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A Fourier Theory for Cast Shadows

  • Ravi Ramamoorthi
  • Melissa Koudelka
  • Peter Belhumeur
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3021)

Abstract

Cast shadows can be significant in many computer vision applications such as lighting-insensitive recognition and surface reconstruction. However, most algorithms neglect them, primarily because they involve non-local interactions in non-convex regions, making formal analysis difficult. While general cast shadowing situations can be arbitrarily complex, many real instances map closely to canonical configurations like a wall, a V-groove type structure, or a pitted surface. In particular, we experiment on 3D textures like moss, gravel and a kitchen sponge, whose surfaces include canonical cast shadowing situations like V-grooves. This paper shows theoretically that many shadowing configurations can be mathematically analyzed using convolutions and Fourier basis functions. Our analysis exposes the mathematical convolution structure of cast shadows, and shows strong connections to recently developed signal-processing frameworks for reflection and illumination. An analytic convolution formula is derived for a 2D V-groove, which is shown to correspond closely to many common shadowing situations, especially in 3D textures. Numerical simulation is used to extend these results to general 3D textures. These results also provide evidence that a common set of illumination basis functions may be appropriate for representing lighting variability due to cast shadows in many 3D textures. We derive a new analytic basis suited for 3D textures to represent illumination on the hemisphere, with some advantages over commonly used Zernike polynomials and spherical harmonics. New experiments on analyzing the variability in appearance of real 3D textures with illumination motivate and validate our theoretical analysis. Empirical results show that illumination eigenfunctions often correspond closely to Fourier bases, while the eigenvalues drop off significantly slower than those for irradiance on a Lambertian curved surface. These new empirical results are explained in this paper, based on our theory.

Keywords

Spherical Harmonic Fourier Theory Eigenvalue Spectrum Cast Shadow Illumination Direction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Basri, R., Jacobs, D.: Lambertian reflectance and linear subspaces. In: ICCV 2001, pp. 383–390 (2001)Google Scholar
  2. 2.
    Dana, K., Nayar, S.: Histogram model for 3d textures. In: CVPR 1998, pp. 618–624 (1998)Google Scholar
  3. 3.
    Dana, K., van Ginneken, B., Nayar, S., Koenderink, J.: Reflectance and texture of realworld surfaces. ACM Transactions on Graphics 18(1), 1–34 (1999)CrossRefGoogle Scholar
  4. 4.
    Epstein, R., Hallinan, P., Yuille, A.: 5 plus or minus 2 eigenimages suffice: An empirical investigation of low-dimensional lighting models. In: IEEE 1995 Workshop Physics-Based Modeling in Computer Vision, pp. 108–116 (1995)Google Scholar
  5. 5.
    Hallinan, P.: A low-dimensional representation of human faces for arbitrary lighting conditions. In: CVPR 1994, pp. 995–999 (1994)Google Scholar
  6. 6.
    Koenderink, J., Doorn, A., Dana, K., Nayar, S.: Bidirectional reflection distribution function of thoroughly pitted surfaces. IJCV 31(2/3), 129–144 (1999)CrossRefGoogle Scholar
  7. 7.
    Koenderink, J., van Doorn, A.: Phenomenological description of bidirectional surface reflection. JOSA A 15(11), 2903–2912 (1998)CrossRefGoogle Scholar
  8. 8.
    Koudelka, M., Magda, S., Belhumeur, P., Kriegman, D.: Acquisition, compression, and synthesis of bidirectional texture functions. In: ICCV 2003 workshop on Texture Analysis and Synthesis (2003)Google Scholar
  9. 9.
    MacRobert, T.M.: Spherical harmonics; an elementary treatise on harmonic functions, with applications. Dover Publications, Mineola (1948)Google Scholar
  10. 10.
    Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, London (1999)zbMATHGoogle Scholar
  11. 11.
    Malzbender, T., Gelb, D., Wolters, H.: Polynomial texture maps. In: SIGGRAPH 2001, pp. 519–528 (2001)Google Scholar
  12. 12.
    Oren, M., Nayar, S.: Generalization of lambert’s reflectance model. In: SIGGRAPH 1994, pp. 239–246 (1994)Google Scholar
  13. 13.
    Ramamoorthi, R.: Analytic PCA construction for theoretical analysis of lighting variability in images of a lambertian object. PAMI 24(10), 1322–1333 (2002)Google Scholar
  14. 14.
    Ramamoorthi, R., Hanrahan, P.: Analysis of planar light fields from homogeneous convex curved surfaces under distant illumination. In: SPIE Photonics West: Human Vision and Electronic Imaging VI, pp. 185–198 (2001)Google Scholar
  15. 15.
    Ramamoorthi, R., Hanrahan, P.: On the relationship between radiance and irradiance: Determining the illumination from images of a convex lambertian object. JOSAA 18(10), 2448–2459 (2001)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Ramamoorthi, R., Hanrahan, P.: A signal-processing framework for inverse rendering. In: SIGGRAPH 2001, pp. 117–128 (2001)Google Scholar
  17. 17.
    Sato, Y.: Illumination distribution from brightness in shadows: adaptive estimation of illumination distribution with unknown reflectance properties in shadow regions. In: ICCV 1999, pp. 875–882 (1999)Google Scholar
  18. 18.
    Sloan, P., Kautz, J., Snyder, J.: Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments. ACM Transactions on Graphics (SIGGRAPH 2002) 21(3), 527–536 (2002)Google Scholar
  19. 19.
    Soler, C., Sillion, F.: Fast calculation of soft shadow textures using convolution. In: SIGGRAPH 1998, pp. 321–332 (1998)Google Scholar
  20. 20.
    Suen, P., Healey, G.: Analyzing the bidirectional texture function. In: CVPR 1998, pp. 753–758 (1998)Google Scholar
  21. 21.
    Torrance, K., Sparrow, E.: Theory for off-specular reflection from roughened surfaces. JOSA 57(9), 1105–1114 (1967)CrossRefGoogle Scholar
  22. 22.
    Westin, S., Arvo, J., Torrance, K.: Predicting reflectance functions from complex surfaces. In: SIGGRAPH 1992, pp. 255–264 (1992)Google Scholar
  23. 23.
    Wolff, L., Nayar, S., Oren, M.: Improved diffuse reflection models for computer vision. IJCV 30, 55–71 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ravi Ramamoorthi
    • 1
  • Melissa Koudelka
    • 2
  • Peter Belhumeur
    • 1
  1. 1.Columbia University 
  2. 2.Yale University 

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