“Refractive Radiative Transfer Equation” by Ament, Bergmann and Weiskopf

  • ©Marco Ament, Christoph Bergmann, and Daniel Weiskopf




    Refractive Radiative Transfer Equation

Session/Category Title: Sound & Light




    We introduce a refractive radiative transfer equation to the graphics community for the physically based rendering of participating media that have a spatially varying index of refraction. We review principles of geometric nonlinear optics that are crucial to discuss a more generic light transport equation. In particular, we present an optical model that has an integral form suitable for rendering. We show rigorously that the continuous bending of light rays leads to a nonlinear scaling of radiance. To obtain physically correct results, we build on the concept of basic radiance—known from discontinuous refraction—to conserve energy in such complex media. Furthermore, the generic model accounts for the reduction in the speed of light due to the index of refraction to render transient effects like the propagation of light echoes. We solve the refractive volume rendering equation by extending photon mapping with transient light transport in a refractive, participating medium. We demonstrate the impact of our approach on the correctness of rendered images of media that are dominated by spatially continuous refraction and multiple scattering. Furthermore, our model enables us to render visual effects like the propagation of light echoes or time-of-flight imagery that cannot be produced with previous approaches.


    1. D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori. 1997. Bouncing light beams and the hamiltonian analogy. Euro. J. Phys. 18, 4, 284–289.
    2. R. Aronson. 1997. Radiative transfer implies a modified reciprocity relation. J. Optic. Soc. Amer. 14, 2, 486–490.
    3. J. Arvo. 1993. Transfer equations in global illumination. InACM SIGGRAPH Course Notes on Global Illumination.
    4. J. Arvo and D. Kirk. 1990. Particle transport and image synthesis. ACM Trans. Comput. Graph. 24, 4, 63–66.
    5. K. E. Atkinson. 1989. An Introduction to Numerical Analysis. Wiley.
    6. M. Berger, T. Trout, and N. Levit. 1990. Ray tracing mirages. IEEE Comput. Graph. Appl. 10, 3, 36–41.
    7. J. F. Blinn. 1982. Light reflection functions for simulation of clouds and dusty surfaces. ACM Trans. Comput. Graph. 16, 3, 21–29.
    8. J. Blondin. 2007. Supernova modelling. http://vis.cs.ucdavis.edu/VisFiles/pages/supernova.php.
    9. H. E. Bond, A. Henden, Z. G. Levay, N. Panagia, W. B. Sparks, S. Starrfield, R. M. Wagner, R. L. M. Corradi, and U. Munari. 2003. An energetic stellar outburst accompanied by circumstellar light echoes. Nature 422, 6930, 405–408.
    10. M. Born and E. Wolf. 1999. Principles of Optics. Cambridge University Press.
    11. M. Cammarano and H. W. Jensen. 2002. Time dependent photon mapping. In Proceedings of the Eurographics Workshop on Rendering. 135–144.
    12. C. Cao, Z. Ren, B Guo, and K. Zhou. 2010. Interactive rendering of non-constant, refractive media using the ray equations of gradient-index optics. Comput. Graph. Forum 29, 4, 1375–1382.
    13. E. Cerezo, F. Pérez, X. Pueyo, F. J. Seron, and F. X. Sillion. 2005. A survey on participating media rendering techniques. Vis. Comput. 21, 5, 303–328.
    14. S. Chandrasekhar. 1960. Radiative Transfer. Dover Publications.
    15. R. L. Cook, T. Porter, and L. Carpenter. 1984. Distributed ray tracing. ACM Trans. Comput. Graph. 18, 3, 137–145.
    16. S. T. Davis and C. Wyman. 2007. Interactive refractions with total internal reflection. In Proceedings of the Graphics Interface Conference. 185–190.
    17. C. De Rousiers, A. Bousseau, K. Subr, N. Holzschuch, and R. Ramamoorthi. 2011. Real-time rough refraction. In Proceedings of the Symposium on Interactive 3D Graphics and Games. 111–118.
    18. A. Ghatak. 2008. Optics. Tata Mcgraw Hill.
    19. H. Goldstein, C. P. Poole, and J. L. Safko. 2001. Classical Mechanics 3rd Ed. Addison Wesley.
    20. M. E. Gröller. 1995. Non-linear raytracing – Visualizing strange worlds. Vis. Comput. 11, 5, 263–274.
