473 (1999).
[CrossRef]

D. Yan, F. Liu, Z. Wang, and A. He, "Moiré tomography by ART," in Laser Interferometry: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 2861, 146-150 (1996).
[CrossRef]

A. He, D. Yan, X. Ni, and H. Wang, "Design and application of real-time holographic large-aperture moire deflector with high sensitivity and high precision," Chin. J. Lasers 18, 827-831 (1991).

D. Yan, A. He, and X. Ni, "New method of asymmetric flow field measurement in hypersonic shock tunnel," Appl. Opt. 30, 770-774 (1991).
[CrossRef] [PubMed]

Y.Song, B. Zhang, and A. He, "The filtered back-projection algorithm of deflection tomography and error analysis," Acta Opt. Sin. (to be published).

Herman, G. T.

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[CrossRef] [PubMed]

Kafri, O.

O. Kafri and I. Glatt, "Moiré deflectometry: a ray deflection approach to optical testing," Opt. Eng. 24, 944-960 (1985).

J. Stricker, E. Keren, and O. Kafri, "Axisymmetric density field measurements by moire deflectometry," AIAA J. 21, 1767-1769 (1983).
[CrossRef]

E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, "Temperature mapping in flames by moiré deflectometry," Appl. Opt. 22, 689-705 (1983).
[CrossRef]

J. Stricker and O. Kafri, "A New Method for Density Gradient Measurements in Compressible Flows," AIAA J. 20, 820-823 (1982).
[CrossRef]

Kak, A. C.

A. H. Andersen and A. C. Kak, "Digital ray tracing in two-dimensional refractive fields," J. Acoust. Soc. Am. 72, 1593-1606 (1982).
[CrossRef]

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Kanjirodan, R.

Keren, E.

J. Stricker, E. Keren, and O. Kafri, "Axisymmetric density field measurements by moire deflectometry," AIAA J. 21, 1767-1769 (1983).
[CrossRef]

E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, "Temperature mapping in flames by moiré deflectometry," Appl. Opt. 22, 689-705 (1983).
[CrossRef]

Kobayashi, T.

Langoju, R.

Lewitt, R. M.

R. M. Lewitt, "Reconstruction algorithms: transform methods," Proc. IEEE 71, 390-408 (1983).
[CrossRef]

Liu, F.

D. Yan, F. Liu, Z. Wang, and A. He, "Moiré tomography by ART," in Laser Interferometry: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 2861, 146-150 (1996).
[CrossRef]

Meier, G.

Middendorf, P.

Montgomery, G. P.

Ni, X.

A. He, D. Yan, X. Ni, and H. Wang, "Design and application of real-time holographic large-aperture moire deflector with high sensitivity and high precision," Chin. J. Lasers 18, 827-831 (1991).

D. Yan, A. He, and X. Ni, "New method of asymmetric flow field measurement in hypersonic shock tunnel," Appl. Opt. 30, 770-774 (1991).
[CrossRef] [PubMed]

Obermeier, F.

Padmaram, R.

Pan, X.

Patnaik, L.

Posner, J. D.

Rangayyan, R.

Reuss, D. L.

Sasaki, O.

Sgulim, S.

E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, "Temperature mapping in flames by moiré deflectometry," Appl. Opt. 22, 689-705 (1983).
[CrossRef]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Soller, C.

Song, Y.

Y. Song, B. Zhang, and A. He, "Bayesian approach to limited-projection reconstruction in moiré tomography," in ICO20: Optical Information Processing, Y. Sheng, S. Zhuang, and Y. Zhang, eds., Proc. SPIE 6027, 6027U (2006).

Y.Song, B. Zhang, and A. He, "The filtered back-projection algorithm of deflection tomography and error analysis," Acta Opt. Sin. (to be published).

Stricker, J.

J. Stricker, "Analysis of 3-D phase objects by moiré deflectometry," Appl. Opt. 23, 3657-3659 (1984).
[CrossRef] [PubMed]

J. Stricker, E. Keren, and O. Kafri, "Axisymmetric density field measurements by moire deflectometry," AIAA J. 21, 1767-1769 (1983).
[CrossRef]

J. Stricker and O. Kafri, "A New Method for Density Gradient Measurements in Compressible Flows," AIAA J. 20, 820-823 (1982).
[CrossRef]

Sweeny, D. W.

Thayyullathil, H.

Vasu, R. Mohan

Vest, C. M.

Wan, X.

Wang, H.

A. He, D. Yan, X. Ni, and H. Wang, "Design and application of real-time holographic large-aperture moire deflector with high sensitivity and high precision," Chin. J. Lasers 18, 827-831 (1991).

Wang, Z.

