Journal of Computer Science and Technology

, Volume 31, Issue 3, pp 547–560 | Cite as

Variance Analysis and Adaptive Sampling for Indirect Light Path Reuse

  • Hao QinEmail author
  • Xin Sun
  • Jun Yan
  • Qi-Ming Hou
  • Zhong Ren
  • Kun Zhou
Regular Paper


In this paper, we study the estimation variance of a set of global illumination algorithms based on indirect light path reuse. These algorithms usually contain two passes — in the first pass, a small number of indirect light samples are generated and evaluated, and they are then reused by a large number of reconstruction samples in the second pass. Our analysis shows that the covariance of the reconstruction samples dominates the estimation variance under high reconstruction rates and increasing the reconstruction rate cannot effectively reduce the covariance. We also find that the covariance represents to what degree the indirect light samples are reused during reconstruction. This analysis motivates us to design a heuristic approximating the covariance as well as an adaptive sampling scheme based on this heuristic to reduce the rendering variance. We validate our analysis and adaptive sampling scheme in the indirect light field reconstruction algorithm and the axis-aligned filtering algorithm for indirect lighting. Experiments are in accordance with our analysis and show that rendering artifacts can be greatly reduced at a similar computational cost.


Monte Carlo method global illumination variance indirect light path reuse 


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  1. [1]
    Lehtinen J, Aila T, Laine S, Durand F. Reconstructing the indirect light field for global illumination. ACM Transactions on Graphics, 2012, 31(4): 51:1–51:10.CrossRefGoogle Scholar
  2. [2]
    Mehta S U, Wang B, Ramamoorthi R, Durand F. Axisaligned filtering for interactive physically-based diffuse indirect lighting. ACM Transactions on Graphics, 2013, 32(4): Article No. 96.Google Scholar
  3. [3]
    Kajiya J T. The rendering equation. ACM SIGGRAPH Computer Graphics, 1986, 20(4): 143–150.CrossRefGoogle Scholar
  4. [4]
    Lafortune E P, Willems Y D. Bi-directional path tracing. In Proc. the 3rd Annual Conference on Computation/Graphics and Visualization Techniques, Aug. 1993, pp.145-153.Google Scholar
  5. [5]
    Veach E, Guibas L. Bidirectional estimators for light transport. In Proc. EGRW, Jun. 1994, pp.147-162.Google Scholar
  6. [6]
    Dutré P. Mathematical frameworks and Monte Carlo algorithms for global illumination in computer graphics [Ph.D. Thesis]. Department of Computer Science, Katholieke Universiteit Leuven, 1996.Google Scholar
  7. [7]
    Ashikhmin M, Premože S, Shirley P, Smits B. A variance analysis of the Metropolis Light Transport algorithm. Computers & Graphics, 2001, 25(2): 287–294.CrossRefGoogle Scholar
  8. [8]
    Veach E, Guibas L J. Metropolis light transport. In Proc. the 24th Annual Conference on Computer Graphics and Interactive Techniques, Aug. 1997, pp.65-76.Google Scholar
  9. [9]
    Kelemen C, Szirmay-Kalos L, Antal G, Csonka F. A simple and robust mutation strategy for the metropolis light transport algorithm. Computer Graphics Forum, 2002, 21(3): 531–540.CrossRefGoogle Scholar
  10. [10]
    Bekaert P, Sbert M, Halton J. Accelerating path tracing by re-using paths. In Proc. the 13th Eurographics Workshop on Rendering, Jun. 2002, pp.125-134.Google Scholar
  11. [11]
    M´endez-Feliu A, Sbert M, Szirmay-Kalos L. Reusing frames in camera animation. J. WSCG, 2006, 13(1/2/3): 97–104.Google Scholar
  12. [12]
    Keller A. Instant radiosity. In Proc. the 24th Annual Conference on Computer Graphics and Interactive Techniques, Aug. 1997, pp.49-56.Google Scholar
  13. [13]
    Walter B, Fernandez S, Arbree A, Bala K, Donikian M, Greenberg D P. Lightcuts: A scalable approach to illumination. ACM Trans. Graphics, 2005, 24(3): 1098–1107.CrossRefGoogle Scholar
  14. [14]
    Jensen H W. Realistic Image Synthesis Using Photon Mapping. A. K. Peters, Ltd., 2009.Google Scholar
  15. [15]
    Knaus C, Zwicker M. Progressive photon mapping: A probabilistic approach. ACM Transactions on Graphics, 2011, 30(3): 25:1–25:13.CrossRefGoogle Scholar
  16. [16]
    Hachisuka T, Ogaki S, Jensen H W. Progressive photon mapping. ACM Transactions on Graphics, 2008, 27(5): 130:1–130:8.CrossRefGoogle Scholar
  17. [17]
    Georgiev I, Křivánek J, Davidovič T, Slusallek P. Light transport simulation with vertex connection and merging. ACM Transactions on Graphics, 2012, 31(6): 192:1–192:10.CrossRefGoogle Scholar
  18. [18]
    Hachisuka T, Pantaleoni J, Jensen H W. A path space extension for robust light transport simulation. ACM Transactions on Graphics, 2012, 31(6): 191:1–191:10.CrossRefGoogle Scholar
  19. [19]
