A frequency analysis of light transport

ABSTRACT

We present a signal-processing framework for light transport. We study the frequency content of radiance and how it is altered by phenomena such as shading, occlusion, and transport. This extends previous work that considered either spatial or angular dimensions, and it offers a comprehensive treatment of both space and angle.We show that occlusion, a multiplication in the primal, amounts in the Fourier domain to a convolution by the spectrum of the blocker. Propagation corresponds to a shear in the space-angle frequency domain, while reflection on curved objects performs a different shear along the angular frequency axis. As shown by previous work, reflection is a convolution in the primal and therefore a multiplication in the Fourier domain. Our work shows how the spatial components of lighting are affected by this angular convolution.Our framework predicts the characteristics of interactions such as caustics and the disappearance of the shadows of small features. Predictions on the frequency content can then be used to control sampling rates for rendering. Other potential applications include precomputed radiance transfer and inverse rendering.

REFERENCES

• 1: Annen, T., Kautz, J., Durand, F., and Seidel, H.-P. 2004. Spherical harmonic gradients for mid-range illumination. In Rendering Techniques 2004 (Proc. EG Symposium on Rendering 2004).
• 2: James Arvo, The irradiance Jacobian for partially occluded polyhedral sources, Proceedings of the 21st annual conference on Computer graphics and interactive techniques, p.343-350, July 1994 [doi>10.1145/192161.192250]
• 3: Ronen Basri , David W. Jacobs, Lambertian Reflectance and Linear Subspaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.2, p.218-233, February 2003 [doi>10.1109/TPAMI.2003.1177153 ]
• 4: Barry G. Becker , Nelson L. Max, Smooth transitions between bump rendering algorithms, Proceedings of the 20th annual conference on Computer graphics and interactive techniques, p.183-190, September 1993 [doi>10.1145/166117.166141]
• 5: Mark R. Bolin , Gary W. Meyer, A frequency based ray tracer, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, p.409-418, September 1995 [doi>10.1145/218380.218497]
• 6: Mark R. Bolin , Gary W. Meyer, A perceptually based adaptive sampling algorithm, Proceedings of the 25th annual conference on Computer graphics and interactive techniques, p.299-309, July 1998 [doi>10.1145/280814.280924]
• 7: Camahort, E., Lerios, A., and Fussell, D. 1998. Uniformly sampled light fields. In Rendering Techniques '98 (Proc. of EG Workshop on Rendering '98), Eurographics, 117--130.
• 8: Jin-Xiang Chai , Shing-Chow Chan , Heung-Yeung Shum , Xin Tong, Plenoptic sampling, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.307-318, July 2000 [doi>10.1145/344779.344932]
• 9: Min Chen , James Arvo, Theory and application of specular path perturbation, ACM Transactions on Graphics (TOG), v.19 n.4, p.246-278, Oct. 2000 [doi>10.1145/380666.380670]
• 10: Do Carmo, M. 1976. Differential Geometry of Curves and Surfaces. Prentice Hall.
• 11: James A. Ferwerda , Peter Shirley , Sumanta N. Pattanaik , Donald P. Greenberg, A model of visual masking for computer graphics, Proceedings of the 24th annual conference on Computer graphics and interactive techniques, p.143-152, August 1997 [doi>10.1145/258734.258818]
• 12: Frolova, D., Simakov, D., and Basri, R. 2004. Accuracy of spherical harmonic approximations for images of Lambertian objects under far and near lighting. In ECCV 2004, European Conference on Computer Vision, 574--587.
• 13: Goodman, J. W. 1996. Introduction To Fourier Optics. McGraw-Hill.
• 14: Steven J. Gortler , Peter Schröder , Michael F. Cohen , Pat Hanrahan, Wavelet radiosity, Proceedings of the 20th annual conference on Computer graphics and interactive techniques, p.221-230, September 1993 [doi>10.1145/166117.166146]
• 15: Halle, M. 1994. Holographic stereograms as discrete imaging systems. In SPIE Proc. Vol. 2176: Practical Holography VIII. S. Benton, Ed., SPIE, 73--84.
• 16: Heckbert, P. 1989. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California at Berkeley, Computer Science Division.
• 17: Holzschuch, N., and Sillion, F. X. 1998. An exhaustive error-bounding algorithm for hierarchical radiosity. Computer Graphics Forum 17, 4.
• 18: Homan Igehy, Tracing ray differentials, Proceedings of the 26th annual conference on Computer graphics and interactive techniques, p.179-186, July 1999 [doi>10.1145/311535.311555]
• 19: Aaron Isaksen , Leonard McMillan , Steven J. Gortler, Dynamically reparameterized light fields, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.297-306, July 2000 [doi>10.1145/344779.344929]
• 20: Alexander Keller, Hierarchical Monte Carlo image synthesis, Mathematics and Computers in Simulation, v.55 n.1-3, p.79-92, Feb. 15, 2001 [doi>10.1016/S0378-4754(00)00248-2 ]
