“Real-time polygonal-light shading with linearly transformed cosines”

  • ©Eric Heitz, Jonathan Dupuy, Stephen Hill, and David Neubelt




    Real-time polygonal-light shading with linearly transformed cosines





    In this paper, we show that applying a linear transformation—represented by a 3 x 3 matrix—to the direction vectors of a spherical distribution yields another spherical distribution, for which we derive a closed-form expression. With this idea, we can use any spherical distribution as a base shape to create a new family of spherical distributions with parametric roughness, elliptic anisotropy and skewness. If the original distribution has an analytic expression, normalization, integration over spherical polygons, and importance sampling, then these properties are inherited by the linearly transformed distributions.By choosing a clamped cosine for the original distribution we obtain a family of distributions, which we call Linearly Transformed Cosines (LTCs), that provide a good approximation to physically based BRDFs and that can be analytically integrated over arbitrary spherical polygons. We show how to use these properties in a realtime polygonal-light shading application. Our technique is robust, fast, accurate and simple to implement.


    1. Arvo, J. 1995. Applications of irradiance tensors to the simulation of non-lambertian phenomena. In Proc. ACM SIGGRAPH, 335–342. Google ScholarDigital Library
    2. Baum, D. R., Rushmeier, H. E., and Winget, J. M. 1989. Improving radiosity solutions through the use of analytically determined form-factors. Computer Graphics (Proc. SIGGRAPH) 23, 3, 325–334. Google ScholarDigital Library
    3. Bingham, C. 1974. An antipodally symmetric distribution on the sphere. The Annals of Statistics 2, 6, 1201–1225.Google ScholarCross Ref
    4. Drobot, M. 2014. Physically based area lights. InGPUPro5. 67–100.Google Scholar
    5. Fisher, R. 1953. Dispersion on a sphere. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 217, 1130, 295–305.Google Scholar
    6. Hill, S., McAuley, S., Burley, B., Chan, D., Fascione, L., Iwanicki, M., Hoffman, N., Jakob, W., Neubelt, D., Pesce, A., and Pettineo, M. 2015. Physically based shading in theory and practice. In ACM SIGGRAPH Courses 2015. Google ScholarDigital Library
    7. Iwasaki, K., Furuya, W., Dobashi, Y., and Nishita, T. 2012. Real-time rendering of dynamic scenes under all-frequency lighting using integral spherical gaussian. Computer Graphics Forum (Proc. of Eurographics) 31, 727–734. Google ScholarDigital Library
    8. Kent, J. T. 1982. The Fisher-Bingham distribution on the sphere. Journal of the Royal Statistical Society. Series B (Methodological) 44, 1, 71–80.Google ScholarCross Ref
    9. Lagarde, S., and de Rousiers, C. 2014. Physically based shading in theory and practice: Moving Frostbite to PBR. In ACM SIGGRAPH Courses 2014. Google ScholarDigital Library
    10. Lambert, J. H. 1760. Photometria, sive de mensura et gradibus luminus, colorum et umbrae.Google Scholar
    11. Lecocq, P., Sourimant, G., and Marvie, J.-E. 2015. Accurate analytic approximations for real-time specular area lighting. In ACM SIGGRAPH 2015 Talks, 68:1–68:1. Google ScholarDigital Library
    12. Phong, B. T. 1975. Illumination for computer generated pictures. Computer Graphics (Proc. SIGGRAPH) 18, 6, 311–317.Google Scholar
    13. Snyder, J. M. 1996. Area light sources for real-time graphics. Tech. Rep. MSR-TR-96-11, Microsoft Research.Google Scholar
    14. Walter, B., Marschner, S. R., Li, H., and Torrance, K. E. 2007. Microfacet models for refraction through rough surfaces. In Proc. Eurographics Symposium on Rendering, 195–206. Google ScholarDigital Library
    15. Wang, L., Lin, Z., Wang, W., and Fu, K. 2008. One-shot approximate local shading. Tech. rep.Google Scholar
    16. Xu, K., Sun, W.-L., Dong, Z., Zhao, D.-Y., Wu, R.-D., and Hu, S.-M. 2013. Anisotropic Spherical Gaussians. ACM Transactions on Graphics (Proc. SIGGRAPH Asia) 32, 6, 209:1–209:11. Google ScholarDigital Library
    17. Xu, K., Cao, Y.-P., Ma, L.-Q., Dong, Z., Wang, R., and Hu, S.-M. 2014. A practical algorithm for rendering interreflections with all-frequency brdfs. ACM Transactions on Graphics 33, 1, 10:1–10:16. Google ScholarDigital Library

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