Abstract

Rayleigh scattering is used frequently in Monte Carlo simulation of multiple scattering. The Rayleigh phase function is quite simple, and one might expect that it should be simple to importance sample it efficiently. However, there seems to be no one good way of sampling it in the literature. This paper provides the details of several different techniques for importance sampling the Rayleigh phase function, and it includes a comparison of their performance as well as hints toward efficient implementation.

© 2011 Optical Society of America

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References

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  8. G. V. G. Baranoski and A. Krishnaswamy, “Light interaction with human skin: from believable images to predictable models,” SIGGRAPH Asia 2008, Course Notes (2008).
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2011

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

2010

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

2008

S. Lallich, F. Enguehard, and D. Baillis, “Radiative properties of silica nanoporous matrices,” Int. J. Thermophys. 29, 1395–1407(2008).
[CrossRef]

2006

M. W. Y. Lam and G. V. G. Baranoski, “A predictive light transport model for the human iris,” Comput. Graph. Forum 25, 359–368 (2006).
[CrossRef]

Q. Liu and F. Weng, “Combined Henyey–Greenstein and Rayleigh phase function,” Appl. Opt. 45, 7475–7479 (2006).
[CrossRef] [PubMed]

2003

G. Marsaglia, “Random number generators,” J. Mod. Appl. Stat. Meth. 2, 2–13 (2003).

2001

2000

S. Seager, B. A. Whitney, and D. D. Sasselov, “Photometric light curves and polarization of close-in extrasolar giant planets,” Astrophys. J. 540, 504–520 (2000).
[CrossRef]

1999

J. F. Hughes and T. Möller, “Building an orthonormal basis from a unit vector,” J. Graphics Tools 4, 33–35 (1999).

1998

M. Matsumoto and T. Nishimaru, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[CrossRef]

1992

B. W. Whitney and L. Hartmann, “Model scattering envelopes of young stellar objects. I. Method and application to circumstellar disks,” Astrophys. J. 395, 529–539 (1992).
[CrossRef]

W. M. Cornette and J. G. Shanks, “Physically reasonable analytical expression for the single-scattering phase function,” Appl. Opt. 31, 3152–3160 (1992).
[CrossRef] [PubMed]

1969

1942

O. E. Lancaster, “Machine method for the extraction of cube root,” J. Am. Stat. Assoc. 37, 112–115 (1942).
[CrossRef]

1940

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Ann. Astrophys. 3, 117–137 (1940). Also in Astrophys. J. 93, 70–83 (1941).

1871

J. W. Strutt, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871). Reprinted in John William Strutt (Baron Rayleigh), Scientific Papers (Cambridge University, 1899), Vol. 1, No. 8, pp. 87–103.

Baillis, D.

S. Lallich, F. Enguehard, and D. Baillis, “Radiative properties of silica nanoporous matrices,” Int. J. Thermophys. 29, 1395–1407(2008).
[CrossRef]

Baranoski, G. V. G.

M. W. Y. Lam and G. V. G. Baranoski, “A predictive light transport model for the human iris,” Comput. Graph. Forum 25, 359–368 (2006).
[CrossRef]

G. V. G. Baranoski and A. Krishnaswamy, Light & Skin Interactions: Simulations for Computer Graphics Applications (Morgan Kaufmann/Elsevier, 2010).

A. Krishnaswamy and G. V. G. Baranoski, “A study on skin optics,” Tech. Rep. CS-2004-01 (University of Waterloo, 2004).

G. V. G. Baranoski and A. Krishnaswamy, “Light interaction with human skin: from believable images to predictable models,” SIGGRAPH Asia 2008, Course Notes (2008).

Beirle, S.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Bigler, J.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford, 1950).

Cornette, W. M.

Deutschmann, T.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Dietrich, A.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Enguehard, F.

S. Lallich, F. Enguehard, and D. Baillis, “Radiative properties of silica nanoporous matrices,” Int. J. Thermophys. 29, 1395–1407(2008).
[CrossRef]

Forster, F. K.

Friedrich, H.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Friess, U.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Ann. Astrophys. 3, 117–137 (1940). Also in Astrophys. J. 93, 70–83 (1941).

Grzegorski, M.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Hartmann, L.

B. W. Whitney and L. Hartmann, “Model scattering envelopes of young stellar objects. I. Method and application to circumstellar disks,” Astrophys. J. 395, 529–539 (1992).
[CrossRef]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Ann. Astrophys. 3, 117–137 (1940). Also in Astrophys. J. 93, 70–83 (1941).

Hibst, R.

Hoberock, J.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Hughes, J. F.

J. F. Hughes and T. Möller, “Building an orthonormal basis from a unit vector,” J. Graphics Tools 4, 33–35 (1999).

Humphreys, G.

M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation (Morgan Kaufmann/Elsevier, 2004).

Kahan, W.

W. Kahan, “Computing a real cube root,” lecture notes (1991), http://www.cims.nyu.edu/~dbindel/class/cs279/qbrt.pdf. Retypeset by D. Bindel, April 2001.

Kalos, M. H.

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods (Wiley, 1986), Vol. 1.
[CrossRef]

Kattawar, G. W.

Kern, C.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Kienle, A.

Knuth, D. E.

D. E. Knuth, The Art of Computer Programming, 2nd ed.(Addison-Wesley, 1998), Vol.  3.

Krishnaswamy, A.

A. Krishnaswamy and G. V. G. Baranoski, “A study on skin optics,” Tech. Rep. CS-2004-01 (University of Waterloo, 2004).

G. V. G. Baranoski and A. Krishnaswamy, “Light interaction with human skin: from believable images to predictable models,” SIGGRAPH Asia 2008, Course Notes (2008).

G. V. G. Baranoski and A. Krishnaswamy, Light & Skin Interactions: Simulations for Computer Graphics Applications (Morgan Kaufmann/Elsevier, 2010).

Kritten, L.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Lallich, S.

S. Lallich, F. Enguehard, and D. Baillis, “Radiative properties of silica nanoporous matrices,” Int. J. Thermophys. 29, 1395–1407(2008).
[CrossRef]

Lam, M. W. Y.

M. W. Y. Lam and G. V. G. Baranoski, “A predictive light transport model for the human iris,” Comput. Graph. Forum 25, 359–368 (2006).
[CrossRef]

Lancaster, O. E.

O. E. Lancaster, “Machine method for the extraction of cube root,” J. Am. Stat. Assoc. 37, 112–115 (1942).
[CrossRef]

Liu, Q.

Luebke, D.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Marsaglia, G.

G. Marsaglia, “Random number generators,” J. Mod. Appl. Stat. Meth. 2, 2–13 (2003).

G. Marsaglia, “Yet another RNG” (1994). Posted to the Usenet group sci.stat.math.

Matsumoto, M.

M. Matsumoto and T. Nishimaru, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[CrossRef]

McAllister, D.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, 1976).

McGuire, M.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Möller, T.

J. F. Hughes and T. Möller, “Building an orthonormal basis from a unit vector,” J. Graphics Tools 4, 33–35 (1999).

Morley, K.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Nishimaru, T.

M. Matsumoto and T. Nishimaru, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[CrossRef]

Parker, S. G.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Pfeilsticker, K.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Pharr, M.

M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation (Morgan Kaufmann/Elsevier, 2004).

Plass, G. N.

Platt, U.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Prados-Román, C.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Pukite, J.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Robison, A.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Sasselov, D. D.

S. Seager, B. A. Whitney, and D. D. Sasselov, “Photometric light curves and polarization of close-in extrasolar giant planets,” Astrophys. J. 540, 504–520 (2000).
[CrossRef]

Seager, S.

S. Seager, B. A. Whitney, and D. D. Sasselov, “Photometric light curves and polarization of close-in extrasolar giant planets,” Astrophys. J. 540, 504–520 (2000).
[CrossRef]

Shanks, J. G.

Stich, M.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

Strutt, J. W.

J. W. Strutt, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871). Reprinted in John William Strutt (Baron Rayleigh), Scientific Papers (Cambridge University, 1899), Vol. 1, No. 8, pp. 87–103.

Tilley, R. J. D.

R. J. D. Tilley, Colour and the Optical Properties of Materials, 2nd ed. (Wiley, 2011).

von Neumann, J.

J. von Neumann, “Various techniques used in connection with random digits,” in Monte Carlo Method, A.S.Householder, ed., Vol. 12 of Applied Mathematics Series (National Bureau of Standards, 1951), Chap. 13, pp. 36–38.

Wagner, T.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Weng, F.

Werner, B.

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Whitlock, P. A.

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods (Wiley, 1986), Vol. 1.
[CrossRef]

Whitney, B. A.

S. Seager, B. A. Whitney, and D. D. Sasselov, “Photometric light curves and polarization of close-in extrasolar giant planets,” Astrophys. J. 540, 504–520 (2000).
[CrossRef]

Whitney, B. W.

B. W. Whitney and L. Hartmann, “Model scattering envelopes of young stellar objects. I. Method and application to circumstellar disks,” Astrophys. J. 395, 529–539 (1992).
[CrossRef]

ACM Trans. Graph.

S. G. Parker, J. Bigler, A. Dietrich, H. Friedrich, J. Hoberock, D. Luebke, D. McAllister, M. McGuire, K. Morley, A. Robison, and M. Stich, “OptiX: a general purpose ray tracing engine,” ACM Trans. Graph. 29, 1–13 (2010).
[CrossRef]

ACM Trans. Model. Comput. Simul.

M. Matsumoto and T. Nishimaru, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[CrossRef]

Ann. Astrophys.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Ann. Astrophys. 3, 117–137 (1940). Also in Astrophys. J. 93, 70–83 (1941).

Appl. Opt.

Astrophys. J.

B. W. Whitney and L. Hartmann, “Model scattering envelopes of young stellar objects. I. Method and application to circumstellar disks,” Astrophys. J. 395, 529–539 (1992).
[CrossRef]

S. Seager, B. A. Whitney, and D. D. Sasselov, “Photometric light curves and polarization of close-in extrasolar giant planets,” Astrophys. J. 540, 504–520 (2000).
[CrossRef]

Comput. Graph. Forum

M. W. Y. Lam and G. V. G. Baranoski, “A predictive light transport model for the human iris,” Comput. Graph. Forum 25, 359–368 (2006).
[CrossRef]

Int. J. Thermophys.

S. Lallich, F. Enguehard, and D. Baillis, “Radiative properties of silica nanoporous matrices,” Int. J. Thermophys. 29, 1395–1407(2008).
[CrossRef]

J. Am. Stat. Assoc.

O. E. Lancaster, “Machine method for the extraction of cube root,” J. Am. Stat. Assoc. 37, 112–115 (1942).
[CrossRef]

J. Graphics Tools

J. F. Hughes and T. Möller, “Building an orthonormal basis from a unit vector,” J. Graphics Tools 4, 33–35 (1999).

J. Mod. Appl. Stat. Meth.

G. Marsaglia, “Random number generators,” J. Mod. Appl. Stat. Meth. 2, 2–13 (2003).

J. Quant. Spectrosc. Radiat. Transfer

T. Deutschmann, S. Beirle, U. Friess, M. Grzegorski, C. Kern, L. Kritten, U. Platt, C. Prados-Román, J. Puķīte, T. Wagner, B. Werner, and K. Pfeilsticker, “The Monte Carlo atmospheric radiative transfer model McArtim: introduction and validation of Jacobians and 3D features,” J. Quant. Spectrosc. Radiat. Transfer 112, 1119–1137 (2011).
[CrossRef]

Opt. Lett.

Philos. Mag.

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[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Spread of direction samples as points on the shape of the Rayleigh phase function. Directions sampled (a) uniformly across the unit sphere and (b) using rejection techniques.

Fig. 2
Fig. 2

Spread of direction samples as points on the shape of the Rayleigh phase function. Directions sampled using Baranoski’s rejection sampling (a) without correction and (b) with correction.

Fig. 3
Fig. 3

Curves illustrating how tight the bound is for the different rejection sampling techniques. The solid curve is the Rayleigh phase function p R = 1 4 π 3 4 ( 1 + cos 2 θ ) , the dashed curve is c pdf = 6 12 ( 1 cos 2 θ ) 1 / 2 , which is used for Baranoski’s rejection sampling, and the dotted curve is c pdf = 3 2 1 4 π , which is used when the unit sphere is sampled uniformly.

Fig. 4
Fig. 4

Spread of direction samples as points on the shape of the Rayleigh phase function. Directions sampled using (a) the simplified rejection technique and (b) the direct sampling technique, which employs Cardan’s formulas.

Tables (1)

Tables Icon

Table 1 Number of Samples in Millions Generated by the Different Importance Sampling Methods in Equal Time ( 1 s on a Laptop Computer)

Equations (36)

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p R ( cos θ ) = 1 4 π 3 4 ( 1 + cos 2 θ ) ,
C s = 128 π 5 3 λ 4 α 2 .
α = n 2 1 n 2 + 2 a 3 .
σ s p R ( ω · ω ) L i d ω ,
σ s = 0 N ( a ) C s ( a ) d a .
σ s = N C s .
d ω = d A s r 2 ,
L s = N 8 π 4 a 6 λ 4 ( n 2 1 n 2 + 2 ) 2 ( 1 + ( ω · ω ) 2 ) L i d ω ,
do   ( x , y , z ) = ( 2 ξ 1 1 , 2 ξ 2 1 , 2 ξ 3 1 ) while   x 2 + y 2 + z 2 > 1 , ω = ( x , y , z ) / ( x 2 + y 2 + z 2 ) 1 / 2 ,
( θ , ϕ ) = ( cos 1 ( 2 ξ 1 1 ) , 2 π ξ 2 ) ,
ξ < f ( x ) c h ( x ) .
max p R = 1 4 π 3 4 ( 1 + 1 2 ) = 3 2 1 4 π ,
do     ω = sample \_ isotropic ( ) while   ξ > 1 2 ( 1 + ( ω · ω ) 2 ) ,
do     cos θ = 2 ξ 1 1 while   ξ 2 > 1 2 ( 1 + cos 2 θ ) , ϕ = 2 π ξ 3 .
ω = ( x , y , z ) = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) ,
ω = x b 1 + y b 2 + z ω .
sin θ = ( 1 cos 2 θ ) 1 / 2 , since     θ [ 0 , π ] .
ccdf ( cos θ ) = 1 π cos 1 ( cos θ ) .
pdf ( cos θ ) = 1 π ( 1 cos 2 θ ) 1 / 2 ,
1 4 π 3 4 ( 1 + cos 2 θ ) = c π ( 1 cos 2 θ ) 1 / 2 sin θ > 0 c = 3 16 sin θ ( 1 + cos 2 θ ) .
do     cos θ = cos ( π ξ 1 ) while   ξ 2 > 9 4 6 ( 1 + cos 2 θ ) ( 1 cos 2 θ ) 1 / 2 , ϕ = 2 π ξ 3
p R ( θ ) = 0 2 π p R ( θ , ϕ ) d ϕ = 0 2 π 1 4 π 3 4 ( 1 + cos 2 θ ) sin θ d ϕ = 3 8 ( 1 + cos 2 θ ) sin θ and p R ( ϕ | θ ) = p R ( θ , ϕ ) p R ( θ ) = 1 2 π .
P R ( ϕ | θ ) = 0 ϕ 1 2 π d ϕ = ϕ 2 π ,
P R ( θ ) = 0 θ 3 8 ( 1 + cos 2 θ ) sin θ d θ = x = cos θ 3 8 1 cos θ ( 1 + x 2 ) d x = 3 8 [ x + 1 3 x 3 ] 1 cos θ = 1 2 3 8 cos θ 1 8 cos 3 θ .
x 3 + a x + b = 0.
a = 3 , b = 4 ( 2 ξ 1 1 ) .
x = [ b 2 + ( b 2 4 + a 3 27 ) 1 / 2 ] 1 / 3 + [ b 2 ( b 2 4 + a 3 27 ) 1 / 2 ] 1 / 3 ,
b 2 4 + a 3 27 = 16 ( 2 ξ 1 1 ) 2 4 + 3 3 27 = 4 ( 2 ξ 1 ) 2 + 1 > 0 ,
cos θ = [ 2 ( 2 ξ 1 1 ) + ( 4 ( 2 ξ 1 1 ) 2 + 1 ) 1 / 2 ] 1 / 3 + [ 2 ( 2 ξ 1 1 ) ( 4 ( 2 ξ 1 1 ) 2 + 1 ) 1 / 2 ] 1 / 3 .
u = [ b 2 ( b 2 4 + a 3 27 ) 1 / 2 ] 1 / 3 ;
x = u a 3 u .
u = [ 2 ( 2 ξ 1 1 ) + ( 4 ( 2 ξ 1 1 ) 2 + 1 ) 1 / 2 ] 1 / 3 , ( θ , ϕ ) = ( cos 1 ( u 1 / u ) , 2 π ξ 2 ) .
cdf ( x i ) = j = 1 i pdf ( x j ) Δ x = pdf ( x i ) Δ x + cdf ( x i 1 ) ,
P R ( cos θ i ) = P R ( x i ) = j = 1 i p R ( x j ) Δ x = 3 8 ( 1 + x i 2 ) Δ x + P R ( x i 1 )
P R ( x i ) = 1 2 3 8 x i 1 8 x i 3 .
cos θ = 1 Δ x ( i P R ( x i ) ξ 1 P R ( x i ) P R ( x i 1 ) ) with P R ( x 0 ) = 0 , ϕ = 2 π ξ 2 ,

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