Abstract

In this paper we present a technique to obtain a diffusion profile of layered turbid media in real time by using the quasi fast Hankel transform (QFHT) and the latest graphics processing unit technique. We apply the QFHT to convolve the diffusion profiles of each layer so as to dramatically reduce the time for the convolution step while maintaining the accuracy. In addition, we also introduce an accelerated technique to generate individual discrete diffusion profiles for each layer through parallel processing. The proposed method is 2 orders of magnitude faster than the existing method, and we validate its efficiency by comparing it with Monte Carlo simulation and another relevant methods.

© 2011 Optical Society of America

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  1. M. Yamaguchi, M. Mitsui, Y. Murakami, H. Fukuda, N. Ohyama, and Y. Kubota, “Multispectral color imaging for dermatology: application in inflammatory and immunologic diseases,” in Proceedings of The Society for Imaging Science and Technology Color Imaging Conference, (The Society for Imaging Science and Technology, 2005), pp. 52–57.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  10. A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnieres, and H. Van Den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37, 779–791 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. C. Donner and H. W. Jensen, “Rapid simulation of steady-state spatially resolved reflectance and transmittance profiles of multilayered turbid materials,” J. Opt. Soc. Am. A 23, 1382–1390(2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  25. C. Donner and H. W. Jensen, “A spectral BSSRDF for shading human skin,” in Proceedings of Eurographics Workshop on Rendering, (Eurographics Association, 2006), pp. 409–418.
  26. J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
    [CrossRef]

2010

A. Liemert and A. Kienle, “Light diffusion in n-layered turbid media: steady-state domain,” J. Biomed. Opt. 15, 025003 (2010).
[CrossRef] [PubMed]

A. Liemert and A. Kienle, “Light diffusion in n-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15, 025002 (2010).
[CrossRef] [PubMed]

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

2009

2008

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed monte carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

2006

2004

2002

J. Y. Hardeberg, F. J. M. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

1998

1997

W. M. Star, “Light dosimetry in vivo,” Phys. Med. Biol. 42, 763–787 (1997).
[CrossRef] [PubMed]

1995

L. Wang, S. Jacques, and L. Zheng, “Mcml—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1992

T. J. Farrell and M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

V. Magni, G. Cerullo, and S. D. Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,” J. Opt. Soc. Am. A 9, 2031–2033 (1992).
[CrossRef]

1984

1980

S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606(1980).
[CrossRef]

1977

1954

1948

Alerstam, E.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed monte carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Alvarez, X.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Andersson-Engels, S.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed monte carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Avrillier, S.

Barakat, R.

Barbosa, N.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Bays, R.

Birkinshaw, M.

M. Birkinshaw, “Radially-symmetric Fourier transforms,” in Astronomical Data Analysis Software and Systems III, Astronomical Society of the Pacific Conference Series (Astronomical Society of the Pacific, 1994), vol.  61, pp. 249–252.

Boas, D. A.

Brettel, H.

J. Y. Hardeberg, F. J. M. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Cerullo, G.

Chen, M.

Chen, W.

Dögnitz, N.

Donner, C.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

C. Donner and H. W. Jensen, “Rapid simulation of steady-state spatially resolved reflectance and transmittance profiles of multilayered turbid materials,” J. Opt. Soc. Am. A 23, 1382–1390(2006).
[CrossRef]

C. Donner and H. W. Jensen, “A spectral BSSRDF for shading human skin,” in Proceedings of Eurographics Workshop on Rendering, (Eurographics Association, 2006), pp. 409–418.

Ettori, D.

Fang, Q.

Farrell, T. J.

T. J. Farrell and M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Fukuda, H.

M. Yamaguchi, M. Mitsui, Y. Murakami, H. Fukuda, N. Ohyama, and Y. Kubota, “Multispectral color imaging for dermatology: application in inflammatory and immunologic diseases,” in Proceedings of The Society for Imaging Science and Technology Color Imaging Conference, (The Society for Imaging Science and Technology, 2005), pp. 52–57.

Gouesbet, G.