• 21: Jason Lawrence , Szymon Rusinkiewicz , Ravi Ramamoorthi, Efficient BRDF importance sampling using a factored representation, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
• 22: Hendrik P. A. Lensch , Michael Goesele , Jan Kautz , Wolfgang Heidrich , Hans-Peter Seidel, Image-Based Reconstruction of Spatially Varying Materials, Proceedings of the 12th Eurographics Workshop on Rendering Techniques, p.103-114, June 25-27, 2001
• 23: Jitendra Malik , Ruth Rosenholtz, Computing Local Surface Orientation and Shape from Texture forCurved Surfaces, International Journal of Computer Vision, v.23 n.2, p.149-168, June 1997 [doi>10.1023/A:1007958829620 ]
• 24: Michael D. McCool, Anisotropic diffusion for Monte Carlo noise reduction, ACM Transactions on Graphics (TOG), v.18 n.2, p.171-194, April 1999 [doi>10.1145/318009.318015]
• 25: Myszkowski, K. 1998. The visible differences predictor: applications to global illumination problems. In Rendering Techniques '98 (Proc. EG Workshop on Rendering '98), Eurographics.
• 26: Victor Ostromoukhov , Charles Donohue , Pierre-Marc Jodoin, Fast hierarchical importance sampling with blue noise properties, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
• 27: A. P. Pentland, A new sense for depth of field, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.9 n.4, p.523-531, July 1987
• 28: Matt Pharr , Greg Humphreys, Physically Based Rendering: From Theory to Implementation, Morgan Kaufmann Publishers Inc., San Francisco, CA, 2004
• 29: Ravi Ramamoorthi , Pat Hanrahan, An efficient representation for irradiance environment maps, Proceedings of the 28th annual conference on Computer graphics and interactive techniques, p.497-500, August 2001 [doi>10.1145/383259.383317]
• 30: Ravi Ramamoorthi , Pat Hanrahan, A signal-processing framework for inverse rendering, Proceedings of the 28th annual conference on Computer graphics and interactive techniques, p.117-128, August 2001 [doi>10.1145/383259.383271]
• 31: Ravi Ramamoorthi , Pat Hanrahan, Frequency space environment map rendering, ACM Transactions on Graphics (TOG), v.21 n.3, July 2002
• 32: Ravi Ramamoorthi , Pat Hanrahan, A signal-processing framework for reflection, ACM Transactions on Graphics (TOG), v.23 n.4, p.1004-1042, October 2004 [doi>10.1145/1027411.1027416]
• 33: Ramamoorthi, R.. Koudelka, M., and Belhumeur, P. 2004. A Fourier theory for cast shadows. In ECCV 2004, European Conference on Computer Vision, 146--162.
• 34: Mikio Shinya , T. Takahashi , Seiichiro Naito, Principles and applications of pencil tracing, ACM SIGGRAPH Computer Graphics, v.21 n.4, p.45-54, July 1987
• 35: Fraņois Sillion , George Drettakis, Feature-based control of visibility error: a multi-resolution clustering algorithm for global illumination, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, p.145-152, September 1995 [doi>10.1145/218380.218434]
• 36: Peter-Pike Sloan , Jan Kautz , John Snyder, Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments, ACM Transactions on Graphics (TOG), v.21 n.3, July 2002
• 37: Cyril Soler , François X. Sillion, Fast calculation of soft shadow textures using convolution, Proceedings of the 25th annual conference on Computer graphics and interactive techniques, p.321-332, July 1998 [doi>10.1145/280814.280927]
• 38: J. Stewart , J. Yu , S. J. Gortler , L. McMillan, A new reconstruction filter for undersampled light fields, Proceedings of the 14th Eurographics workshop on Rendering, June 25-27, 2003, Leuven, Belgium
• 39: C. Tomasi , R. Manduchi, Bilateral Filtering for Gray and Color Images, Proceedings of the Sixth International Conference on Computer Vision, p.839, January 04-07, 1998
• 40: Ward, G. J., and Heckbert, P. 1992. Irradiance gradients. In Proc. of EG Workshop on Rendering '92, Eurographics, 85--98.
• 41: Gregory J. Ward , Francis M. Rubinstein , Robert D. Clear, A ray tracing solution for diffuse interreflection, ACM SIGGRAPH Computer Graphics, v.22 n.4, p.85-92, Aug. 1988
• 42: Daniel N. Wood , Daniel I. Azuma , Ken Aldinger , Brian Curless , Tom Duchamp , David H. Salesin , Werner Stuetzle, Surface light fields for 3D photography, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.287-296, July 2000 [doi>10.1145/344779.344925]

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.4 IMAGE PROCESSING AND COMPUTER VISION
      I.4.1 Digitization and Image Capture
          Subjects: Sampling

Additional Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.3 Picture/Image Generation
          Subjects: Display algorithms
  I.4 IMAGE PROCESSING AND COMPUTER VISION
      I.4.1 Digitization and Image Capture
          Subjects: Reflectance
  I.5 PATTERN RECOGNITION
      I.5.4 Applications
          Subjects: Signal processing

General Terms:
Algorithms, Experimentation, Performance

Keywords:
Fourier analysis, light transport, signal processing