A Variational Approach to Shape-from-Shading Under Natural Illumination

  • Yvain QuéauEmail author
  • Jean Mélou
  • Fabien Castan
  • Daniel Cremers
  • Jean-Denis Durou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10746)


A numerical solution to shape-from-shading under natural illumination is presented. It builds upon an augmented Lagrangian approach for solving a generic PDE-based shape-from-shading model which handles directional or spherical harmonic lighting, orthographic or perspective projection, and greylevel or multi-channel images. Real-world applications to shading-aware depth map denoising, refinement and completion are presented.


Natural Illumination Spherical Harmonic Lighting Augmented Lagrangian Approach Local Gradient Estimate Depth Refinement 
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  1. 1.
    Durou, J.D., Falcone, M., Sagona, M.: Numerical methods for shape-from-shading: a new survey with benchmarks. Comput. Vis. Image Underst. 109, 22–43 (2008)CrossRefGoogle Scholar
  2. 2.
    Horn, B.K.P., Brooks, M.J.: The variational approach to shape from shading. Comput. Vis. Graph. Image Process. 33, 174–208 (1986)CrossRefzbMATHGoogle Scholar
  3. 3.
    Lions, P.L., Rouy, E., Tourin, A.: Shape-from-shading, viscosity solutions and edges. Numer. Math. 64, 323–353 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Han, Y., Lee, J.Y., Kweon, I.S.: High quality shape from a single RGB-D image under uncalibrated natural illumination. In: Proceedings of the ICCV (2013)Google Scholar
  5. 5.
    Basri, R., Jacobs, D.P.: Lambertian reflectances and linear subspaces. IEEE Trans. Pattern Anal. Mach. Intell. 25, 218–233 (2003)CrossRefGoogle Scholar
  6. 6.
    Quéau, Y., Mecca, R., Durou, J.D.: Unbiased photometric stereo for colored surfaces: a variational approach. In: Proceedings of the CVPR (2016)Google Scholar
  7. 7.
    Johnson, M.K., Adelson, E.H.: Shape estimation in natural illumination. In: Proceedings of the CVPR (2011)Google Scholar
  8. 8.
    Huang, R., Smith, W.A.P.: Shape-from-shading under complex natural illumination. In: Proceedings of the ICIP (2011)Google Scholar
  9. 9.
    Or-El, R., Rosman, G., Wetzler, A., Kimmel, R., Bruckstein, A.: RGBD-fusion: real-time high precision depth recovery. In: Proceedings of the CVPR (2015)Google Scholar
  10. 10.
    Frolova, D., Simakov, D., Basri, R.: Accuracy of spherical harmonic approximations for images of Lambertian objects under far and near lighting. In: Pajdla, T., Matas, J. (eds.) ECCV 2004. LNCS, vol. 3021, pp. 574–587. Springer, Heidelberg (2004). CrossRefGoogle Scholar
  11. 11.
    Barron, J.T., Malik, J.: Shape, illumination, and reflectance from shading. IEEE Trans. Pattern Anal. Mach. Intell. 37, 1670–1687 (2015)CrossRefGoogle Scholar
  12. 12.
    Cristiani, E., Falcone, M.: Fast semi-Lagrangian schemes for the eikonal equation and applications. SIAM J. Numer. Anal. 45, 1979–2011 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Breuß, M., Cristiani, E., Durou, J.D., Falcone, M., Vogel, O.: Perspective shape from shading: ambiguity analysis and numerical approximations. SIAM J. Imaging Sci. 5, 311–342 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Quéau, Y., Durou, J.D., Aujol, J.F.: Normal integration: a survey. J. Math. Imaging Vis. (2017)Google Scholar
  15. 15.
    Graber, G., Balzer, J., Soatto, S., Pock, T.: Efficient minimal-surface regularization of perspective depth maps in variational stereo. In: Proceedings of the CVPR (2015)Google Scholar
  16. 16.
    Glowinski, R., Marroco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires. Revue française d’automatique, informatique, recherche opérationnelle. Analyse numérique 9, 41–76 (1975)CrossRefzbMATHGoogle Scholar
  17. 17.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2011)CrossRefzbMATHGoogle Scholar
  18. 18.
    Li, G., Pong, T.K.: Global convergence of splitting methods for nonconvex composite optimization. SIAM J. Optim. 25, 2434–2460 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hong, M., Luo, Z.Q., Razaviyayn, M.: Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. SIAM J. Optim. 26, 337–364 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zollhöfer, M., Dai, A., Innman, M., Wu, C., Stamminger, M., Theobalt, C., Nießner, M.: Shading-based refinement on volumetric signed distance functions. ACM Trans. Graph. 34, 96:1–96:14 (2015)CrossRefzbMATHGoogle Scholar
  21. 21.
    Jancosek, M., Pajdla, T.: Multi-view reconstruction preserving weakly-supported surfaces. In: Proceedings of the CVPR (2011)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Yvain Quéau
    • 1
    Email author
  • Jean Mélou
    • 2
    • 3
  • Fabien Castan
    • 3
  • Daniel Cremers
    • 1
  • Jean-Denis Durou
    • 2
  1. 1.Department of InformaticsTechnical University of MunichMunichGermany
  2. 2.IRIT, UMR CNRS 5505, Université de ToulouseToulouseFrance
  3. 3.Mikros ImageLevallois-PerretFrance

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