    21. D. Gutierrez, A. Munoz, O. Anson, and F. J. Seron. 2005. Nonlinear volume photon mapping. In Proceedings of the Eurographics Symposium on Rendering Techniques. 291–300.
    22. D. Gutierrez, F. J. Seron, A. Munoz, and O. Anson. 2006. Simulation of atmospheric phenomena. ACM Trans. Comput. Graph. 30, 6, 994–1010.
    23. D. Gutierrez, F. J. Seron, A. Munoz, and O. Anson. 2008. Visualizing underwater ocean optics. Comput. Graph. Forum 27, 2, 547–556.
    24. T. Hachisuka and H. W. Jensen. 2009. Stochastic progressive photon mapping. ACM Trans. Graph. 28, 5, 141:1–141:8.
    25. T. Hachisuka, S. Ogaki, and H. W. Jensen. 2008. Progressive photon mapping. ACM Trans. Graph. 27, 5, 130:1–130:8.
    26. P. Hanrahan and W. Krueger. 1993. Reflection from layered surfaces due to subsurface scattering. In Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’93). 165–174.
    27. E. G. Harris. 1965. Radiative transfer in dispersive media. Phys. Rev. 138, B479–B485.
    28. P. S. Heckbert and P. Hanrahan. 1984. Beam tracing polygonal objects. ACM Trans. Comput. Graph. 18, 3, 119–127.
    29. F. Heide, M. B. Hullin, J. Gregson, and W. Heidrich. 2013. Low-budget transient imaging using photonic mixer devices. ACM Trans. Graph. 32, 4, 45:1–45:10.
    30. P.-K. Hsiung, R. H. Thibadeau, and M. Wu. 1990. T-buffer: Fast visualization of relativistic effects in space-time. ACM Trans. Comput. Graph. 24, 2, 83–88.
    31. I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel. 2007. Eikonal rendering: Efficient light transport in refractive objects. ACM Trans. Graph. 26, 3.
    32. W. Jakob. 2010. Mitsuba renderer. http://www.mitsuba-renderer.org.
    33. A. Jarabo. 2012. Femto-photography: Visualizing light in motion. M.S. thesis, Universidad Zaragoza.
    34. W. Jarosz, D. Nowrouzezahrai, I. Sadeghi, and H. W. Jensen. 2011. A comprehensive theory of volumetric radiance estimation using photon points and beams. ACM Trans. Graph. 30, 1, 5:1–5:19.
    35. W. Jarosz, M. Zwicker, and H. W. Jensen. 2008. The beam radiance estimate for volumetric photon mapping. Comput. Graph. Forum 27, 2, 557–566.
    36. H. W. Jensen. 2005. Realistic Image Synthesis Using Photon Mapping 2nd Ed. AK Peters.
    37. H. W. Jensen and P. H. Christensen. 1998. Efficient simulation of light transport in scences with participating media using photon maps. In Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’98). 311–320.
    38. D. Jönsson, E. Sundén, A. Ynnerman, and T. Ropinski. 2012. Interactive volume rendering with volumetric illumination. In Proceedings of the Eurographics — State of the Art Reports Conference. 53–74.
    39. J. T. Kajiya. 1986. The rendering equation. ACM Trans. Comput. Graph. 20, 4, 143–150.
    40. J. T. Kajiya and B. P. von Herzen. 1984. Ray tracing volume densities. ACM Trans. Comput. Graph. 18, 3, 165–174.
    41. T. Khan and H. Jiang. 2003. A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices. J. Optics A: Pure Appl. Optics 5, 2, 137–141.
    42. A. Kirmani, T. Hutchison, J. Davis, and R. Raskar. 2011. Looking around the corner using ultrafast transient imaging. Int. J. Comput. Vis. 95, 1, 13–28.
    43. O. Krause, M. Tanaka, T. Usuda, T. Hattori, M. Goto, S. Birkmann, and K. Nomoto. 2008. Tycho Brahe’s 1572 supernova as a standard type ia explosion revealed from its light echo spectrum. Nature 456, 7222, 617–619.
    44. E. P. Lafortune and Y. D. Willems. 1996. Rendering participating media with bidirectional path tracing. In Proceedings of the Eurographics Workshop on Rendering Techniques. 91–100.
    45. P. Latorre, F. J. Seron, and D. Gutierrez. 2012. Birefringence: Calculation of refracted ray paths in biaxial crystals. Vis. Comput. 28, 4, 341–356.
    46. M. Lax and D. F. Nelson. 1975. Radiance theorem and optical invariants in anisotropic media. J. Optical Soc. Amer. A 65, 6, 668–675.
    47. D. R. Lide. 2007. CRC Handbook of Chemistry and Physics. CRC Press.
    48. S. Liebes. 1969. Brightness-on the ray invariance of b/n2. Amer. J. Phys. 37, 932–934.
    49. J. Liouville. 1838. Note sur la theorie de la variation des constants arbitraires. J. de Mathématiques Pures et Appliquées 3, 1, 342–349.