    Mitchell D P. Generating antialiased images at low sampling densities. ACM SIGGRAPH Computer Graphics, 1987, 21(4): 65–72.CrossRefGoogle Scholar
  20. [20]
    Mitchell D P. Spectrally optimal sampling for distribution ray tracing. ACM SIGGRAPH Computer Graphics, 1991, 25(4): 157–164.CrossRefGoogle Scholar
  21. [21]
    Rigau J, Feixas M, Sbert M. Refinement criteria based on f-divergences. In Proc. the 14th Eurographics Workshop on Rendering, Jun. 2003, pp.260-269.Google Scholar
  22. [22]
    Hachisuka T, Jarosz W, Weistroffer R P, Dale K, Humphreys G, Zwicker M, Jensen H W. Multidimensional adaptive sampling and reconstruction for ray tracing. ACM Transactions on Graphics, 2008, 27(3): 33:1–33:10.CrossRefGoogle Scholar
  23. [23]
    Lee M E, Redner R A, Uselton S P. Statistically optimized sampling for distributed ray tracing. ACM SIGGRAPH Computer Graphics, 1985, 19(3): 61–68.CrossRefGoogle Scholar
  24. [24]
    Xu Q, Sbert M, Feixas M, Scopigno R. A new refinement criterion for adaptive sampling in path tracing. In Proc. the Int. Symp. Industrial Electronics (ISIE), 2010, pp.1556-1561.Google Scholar
  25. [25]
    Overbeck R S, Donner C, Ramamoorthi R. Adaptive wavelet rendering. ACM Transactions on Graphics, 2009, 28(5): 140:1–140:12.CrossRefGoogle Scholar
  26. [26]
    Durand F, Holzschuch N, Soler C, Chan E, Sillion F X. A frequency analysis of light transport. ACM Transactions on Graphics, 2005, 24(3): 1115–1126.CrossRefGoogle Scholar
  27. [27]
    Egan K, Tseng Y T, Holzschuch N et al. Frequency analysis and sheared reconstruction for rendering motion blur. ACM Trans. Graphics, 2009, 28(3): 93:1–93:13.CrossRefGoogle Scholar
  28. [28]
    Soler C, Subr K, Durand F, Holzschuch N, Sillion F. Fourier depth of field. ACM Transactions on Graphics, 2009, 28(2): 18:1–18:12.CrossRefGoogle Scholar
  29. [29]
    Egan K, Hecht F, Durand F, Ramamoorthi R. Frequency analysis and sheared filtering for shadow light fields of complex occluders. ACM Transactions on Graphics, 2011, 30(2): 9:1–9:13.CrossRefGoogle Scholar
  30. [30]
    Mehta S U, Wang B, Ramamoorthi R. Axis-aligned filtering for interactive sampled soft shadows. ACM Transactions on Graphics, 2012, 31(6): 163:1–163:10.CrossRefGoogle Scholar
  31. [31]
    Belcour L, Soler C, Subr K, Holzschuch N, Durand F. 5D covariance tracing for efficient defocus and motion blur. ACM Transactions on Graphics, 2013, 32(3): 31:1–31:18.CrossRefzbMATHGoogle Scholar
  32. [32]
    Fredo D. A frequency analysis of Monte-Carlo and other numerical integration schemes. Technical Report, MIT-CSAIL-TR-2011-052, 2011., Feb. 2016.
  33. [33]
    Ramamoorthi R, Anderson J, Meyer M, Nowrouzezahrai D. A theory of Monte Carlo visibility sampling. ACM Transactions on Graphics, 2012, 31(5): 121:1–121:16.CrossRefGoogle Scholar
  34. [34]
    Donoho D L, Johnstone I M. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 1994, 81(3): 425–455.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Xu R, Pattanaik S N. Non-iterative, robust Monte Carlo noise reduction. IEEE CGA, 2005, 25(2): 31–35.Google Scholar
  36. [36]
    Rousselle F, Knaus C, Zwicker M. Adaptive rendering with non-local means filtering. ACM Transactions on Graphics, 2012, 31(6): 195:1–195:11.CrossRefGoogle Scholar
  37. [37]
    Dammertz H, Sewtz D, Hanika J, Lensch H P A. Edgeavoiding Á-Trous wavelet transform for fast global illumination filtering. In Proc. the Conference on High Performance Graphics, Jun. 2010, pp.67-75.Google Scholar
  38. [38]
    Shirley P, Aila T, Cohen J, Enderton E, Laine S, Luebke D, McGuire M. A local image reconstruction algorithm for stochastic rendering. In Proc. the Symposium on Interactive 3D Graphics and Games, Feb. 2011, pp.9-14.Google Scholar
  39. [39]
    Sen P, Darabi S. On filtering the noise from the random parameters in Monte Carlo rendering. ACM Transactions on Graphics, 2012, 31(3): 18:1–18:15.CrossRefGoogle Scholar
  40. [40]
    Li T M, Wu Y T, Chuang Y Y. SURE-based optimization for adaptive sampling and reconstruction. ACM Transactions on Graphics, 2012, 31(6): 194:1–194:9.Google Scholar
  41. [41]
    Xu Q, Sbert M. A new way to re-using paths. In Lecture Notes in Computer Science 4706, Gervasi O, Gavrilova M (eds.), Springer Berlin Heidelberg, 2007, pp.741-750.Google Scholar
  42. [42]
    Sbert M. Error and complexity of random walkMonte Carlo radiosity. IEEE Transactions on Visualization and Computer Graphics, 1997, 3(1): 23–38.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Hao Qin
    • 1
    Email author
  • Xin Sun
    • 2
  • Jun Yan
    • 1
  • Qi-Ming Hou
    • 1
  • Zhong Ren
    • 1
  • Kun Zhou
    • 1
  1. 1.State Key Laboratory of Computer Aided Design and Computer GraphicsZhejiang UniversityHangzhouChina
  2. 2.Microsoft Research AsiaBeijingChina

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