D. Yan, F. Liu, Z. Wang, and A. He, "Moiré tomography by ART," in Laser Interferometry: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 2861, 146-150 (1996).
[CrossRef]

Wenskus, R.

Wu, D.

Yan, D.

D. Yan, F. Liu, Z. Wang, and A. He, "Moiré tomography by ART," in Laser Interferometry: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 2861, 146-150 (1996).
[CrossRef]

A. He, D. Yan, X. Ni, and H. Wang, "Design and application of real-time holographic large-aperture moire deflector with high sensitivity and high precision," Chin. J. Lasers 18, 827-831 (1991).

D. Yan, A. He, and X. Ni, "New method of asymmetric flow field measurement in hypersonic shock tunnel," Appl. Opt. 30, 770-774 (1991).
[CrossRef] [PubMed]

Yi, J.

Yu, S.

Zhang, B.

Y. Song, B. Zhang, and A. He, "Bayesian approach to limited-projection reconstruction in moiré tomography," in ICO20: Optical Information Processing, Y. Sheng, S. Zhuang, and Y. Zhang, eds., Proc. SPIE 6027, 6027U (2006).

Y.Song, B. Zhang, and A. He, "The filtered back-projection algorithm of deflection tomography and error analysis," Acta Opt. Sin. (to be published).

Zhu, D.

D. Zhu, Laser Metrology for Thermal Physics (Science, 1990).

AIAA J.

J. Stricker, E. Keren, and O. Kafri, "Axisymmetric density field measurements by moire deflectometry," AIAA J. 21, 1767-1769 (1983).
[CrossRef]

J. Stricker and O. Kafri, "A New Method for Density Gradient Measurements in Compressible Flows," AIAA J. 20, 820-823 (1982).
[CrossRef]

Appl. Opt.

H. Thayyullathil, R. Langoju, R. Padmaram, R. Mohan Vasu, R. Kanjirodan, and L. Patnaik, "Three-dimensional optical tomographic imaging of supersonic jets through inversion of phase data obtained through the transport-of-intensity equation," Appl. Opt. 43, 4133-4141 (2004).
[CrossRef]

E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, "Temperature mapping in flames by moiré deflectometry," Appl. Opt. 22, 689-705 (1983).
[CrossRef]

D. W. Sweeny and C. M. Vest, "Reconstruction of three-dimensional refractive index field from multi-direction interferometric data," Appl. Opt. 12, 2649-2664 (1973).
[CrossRef]

G. P. Montgomery, Jr., and D. L. Reuss, "Effects of refraction on axisymmetric flame temperatures measured by holographic interferometry," Appl. Opt. 21, 1373-1380 (1982).
[CrossRef] [PubMed]

J. Stricker, "Analysis of 3-D phase objects by moiré deflectometry," Appl. Opt. 23, 3657-3659 (1984).
[CrossRef] [PubMed]

R. Rangayyan, A. P. Dhawan, and R. Gordon, "Algorithms for limited-view computed tomography: an annotated bibliography and a challenge," Appl. Opt. 24, 4000-4012 (1985).
[CrossRef] [PubMed]

G. W. Faris and R. L. Byer, "Three-dimensional beam-deflection optical tomography of a supersonic jet," Appl. Opt. 27, 5202-5212 (1988).
[CrossRef] [PubMed]

D. Yan, A. He, and X. Ni, "New method of asymmetric flow field measurement in hypersonic shock tunnel," Appl. Opt. 30, 770-774 (1991).
[CrossRef] [PubMed]

O. Sasaki and T. Kobayashi, "Beam-deflection optical tomography of the refractive-index distribution based on the Rytov approximation," Appl. Opt. 32, 746-751 (1993).
[CrossRef] [PubMed]

C. Soller, R. Wenskus, P. Middendorf, G. Meier, and F. Obermeier, "Interferometric tomography for flow visualization of density fields in supersonic jets and convective flow," Appl. Opt. 33, 2921-2932 (1994).
[CrossRef] [PubMed]

D. Wu and A. He, "Measurement of three-dimensional temperature fields with interferometric tomography," Appl. Opt. 38, 3468-3473 (1999).
[CrossRef]

A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, and D. Griffin, "Three-dimensional rainbow schlieren tomography of a temperature field in gas flows," Appl. Opt. 37, 479-485 (1998).
[CrossRef]

M. Anastasio and X. Pan, "Full- and minimal-scan reconstruction algorithms for fan-beam diffraction tomography," Appl. Opt. 40, 3334-3345 (2001).
[CrossRef]

J. D. Posner and D. Dunn-Rankin, "Temperature field measurements of small, nonpremixed flames with use of an Abel inversion of holographic interferograms," Appl. Opt. 42, 952-959 (2003).
[CrossRef] [PubMed]

Chin. J. Lasers

A. He, D. Yan, X. Ni, and H. Wang, "Design and application of real-time holographic large-aperture moire deflector with high sensitivity and high precision," Chin. J. Lasers 18, 827-831 (1991).

J. Acoust. Soc. Am.

A. H. Andersen and A. C. Kak, "Digital ray tracing in two-dimensional refractive fields," J. Acoust. Soc. Am. 72, 1593-1606 (1982).
[CrossRef]

J. Opt. Soc. Am. A

J. Theor. Biol.

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[CrossRef] [PubMed]

Opt. Eng.

O. Kafri and I. Glatt, "Moiré deflectometry: a ray deflection approach to optical testing," Opt. Eng. 24, 944-960 (1985).

Opt. Lett.

Proc. IEEE

R. M. Lewitt, "Reconstruction algorithms: transform methods," Proc. IEEE 71, 390-408 (1983).
[CrossRef]

Proc. SPIE

Y. Song, B. Zhang, and A. He, "Bayesian approach to limited-projection reconstruction in moiré tomography," in ICO20: Optical Information Processing, Y. Sheng, S. Zhuang, and Y. Zhang, eds., Proc. SPIE 6027, 6027U (2006).

D. Yan, F. Liu, Z. Wang, and A. He, "Moiré tomography by ART," in Laser Interferometry: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 2861, 146-150 (1996).
[CrossRef]

Other

D. Zhu, Laser Metrology for Thermal Physics (Science, 1990).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Y.Song, B. Zhang, and A. He, "The filtered back-projection algorithm of deflection tomography and error analysis," Acta Opt. Sin. (to be published).

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Figures (16)

Fig. 1
Fig. 1

Coordinate setup.

Fig. 2
Fig. 2

Grid divisions of the iterative algorithms.

Fig. 3
Fig. 3

(Color online) Local basis functions.

Fig. 4
Fig. 4

Calculation of p ij .

Fig. 5
Fig. 5

(Color online) Three-peak Gaussian simulated temperature field.

Fig. 6
Fig. 6

Simulated deflection projection of 30° and corresponding phase projection obtained with Eq. (3).

Fig. 7
Fig. 7

(Color online) Simulated deflection projection of 30° with system noise added and corresponding phase projection obtained with Eq. (3).

Fig. 8
Fig. 8

(Color online) Reconstructions of the three-peak Gaussian temperature fields without noise: (a) DFBP, (b) DCIR-a, (c) DRIR-c.

Fig. 9
Fig. 9

(Color online) Reconstructions of the three-peak Gaussian temperature fields with system noise: (a) DFBP, (b) DCIR-b, (c) DRIR-a.

Fig. 10
Fig. 10

(Color online) Convergence of the DCIR and DRIR algorithms.

Fig. 11
Fig. 11

(Color online) Convergence of the DRIR algorithms with three basis functions (without noise).

Fig. 12
Fig. 12

Optical schematic diagram of the experimental device.

Fig. 13
Fig. 13

Moiré pattern of the flow field generated by a blunted cone in the hypersonic wind tunnel.

Fig. 14
Fig. 14

(Color online) Density distribution of the cross section ( 4.15   cm below the bottom surface of the blunted cone).

Fig. 15
Fig. 15

(Color online) Comparisons between the inverse Abel transform and the DRIR algorithm.

Fig. 16
Fig. 16

(Color online) Reconstruction of the density field of the axial symmetry flow field: (a) three-dimensional reconstruction, (b) xy cross section at z = 3   cm , (c) xy cross section at z = 1.5   cm , (d) xz cross section at y = 15   cm . (The reconstruction area is below the bottom surface of the blunted cone.)

Tables (2)

Tables Icon

Table 1 Results of the Calculation of the Partial Derivative a

Tables Icon

Table 2 Reconstruction Errors a

Equations (119)