    50. M. Magnor, K. Hildebrand, A. Lintu, and A. Hanson. 2005. Reflection nebula visualization. In Proceedings of the IEEE Visualization Conference 255–262.
    51. L. Martí-López, J. Bouza-Domínguez, J. C. Hebden, S. R. Arridge, and R. A. Martínez-Celorio. 2003. Validity conditions for the radiative transfer equation. J. Optical Soc. Amer. A 20, 11, 2046–2056.
    52. L. Martí- López, J. Bouza-Domínguez, R. A. Martínez-Celorio, and J. C. Hebden. 2006. An investigation of the ability of modified radiative transfer equations to accommodate laws of geometrical optics. Optics Comm. 266, 1, 44–49.
    53. T. Müller, S. Grottel, and D. Weiskopf. 2010. Special relativistic visualization by local ray tracing. IEEE Trans. Vis. Comput. Graph. 16, 6, 1243–1250.
    54. F. K. Musgrave. 1990. A note on ray tracing mirages. IEEE Comput. Graph. Appl. 10, 6, 10–12.
    55. N. Naik, S. Zhao, A. Velten, R. Raskar, and K. Bala. 2011. Single view reflectance capture using multiplexed scattering and time-offlight imaging. ACM Trans. Graph. 30, 6, 171:1–171:10.
    56. S. G. Narasimhan, M. Gupta, C. Donner, R. Ramamoorthi, S. K. Nayar, and H. W. Jensen. 2006. Acquiring scattering properties of participating media by dilution. ACM Trans. Graph. 25, 3, 1003–1012.
    57. F. E. Nicodemus. 1963. Radiance. Amer. J. Phys. 31, 5, 368–377.
    58. F. E. Nicodemus. 1976. Self-study manual on optical radiation measurements: Part I–Concepts, chapters 1 to 3. NBS Tech. not. 910-1, NBS (The US National Bureau of Standards).
    59. M. M. Oliveira and M. Brauwers. 2007. Real-time refraction through deformable objects. In Proceedings of the Symposium on Interactive 3D Graphics and Games. 89–96.
    60. R. Pandharkar, A. Velten, A. Bardagjy, E. Lawson, M. Bawendi, and R. Raskar. 2011. Estimating motion and size of moving non-line-of-sight objects in cluttered environments. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 265–272.
    61. F. Patat. 2005. Reflections on reflexions – I. Light echoes in type Ia supernovae. Monthly Not. Royal Astron. Soc. 357, 4, 1167–1177.
    62. M. Pauly, T. Kollig, and A. Keller. 2000. Metropolis light transport for participating media. In Proceedings of the Eurographics Workshop on Rendering Techniques. 11–22.
    63. M. Pharr and G. Humphreys. 2004. Physically Based Rendering. Elsevier.
    64. G. C. Pomraning. 2005. The Equations of Radiation Hydrodynamics. Dover Publications.
    65. R. W. Preisendorfer. 1957a. A mathematical foundation for radiative transfer theory. J. Math. Mech. 6, 4, 685–730.
    66. R. W. Preisendorfer. 1957b. Radiative transfer axioms. Tech. rep. 57-44, Scripps Institution of Oceanography, University of California, San Diego.
    67. R. W. Preisendorfer. 1965. Radiative Transfer on Discrete Spaces. Pergamon Press.
    68. M. Premaratne, E. Premaratne, and A. Lowery. 2005. The photon transport equation for turbid biological media with spatially varying isotropic refractive index. Optics Express 13, 2, 389–399.
    69. A. Rest, R. J. Foley, B. Sinnott, D. L. Welch, C. Badenes, A. V. Filippenko, M. Bergmann, W. A. Bhatti, S. Blondin, P. Challis, G. Damke, H. Finley, M. E. Huber, D. Kasen, R. P. Kirshner, T. Matheson, P. Mazzali, D. Minniti, R. Nakajima, G. Narayan, K. Olsen, D. Sauer, R. C. Smith, and N. B. Suntzeff. 2011. Direct confirmation of the asymmetry of the CAS A supernova with light echoes. Astrophys. J. 732, 3, 1–13.
    70. H. E. Rushmeier. 1988. Realistic image synthesis for scenes with radiatively participating media. Ph.D. thesis, Cornell University.
    71. I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen. 2012. Physically-based simulation of rainbows. ACM Trans. Graph. 31, 1, 3:1–3:12.
    72. A. Smith, J. Skorupski, and J. Davis. 2008. Transient rendering. Tech. rep. UCSC-SOE-08-26, School of Engineering, University of California, Santa Cruz.
    73. J. Stam and E. Languénou. 1996. Ray tracing in non-constant media. In Proceedings of the Eurographics Workshop on Rendering Techniques. 225–234.
    74. Stanford. 2013. The Stanford 3d scanning repository. http://graphics.stanford.edu/data/3Dscanrep/.
