International Journal of Computer Vision

, Volume 107, Issue 2, pp 139–154 | Cite as

A Closed-Form, Consistent and Robust Solution to Uncalibrated Photometric Stereo Via Local Diffuse Reflectance Maxima

  • Thoma PapadhimitriEmail author
  • Paolo Favaro


Images of an object under different illumination are known to provide strong cues about the object surface. A mathematical formalization of how to recover the normal map of such a surface leads to the so-called uncalibrated photometric stereo problem. In the simplest instance, this problem can be reduced to the task of identifying only three parameters: the so-called generalized bas-relief (GBR) ambiguity. The challenge is to find additional general assumptions about the object, that identify these parameters uniquely. Current approaches are not consistent, i.e., they provide different solutions when run multiple times on the same data. To address this limitation, we propose exploiting local diffuse reflectance (LDR) maxima, i.e., points in the scene where the normal vector is parallel to the illumination direction (see Fig. 1). We demonstrate several noteworthy properties of these maxima: a closed-form solution, computational efficiency and GBR consistency. An LDR maximum yields a simple closed-form solution corresponding to a semi-circle in the GBR parameters space (see Fig. 2); because as few as two diffuse maxima in different images identify a unique solution, the identification of the GBR parameters can be achieved very efficiently; finally, the algorithm is consistent as it always returns the same solution given the same data. Our algorithm is also remarkably robust: It can obtain an accurate estimate of the GBR parameters even with extremely high levels of outliers in the detected maxima (up to 80 % of the observations). The method is validated on real data and achieves state-of-the-art results.