Gutierrez, D.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Hardeberg, J. Y.

J. Y. Hardeberg, F. J. M. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Huang, M.

Huang, W.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Jacques, S.

L. Wang, S. Jacques, and L. Zheng, “Mcml—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Jensen, H. W.

C. Donner and H. W. Jensen, “Rapid simulation of steady-state spatially resolved reflectance and transmittance profiles of multilayered turbid materials,” J. Opt. Soc. Am. A 23, 1382–1390(2006).
[CrossRef]

C. Donner and H. W. Jensen, “A spectral BSSRDF for shading human skin,” in Proceedings of Eurographics Workshop on Rendering, (Eurographics Association, 2006), pp. 409–418.

Jimenez, J.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Kienle, A.

A. Liemert and A. Kienle, “Light diffusion in n-layered turbid media: steady-state domain,” J. Biomed. Opt. 15, 025003 (2010).
[CrossRef] [PubMed]

A. Liemert and A. Kienle, “Light diffusion in n-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15, 025002 (2010).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnieres, and H. Van Den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37, 779–791 (1998).
[CrossRef]

Kubelka, P.

Kubota, Y.

M. Yamaguchi, M. Mitsui, Y. Murakami, H. Fukuda, N. Ohyama, and Y. Kubota, “Multispectral color imaging for dermatology: application in inflammatory and immunologic diseases,” in Proceedings of The Society for Imaging Science and Technology Color Imaging Conference, (The Society for Imaging Science and Technology, 2005), pp. 52–57.

Letoulouzan, J. N.

Liemert, A.

A. Liemert and A. Kienle, “Light diffusion in n-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15, 025002 (2010).
[CrossRef] [PubMed]

A. Liemert and A. Kienle, “Light diffusion in n-layered turbid media: steady-state domain,” J. Biomed. Opt. 15, 025003 (2010).
[CrossRef] [PubMed]

Magni, V.

Maheu, B.

Matts, P.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Mitsui, M.

M. Yamaguchi, M. Mitsui, Y. Murakami, H. Fukuda, N. Ohyama, and Y. Kubota, “Multispectral color imaging for dermatology: application in inflammatory and immunologic diseases,” in Proceedings of The Society for Imaging Science and Technology Color Imaging Conference, (The Society for Imaging Science and Technology, 2005), pp. 52–57.

Murakami, Y.

M. Yamaguchi, M. Mitsui, Y. Murakami, H. Fukuda, N. Ohyama, and Y. Kubota, “Multispectral color imaging for dermatology: application in inflammatory and immunologic diseases,” in Proceedings of The Society for Imaging Science and Technology Color Imaging Conference, (The Society for Imaging Science and Technology, 2005), pp. 52–57.

Nghiem, H. L.

Ohyama, N.

M. Yamaguchi, M. Mitsui, Y. Murakami, H. Fukuda, N. Ohyama, and Y. Kubota, “Multispectral color imaging for dermatology: application in inflammatory and immunologic diseases,” in Proceedings of The Society for Imaging Science and Technology Color Imaging Conference, (The Society for Imaging Science and Technology, 2005), pp. 52–57.

Orvalho, V.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Parshall, E.

Patterson, M. S.

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnieres, and H. Van Den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37, 779–791 (1998).
[CrossRef]

T. J. Farrell and M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Sablong, R.

Sandler, B. H.

Schmitt, F. J. M.

J. Y. Hardeberg, F. J. M. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Scully, T.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Sheng, S.-C.

S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606(1980).
[CrossRef]

Siegman, A. E.

S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606(1980).
[CrossRef]

A. E. Siegman, “Quasi fast hankel transform,” Opt. Lett. 1, 13–15 (1977).
[CrossRef] [PubMed]

Silvestri, S. D.

Star, W. M.

W. M. Star, “Light dosimetry in vivo,” Phys. Med. Biol. 42, 763–787 (1997).
[CrossRef] [PubMed]

Svensson, T.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed monte carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Tinet, E.

Tualle, J.-M.

Van Den Bergh, H.

Vieira, T.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Wagnieres, G.