    75. W. M. Strouse. 1972. Bouncing light beam. Amer. J. Phys. 40, 6, 913–914.
    76. X. Sun, K. Zhou, S. Lin, and B. Guo. 2010. Line space gathering for single scattering in large scenes. ACM Trans. Graph. 29, 4, 54:1–54:8.
    77. X. Sun, K. Zhou, E. Stollnitz, J. Shi, and B. Guo. 2008. Interactive relighting of dynamic refractive objects. ACM Trans. Graph. 27, 3, 35:1–35:9.
    78. L. Szirmay-Kalos, B. Tóth, and M. Magdics. 2011. Free path sampling in high resolution inhomogeneous participating media. Comput. Graph. Forum 30, 1, 85–97.
    79. R. C. Tolman. 2010. The Principles of Statistical Mechanics. Dover Publications.
    80. J.-M. Tualle and E. Tinet. 2003. Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index. Optics Comm. 228, 13, 33–38.
    81. E. Veach. 1996. Non-symmetric scattering in light transport algorithms. In Proceedings of the Eurographics Workshop on Rendering Techniques. 81–90.
    82. A. Velten, E. Lawson, A. Bardagjy, M. Bawendi, and R. Raskar. 2011a. Slow art with a trillion frames per second camera. ACM SIGGRAPH 2011 Posters. 13:1–13:1.
    83. A. Velten, R. Raskar, and M. Bawendi. 2011b. Picosecond camera for time-of-flight imaging. Conference Paper Imaging Systems and Applications. http://www.opticsinfobase.org/abstract.cfm?uri= ISA-2011-IMB4.
    84. A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar. 2012. Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging. Nature Comm. 3, 745.
    85. A. Velten, D. Wu, A. Jarabo, B. Masia, C. Barsi, C. Joshi, E. Lawson, M. Bawendi, D. Gutierrez, and R. Raskar. 2013. Femto-photography: Capturing and visualizing the propagation of light. ACM Trans. Graph. 32, 4, 44:1–44:8.
    86. B. Walter, S. R. Marschner, H. Li, and K. E. Torrance. 2007. Microfacet models for refraction through rough surfaces. In Proceedings of the Eurographics Conference on Rendering Techniques. 195–206.
    87. B. Walter, S. Zhao, N. Holzschuch, and K. Bala. 2009. Single scattering in refractive media with triangle mesh boundaries. ACM Trans. Graph. 28, 3, 92:1–92:8.
    88. M. Watt. 1990. Light-water interaction using backward beam tracing. ACM SIGGRAPH Comput. Graph. 24, 4, 377–385.
    89. A. Weidlich and A. Wilkie. 2008. Realistic rendering of birefringency in uniaxial crystals. ACM Trans. Graph. 27, 1, 6:1–6:12.
    90. A. Weidlich and A. Wilkie. 2009a. Anomalous dispersion in predictive rendering. Comput. Graph. Forum 28, 4, 1065–1072.
    91. A. Weidlich and A. Wilkie. 2009b. Rendering the effect of labradoescence. In Proceedings of the Graphics Interface Conference. 79–85.
    92. D. Weiskopf, U. Kraus, and H. Ruder. 1999. Searchlight and Doppler effects in the visualization of special relativity: A corrected derivation of the transformation of radiance. ACM Trans. Graph. 18, 3, 278–292.
    93. D. Weiskopf, T. Schafhitzel, and T. Ertl. 2004. GPU-based nonlinear ray tracing. Comput. Graph. Forum 23, 3, 625–633.
    94. A. Wilkie, A. Weidlich, M. Magnor, and A. Chalmers. 2009. Predictive rendering. In ACM SIGGRAPH ASIA Courses. 12:1–12:428.
    95. D. Wu, M. O’Toole, A. Velten, A. Agrawal, and R. Raskar. 2012a. Decomposing global light transport using time of flight imaging. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 366–373.
    96. D. Wu, G. Wetzstein, C. Barsi, T. Willwacher, M. O’Toole, N. Naik, Q. Dai, K. Kutulakos, and R. Raskar. 2012b. Frequency analysis of transient light transport with applications in bare sensor imaging. In Proceedings of the European Conference on Computer Vision. 542–555.
    97. C. Wyman. 2005. An approximate image-space approach for interactive refraction. ACM Trans. Graph. 24, 3, 1050–1053.
    98. C. Wyman and S. Davis. 2006. Interactive image-space techniques for approximating caustics. In Proceedings of the Symposium on Interactive 3D Graphics and Games. 153–160.
    99. Y. Zhao, Y. Han, Z. Fan, F. Qiu, Y.-C. Kuo, A. E. Kaufman, and K. Mueller. 2007. Visual simulation of heat shimmering and mirage. IEEE Trans. Vis. Comput. Graph. 13, 1, 179–189.

ACM Digital Library Publication:

Overview Page: