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Towards efficient image irradiance modelling of convex Lambertian surfaces under single viewpoint and frontal illumination

Towards efficient image irradiance modelling of convex Lambertian surfaces under single viewpoint and frontal illumination

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Under local illumination assumption, phenomenological appearance models capture surface appearance through the mathematical modelling of the reflection process. Theoretically, due to the arbitrariness of the lighting function, the space of all possible images of a fixed-pose object under all possible illumination conditions is infinite dimensional. Nonetheless, due to their low- frequency nature, irradiance signals can be represented using low-order basis functions, where spherical harmonics (SH) has been extensively adopted. When capturing image irradiance from a single viewpoint, the visible part of the object's surface constructs the upper hemisphere of the surface normals where the SH is no longer orthonormal. In this paper, we propose the use of hemispherical harmonics (HSH) to model image irradiance of convex Lambertian objects perceived from single viewpoint under unknown distant as well as near illumination. We prove analytically, and validate experimentally, that the Lambertian reflectance kernel has a more compact harmonic expansion in the hemispherical domain when compared to its spherical counterpart. Our experiments illustrate that, despite of having poor approximation accuracy under very close lights, such behavior improves exponentially with little increase in the distance to the light source relative to the object size.


    1. 1)
      • 15. Kythe, P.K., Schaferkotter, M.R.: ‘Handbook of computational methods for integration’ (Chapman & Hall/CRC, 2004).
    2. 2)
      • 8. Nillius, P., Eklundh, J.: ‘Low-dimensional representations of shaded surfaces under varying illumination’. Computer Vision and Pattern Recognition, 2003. Proc.. 2003 IEEE Computer Society Conf., 2003, vol. 2, pp. II-185II-192.
    3. 3)
      • 11. Elhabian, S., Rara, H., Farag, A.: ‘Towards accurate and efficient representation of image irradiance of convex-Lambertian objects under unknown near lighting’. Int. Conf. Computer Vision (ICCV), Barcelona, Spain, 2011.
    4. 4)
      • 18. Azuma, R.: ‘A survey of augmented reality’, Presence: Teleoperators Virtual Environ., 1997, 6, (4), pp. 355385.
    5. 5)
      • 5. Green, R.: ‘Spherical harmonic lighting: the gritty details’. Archives of the Game Developers Conf., 2003, vol. 2, pp. 23.
    6. 6)
      • 1. Basri, R., Jacobs, D.W.: ‘Lambertian reflectance and linear subspaces’, IEEE Trans. Pattern Anal. Mach. Intell., 2003, 25, (2), pp. 218233 (doi: 10.1109/TPAMI.2003.1177153).
    7. 7)
      • 17. Okabe, T., Sato, I., Sato, Y.: ‘Spherical harmonics vs. haar wavelets: basis for recovering illumination from cast shadows’. Proc. 2004 IEEE Computer Society Conf. Computer Vision and Pattern Recognition, 2004. CVPR 2004, IEEE, Washington, DC, USA, 2004, vol. 1, pp. 5057.
    8. 8)
      • 9. Koenderink, J., van Doorn, A.: ‘Phenomenological description of bidirectional surface reflection’, J. Opt. Soc. Am., 1998, 15, (11), pp. 29032912 (doi: 10.1364/JOSAA.15.002903).
    9. 9)
      • 12. Elhabian, S., Rara, H., Farag, A.: ‘Modeling Lambertian surfaces under unknown distant illumination using hemispherical harmonics’. Eighth Canadian Conf. Computer and Robot Vision (CRV), St. John's, Newfoundland, Canada, 2011, pp. 293300.
    10. 10)
      • 13. Arfken, G.B., Weber, H.J.: ‘Mathematical methods for physicists: a comprehensive guide’ (Elsevier Academic Press – Mathematics, 2005, 6th edn.) Chapter 12.
    11. 11)
      • 3. D'Zmura, M.: ‘Shading ambiguity: reflectance and illumination’, Comput. Models Vis. Process., 1991, pp. 187207.
    12. 12)
      • 14. Makhotkin, O.A.: ‘Analysis of radiative transfer between surfaces by hemispherical harmonics’, J. Quant. Spectrosc. Radiat. Transf., 1996, 56, (6), pp. 869879 (doi: 10.1016/S0022-4073(96)00040-4).
    13. 13)
      • 16. Frolova, D., Simakov, D., Basri, R.: ‘Accuracy of spherical harmonic approximations for images of Lambertian objects under far and near lighting’. European Conf. Computer Vision, Zofin Palace, Slovansky ostrov, Prague 1, Czech Republic, 2004, pp. 574587.
    14. 14)
      • 20. Basri, R., Jacobs, D., Kemelmacher, I.: ‘Photometric stereo with general, unknown lighting’, Int. J. Comput. Vision, 2007, 72, (3), pp. 239257 (doi: 10.1007/s11263-006-8815-7).
    15. 15)
      • 4. Horn, B.: ‘Robot vision’ (McGraw-Hill, 1986).
    16. 16)
      • 10. Gautron, P., Krivanek, J., Pattanaik, S.N., Bouatouch, K.: ‘A novel hemispherical basis for accurate and efficient rendering’. Proc. Fifteenth Eurographics Conf. Rendering Techniques, Eurographics Association, 2004, pp. 321330.
    17. 17)
      • 2. Cabral, B., Max, N., Springmeyer, R.: ‘Bidirectional reflection functions from surface bump maps’, SIGGRAPH Comput. Graph., 1987, 21, (4), pp. 273281 (doi: 10.1145/37402.37434).
    18. 18)
      • 7. Ramamoorthi, R.: ‘Analytic PCA construction for theoretical analysis of lighting variability in images of a lambertian object’, IEEE Trans. Pattern Anal. Mach. Intell., 2002, 24, (10), pp. 13221333 (doi: 10.1109/TPAMI.2002.1039204).
    19. 19)
      • 6. Ramamoorthi, R., Hanrahan, P.: ‘On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object’, J. Opt. Soc. Am. A, 2001, 18, (10), pp. 24482459. Available at (doi: 10.1364/JOSAA.18.002448).
    20. 20)
      • 19. Mei, X., Ling, H., Jacobs, D.: ‘Illumination recovery from image with cast shadows via sparse representation’, IEEE Trans. Image Process., 2011, 20, (8), pp. 23662377 (doi: 10.1109/TIP.2011.2118222).

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