Wang, L.

L. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J. Opt. Soc. Am. A 15, 936–944 (1998).
[CrossRef]

L. Wang, S. Jacques, and L. Zheng, “Mcml—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Wang, S. M.

X. C. Wang and S. M. Wang, “Light transport model in a n-layered mismatched tissue,” Waves Random Complex Media 16, 121–135 (2006).
[CrossRef]

Wang, X. C.

X. C. Wang and S. M. Wang, “Light transport model in a n-layered mismatched tissue,” Waves Random Complex Media 16, 121–135 (2006).
[CrossRef]

Weyrich, T.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Yamaguchi, M.

M. Yamaguchi, M. Mitsui, Y. Murakami, H. Fukuda, N. Ohyama, and Y. Kubota, “Multispectral color imaging for dermatology: application in inflammatory and immunologic diseases,” in Proceedings of The Society for Imaging Science and Technology Color Imaging Conference, (The Society for Imaging Science and Technology, 2005), pp. 52–57.

Yu, L.

Zheng, L.

L. Wang, S. Jacques, and L. Zheng, “Mcml—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zhu, Z.

ACM Trans. Graph.

J. Jimenez, T. Scully, N. Barbosa, C. Donner, X. Alvarez, T. Vieira, P. Matts, V. Orvalho, D. Gutierrez, and T. Weyrich, “A practical appearance model for dynamic facial color,” ACM Trans. Graph. 29, Art.. 18661671, (2010).
[CrossRef]

Appl. Opt.

Comput. Methods Programs Biomed.

L. Wang, S. Jacques, and L. Zheng, “Mcml—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

J. Biomed. Opt.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed monte carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

A. Liemert and A. Kienle, “Light diffusion in n-layered turbid media: steady-state domain,” J. Biomed. Opt. 15, 025003 (2010).
[CrossRef] [PubMed]

A. Liemert and A. Kienle, “Light diffusion in n-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15, 025002 (2010).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Med. Phys.

T. J. Farrell and M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Opt. Eng.

J. Y. Hardeberg, F. J. M. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

W. M. Star, “Light dosimetry in vivo,” Phys. Med. Biol. 42, 763–787 (1997).
[CrossRef] [PubMed]

Phys. Rev. A

S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606(1980).
[CrossRef]

Waves Random Complex Media

X. C. Wang and S. M. Wang, “Light transport model in a n-layered mismatched tissue,” Waves Random Complex Media 16, 121–135 (2006).
[CrossRef]

Other

M. Birkinshaw, “Radially-symmetric Fourier transforms,” in Astronomical Data Analysis Software and Systems III, Astronomical Society of the Pacific Conference Series (Astronomical Society of the Pacific, 1994), vol.  61, pp. 249–252.

M. Yamaguchi, M. Mitsui, Y. Murakami, H. Fukuda, N. Ohyama, and Y. Kubota, “Multispectral color imaging for dermatology: application in inflammatory and immunologic diseases,” in Proceedings of The Society for Imaging Science and Technology Color Imaging Conference, (The Society for Imaging Science and Technology, 2005), pp. 52–57.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

C. Donner and H. W. Jensen, “A spectral BSSRDF for shading human skin,” in Proceedings of Eurographics Workshop on Rendering, (Eurographics Association, 2006), pp. 409–418.

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Figures (11)

Fig. 1
Fig. 1

Configuration of point sources for a thin layer based on the multipole approximation. The sign of the sources indicate whether the source is real (+) or virtual (−).

Fig. 2
Fig. 2

Geometry of a light path in layered turbid media and notations of various reflectance and transmittance. Incident light is denoted by the thick solid arrow, transmittance path by the thin solid arrows, and reflectance path by the dotted arrows.

Fig. 3
Fig. 3

Example of a general diffusion profile generated by the dipole approximation and exponentially distributed sampling points on the profile. The used optical parameters are σ s = 1.0 mm 1 , σ a = 0.01 mm 1 , and η = 1.0 , respectively.

Fig. 4
Fig. 4

Comparison of the time between the sampling process and the convolution process.