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

φ p ( y , θ ) = n ( r , ψ ) δ ( y r   sin ( ψ θ ) ) d r d ψ .
φ d ( y , θ ) = 1 n 0 n y δ ( y r   sin ( ψ θ ) ) d r d ψ ,
n 0
n c ( r , ψ )
n ( r , ψ ) = n 0 n c ( r , ψ )
φ p
φ d
φ p ( y , θ ) = n 0 2 [ φ d ( y , θ ) sgn ( y ) ] .
x y
φ d ( y , θ ) = 1 n 0 + n y | y = y cos θ x sin θ d x .
n / y
φ d
x
W × W
n ^ ( x , y )
b i ( x , y )
n ^ ( x , y ) = i = 0 N 1 n i b i ( x , y ) .
b i
b i
B × B
b i
N = ( W + B 1 )
( W + B 1 )
( x i , y i )
b i ( x , y ) = b ( x x i , y y i ) = ϕ ( x x i ) ϕ ( y y i ) .
φ d ( y , θ ) = - 1 n 0 i = 0 N 1 n i + b i ( x , y ) y | y = y cos θ x sin θ d x .
x y
x y
x = x   cos   θ - y   sin   θ ,
y = x   sin   θ + y   cos   θ .
b i ( x , y ) / y
b i ( x , y ) y = ϕ ( x x i ) d ϕ ( y y i ) d y y y + ϕ ( y y i ) × d ϕ ( x x i ) d x x y = ϕ ( x x i ) d ϕ ( y y i ) d y cos   θ ϕ ( y y i ) × d ϕ ( x x i ) d x sin   θ .
ϕ ( x )
b i ( x , y ) / y
( x , y )
M = l × q
φ j
n 0 φ j = i = 0 N 1 n i + b i ( x , y ) y | y = y cos θ x sin θ d x ,
j = 0 , ,   M 1 .
h = P n .
h j
n i
b i ( x , y )
M × N
p i j
b i ( x , y )
p i j
b i ( x , y )
p i j
b i ( x , y ) / y
b i ( x , y )
p i j = k = 0 K 1 b i ( x , y ) y | y = y cos θ x sin θ Δ s ,
M N
M < N
n ( 0 )
n ( 0 , 0 ) = n ( 0 ) ,
n i ( K , j + 1 ) = n i ( K , j ) + ω h j i = 0 N 1 p i j n i ( K , j ) i = 0 N 1 ( p i j ) 2 p ij ,
0 i N 1 , 0 j M 1 ,
n ( K + 1 , 0 ) = n ( K , M 1 ) ,
ω ( 0 , 2 )
t ^ ( x , y )
6.5 cm × 6.5   cm
50 × 50
t ^ ( x , y ) = 300   exp [ ( x 2.6 ) 2 + ( y 2.21 ) 2 0.8 ] + 200   exp [ ( x 4.68 ) 2 + ( y 3.12 ) 2 0.8 ] + 100   exp [ ( x 2.73 ) 2 + ( y 4.29 ) 2 0.8 ]
for   0 < x < 6.5 0 < y < 6.5 ,
t ^ ( x , y ) = 0   outside   the   region .
n c
n c 1 = K ρ ,
n c
n c 1 = 0.292015 × 10 3 1 + 0.368184 × 10 2 t .
n ^ c ( x , y )
t ^ ( x , y )
φ j
p i j
φ j = φ j + N ( μ , σ 2 ) ,
N ( μ , σ 2 )
σ 2
N ( 0 , 0 )
N ( 0 , ( 0.006 ) 2 )
N ( 0.000036 , ( 0.006 ) 2 )
RMSE = [ i = 0 N 1 ( t i t ^ i ) 2 ] 1 / 2 N · t ^ max ,
MAE = i = 0 N 1 | t i t ^ i | N · t ^ max ,
PVE = | t max t ^ max | t ^ max .
t ^ i
( x i + d / 2 , y i + d / 2 )
t ^ max
t i
t max
n 0
0 n i ( K ) ( n 0 1 )
Φ 500   mm
Φ 50   mm
0.05   mm
+ α / 2
p = a 2   sin ( α / 2 ) .
p m = φ p Δ / a ,
φ
φ = ( f M2 / f L2 ) φ ,
f M2
f L2
4.15   cm
30   cm × 30   cm
200 × 200
26.1   cm
24.6   cm
ϕ ( x x i )
b i / y
B i , 3 ( x )
[ B i , 3 ( x ) ( B i , 2 ( y ) - B i + 1 , 2 ( y ) ) cos θ - B i , 3 ( y ) ( B i , 2 ( x ) - B i + 1 , 2 ( x ) ) sin θ ] / d
exp [ c 1 ( x x i 2 d ) ]
2 c 1   exp [ c 1 ( x x i 2 d ) ] exp [ c 1 ( y y i 2 d ) ] [ ( x x i 2 d ) sin   θ ( y y i 2 d ) cos   θ ]
0.5 { 1 + cos [ c 2 ( x x i 2 d ) ] }
c 2 4 { 1 + cos [ c 2 ( y y i 2 d ) ] } sin [ c 2 ( x x i 2 d ) ] sin   θ c 2 4 { 1 + cos [ c 2 ( x x i 2 d ) ] } × sin [ c 2 ( y y i 2 d ) ] sin   θ
c 1 = ln ( 0 .001 ) / ( 2 d ) 2
c 2 = π / ( 2 d )
( 4.15   cm
z = 3   cm
z = 1.5   cm
y = 15   cm

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