Photometric stereo Uncalibrated Diffuse Maxima GBR Ambiguity 


  1. Agrawal, A., & Raskar, R. (2006). What is the range of surface reconstructions from a gradient field. In ECCV (pp. 578–591). Berlin:Springer.Google Scholar
  2. Alldrin, N., & Kriegman, D. (2007). Toward reconstructing surfaces with arbitrary isotropic reflectance: A stratified photometric stereo approach. International Journal of Computer Vision, 21, 1–8.Google Scholar
  3. Alldrin, N., Mallick, S., & Kriegman, D. (2007). Resolving the generalized bas-relief ambiguity by entropy minimization. In Computer Vision and Pattern Recognition (pp. 1–7, 17–22).Google Scholar
  4. Basri, R., & Jacobs, D. (2001). Photometric stereo with general, unknown lighting. In IEEE Conference on Computer Vision and Pattern Recognition (pp. 374–381).Google Scholar
  5. Belhumeur, P. N., Kriegman, D. J., & Yuille, A. L. (1999). The bas-relief ambiguity. International Journal of Computer Vision, 35(1), 33–44.Google Scholar
  6. Chandraker, M., Bai, J., & Ramamoorthi, R. (2011). A theory of photometric reconstruction for unknown isotropic reflectances. In IEEE Conference on Computer Vision and Pattern Recognition.Google Scholar
  7. Chandraker, M. K., Kahl, F., & Kriegman, D. J. (2005). Reflections on the generalized bas-relief ambiguity. In Conference on Computer Vision and, Pattern Recognition (pp. 788–795).Google Scholar
  8. Drbohlav, O., & Chantler, M. (2005). Can two specular pixels calibrate photometric stereo?. In Proceedings of International Conference on Computer Vision (pp. 1850–1857).Google Scholar
  9. Drbohlav, O., & Sara, R. (2002). Specularities reduce ambiguity of uncalibrated photometric stereo. In European Conference on Computer Vision (pp. 46–62).Google Scholar
  10. Favaro, P., & Papadhimitri, T. (2012). A closed-form solution to uncalibrated photometric stereo via diffuse maxima. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 821–828).Google Scholar
  11. Georghiades, A. (2003). Incorporating the torrance and sparrow model of reflectance in uncalibrated photometric stereo. In Proceedings of International Conference on Computer Vision (pp. 816–823).Google Scholar
  12. Georghiades, A. S., Belhumeur, P. N., & Kriegman, D. J. (2001). From few to many: Illumination cone models for face recognition under variable lighting and pose. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23, 643–660.CrossRefGoogle Scholar
  13. Hayakawa, H. (1994). Photometric stereo under a light-source with arbitrary motion. Journal of the Optical Society of America, 11(11), 3079–3089.CrossRefMathSciNetGoogle Scholar
  14. Hertzmann, A., & Seitz, S. (2005). Example-based photometric stereo: Shape reconstruction with general, varying brdfs. Pattern Analysis and Machine Intelligence, 27(8), 1254–1264.CrossRefGoogle Scholar
  15. Koppal, S. J., & Narasimhan, S. G. (2006). Clustering appearance for scene analysis. IEEE Conference on Computer Vision and Pattern Recognition, 2, 1323–1330.Google Scholar
  16. Lagger, P., & Fua, P. (2008). Retrieving multiple light sources in the presence of specular reflections and texture. Computer Vision and Image Understanding, 111, 207–218.CrossRefGoogle Scholar
  17. Lin, Z., Chen, M., & Ma, Y. (2009). The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. UIUC Technical, Report UILU-ENG-09-2215.Google Scholar
  18. Okabe, T., Sato, I., & Sato, Y. (2009). Attached shadow coding: Estimating surface normals from shadows under unknown reflectance and lighting conditions. In Proceedings of International Conference on Computer Vision (pp. 1693–1700).Google Scholar
  19. Oren, M., & Nayar, S. K. (1997). A theory of specular surface geometry. International Journal of Computer Vision, 24, 105–124.CrossRefGoogle Scholar
  20. Sato, I., Okabe, T., Yu, Q., & Sato, Y. (2007). Shape reconstruction based on similarity in radiance changes under varying illumination. In Proceedings of International Conference on Computer Vision (pp. 1–8, 14–21).Google Scholar
  21. Shi, B., Matsushita, Y., Wei, Y., Xu, C., & Tan, P. (2010). Self-calibrating photometric stereo. In IEEE Conference on Computer Vision and, Pattern Recognition (pp. 1118–1125).Google Scholar
  22. Sunkavalli, H. P. K., & Zickler, T. (2010). Visibility subspaces: Uncalibrated photometric stereo with shadows. In European Conference on Computer Vision, Part II (pp. 251–264).Google Scholar
  23. Tan, P., Mallick, S., Quan, L., Kriegman, D., & Zickler, T. (2007). Isotropy, reciprocity and the generalized bas-relief ambiguity. Computer Vision and Pattern Recognition Conference, 1(8), 17–22.Google Scholar
  24. Woodham, R. (1980). Photometric method for determining surface orientation from multiple images. Optical Engineering, 19(1), 139–144.CrossRefGoogle Scholar
  25. Wright, J., Ganesh, A., Rao, S., Peng, Y., & Ma, Y. (2009). Robust principal component analysis: Exact recovery of corrupted low-rank matrices by convex optimization. In Proceedings of Neural Information Processing Systems (NIPS).Google Scholar
  26. Wu, L., Ganesh, A., Shi, B., Matsushita, Y., Wang, Y., & Ma, Y. (2011). Robust photometric stereo via low-rank matrix completion and recovery. In Asian Conference on Computer Vision (pp. 703–717).Google Scholar
  27. Wu, T., & Tang, C. (2005). Dense photometric stereo using a mirror sphere and graph cut. Computer Vision and, Pattern Recognition, 1, 140–147.Google Scholar
  28. Yuille, A., & Snow, D. (1997). Shape and albedo from multiple images using integrability. In Computer Vision and, Pattern Recognition (pp. 158–164). Google Scholar
  29. Zickler, T., Belhumeur, P., & Kriegman, D. (2002). Helmholtz stereopsis: Exploiting reciprocity for surface reconstruction. International Journal of Computer Vision, 49(2–3), 215–227.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of BernBernSwitzerland

Personalised recommendations