Fig. 5
Fig. 5

Diagram for the sampling process using the GPU. The dashed arrow lines indicate for common data flow from the CPU controlled sector to the GPU controlled sector.

Fig. 6
Fig. 6

Comparison of the convolution results with and without the lower-end correction. The used material is the same as Fig. 5b of Donner and Jensen [15]. The optical parameters for the upper layer are σ s = 1.3 mm 1 , σ a = 0.005 mm 1 , η = 1.4 , and d = 10 mm , the below layer σ s = 1.0 mm 1 , σ a = 0.022 mm 1 , η = 1.4 , and d = . The number of the photons for Monte Carlo simulation is 4 million.

Fig. 7
Fig. 7

Comparison of convolution results with and without the lower-end correction for the case of using the small number of samples (1024). The same optical properties in Fig. 6 are used.

Fig. 8
Fig. 8

(a) Comparison of the proposed and existing method of Donner and Jensen [15]. (b) Relative error comparison with the Monte Carlo of the proposed and existing method.

Fig. 9
Fig. 9

Comparison of the results by the proposed method with those by the Monte Carlo simulation for a (a) matched and (b) mismatched refractive index. Both materials are semi-infinite media and have the same optical parameters and thickness of the upper layers except for the refractive indices. The optical parameters for the upper layer are σ s = 1.0 mm 1 , σ a = 0.005 mm 1 , and d = 10 mm , the below layer σ s = 0.5 mm 1 , σ a = 0.005 mm 1 , and d = . The relative refractive indices are 1.0 and 1.0 for the left and 1.2 and 1.5 for the right, respectively.

Fig. 10
Fig. 10

Performance comparison of the CPU and GPU as a function of the number of used samples when generating discrete diffusion profiles from 2-layered media with finite thickness.

Fig. 11
Fig. 11

Comparison of elapsed time by the proposed method with Donner and Jensen [15] when generating a reflectance profile of 2-layered finite media as a function of the number of used samples.

Tables (2)

Tables Icon

Table 1 Notations and Formula for the Multipole Source Approximation

Tables Icon

Table 2 Detailed Comparison of Time for Various Methods a

Equations (12)

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

R ( r ) = α 4 π i = n n [ z r , i ( σ tr + 1 d r , i ) e σ tr d r , i d r , i 2 + z v , i ( σ tr + 1 d v , i ) e σ tr d v , i d v , i 2 ] .
T ( r ) = α 4 π i = n n [ ( d z r , i ) ( σ tr + 1 d r , i ) e σ tr d r , i d r , i 2 ( d z v , i ) ( σ tr + 1 d v , i ) e σ tr d v , i d v , i 2 ] .
R = R u + + T u R b + T u + + T u R b + R u R b + T u + + T u R b + R u R b + R u R b + T u + + ,
T = T u T b + T u R b + R u T b + T u R b + R u R b + R u T b + .
R ( r ) = R u + + T u * R b + * T u + + T u * R b + * R u * R b + * T u + + T u * R b + * R u * R b + * R u * R b + * T u + + .
T ( r ) = T u * T b + T u * R b + * R u * T b + T u * R b + * R u * R b + * R u * T b + .
R ( r ) = iFT [ R u + + T u R b + T u + + T u R b + R u R b + T u + + T u R b + R u R b + R u R b + T u + + ] = iFT [ R u + + T u R b + T u + 1 R b + R u ] ,
T ( r ) = iFT [ T u T b + T u R b + R u T b + T u R b + R u R b + R u T b + ] = iFT [ T u T b 1 R b + R u ] ,
g ( ρ ) = 2 π 0 r f ( r ) J 0 ( 2 π ρ r ) d r ,
g ^ ( y ) = f ^ ( x ) j ^ ( x + y ) d x ,
g m = n = 0 N 1 f n j n + m = FFT [ FFT ( f m ) × iFFT ( j m ) ] ,
r n = r 0 e α n , ρ m = ρ 0 e α m , ( n , m = 0 , 1 , 2 , , N 1 ) ,

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