Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

Regularized Quadratic Penalty Methods for Shape from Shading

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Shape from shading (SFS) denotes the problem of reconstructing a 3D surface, starting from a single shaded image which represents the surface itself. Minimization techniques are commonly used for solving the SFS problem, where the objective function is a weighted combination of the brightness error, plus one or more terms aiming to obtain a valid solution. We present a regularized quadratic penalty method where quadratic penalization is used to adaptively adjust the smoothing weights, and regularization improves the robustness and reliability of the procedure. A nonmonotone Barzilai–Borwein method is employed to efficiently solve the arising subproblems. Numerical results are provided showing the reliability of the proposed approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Birgin, E., Martínez, J., Raydan, M.: Spectral projected gradient methods: review and perspectives. J. Stat. Softw. 60, 1–21 (2014)

  2. Camilli, F., Falcone, M.: An approximation scheme for the maximal solutions of the shape from shading model. In: Proceedings of the 3rd IEEE International Conference on Image Processing, Lausanne, Switzerland, vol. I, pp. 49–52 (1996)

  3. Dai, Y.H., Fletcher, R.: Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming. Numerische Mathematik 100, 21–47 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Daniel, P., Durou, J.-D.: From deterministic to stochastic methods for shape from shading. In: Proceedings of the 4th Asian Conference on Computer Vision, Taipei, Taiwan, pp. 187–192 (2000)

  5. De Asmundis, R., di Serafino, D., Riccio, F., Toraldo, G.: On spectral properties of steepest descent methods. IMA J. Numer. Anal. 33, 1416–1435 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Asmundis, R., di Serafino, D., Hager, W., Toraldo, G., Zhang, H.: An efficient gradient method using the Yuan steplength. Comput. Optim. Appl. 59, 541–563 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. Durou, J.-D., Aujol, J.-F., Couteille, F.: Integrating the normal field of a surface in the presence of discontinuities. In: Cremers, D., Boykov, Y., Blake, A., Schmidt, F.R. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science, vol. 5681, pp. 261–273. Springer, Berlin, Heidelberg (2009)

  9. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1996)

    Book  MATH  Google Scholar 

  10. Falcone, M., Sagona, M., Seghini, A.: A scheme for the shape from shading model with “black shadows”. In: Brezzi, F., Buffa, A., Corsaro, S., Murli, A. (eds.) Numerical Mathematics and Advanced Applications. pp. 503–512. Springer, Milano (2003)

  11. Fletcher, R.: On the Barzilai–Borwein method. In: Qi, L., Teo, K., Yang, X., Pardalos, P.M., Hearn, D. (eds.) Optimization and Control with Applications. Applied Optimization, vol. 96, pp. 235–256. Springer, New York (2005)

    Chapter  Google Scholar 

  12. Frankot, R.T., Chellappa, R.: A method for enforcing integrability in shape from shading algorithms. IEEE Trans. Pattern Anal. Mach. Intell. 10, 439–451 (1988)

    Article  MATH  Google Scholar 

  13. Governi, L., Furferi, R., Puggelli, L., Volpe, Y.: Improving surface reconstruction in shape from shading using easy-to-set boundary conditions. Int. J. Comput. Vis. Robot. 3, 225–247 (2013)

    Article  Google Scholar 

  14. Governi, L., Furferi, R., Carfagni, M., Puggelli, L., Volpe, Y.: Digital bas-relief design: a novel shape from shading-based method. Comput. Aided Des. Appl. 11, 153–164 (2014)

    Article  Google Scholar 

  15. Horn, B.K.P.: Shape from Shading: A Method for Obtaining the Shape of a Smooth Opaque Object from One View, MAC-TR-79 and AI-TR-232. Massachusetts Institute of Technology, Artificial Intelligence Laboratory, Cambridge (1970)

    Google Scholar 

  16. Horn, B.K.P.: Obtaining shape from shading. In: Winston, P.H. (ed.) The Psychology of Computer Vision. McGraw Hill, New York (1975)

    Google Scholar 

  17. Horn, B.K.P.: Height and gradient from shading. Int. J. Comput. Vis. 5, 37–75 (1990)

    Article  Google Scholar 

  18. Horn, B.K.P., Brooks, M.J.: The variational approach to shape from shading. Comput. Vis. Graph. Image Proc. 33, 174–208 (1986)

    Article  MATH  Google Scholar 

  19. Horn, B.K.P., Brooks, M.J.: Shape from Shading. MIT Press, Cambridge (1989)

    MATH  Google Scholar 

  20. Kozera, R.: Uniqueness in shape from shading revisited. J. Math. Imaging Vis. 7, 123–138 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Leclerc, Y.G., Bobick, A.F.: The direct computation of height from shading. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Maui, HI, USA, pp. 552–558 (1991)

  22. Nocedal, J., Wright, S.J.: Numerical Optimization. Series in Operations Research and Financial Engineering. Springer, New York (2006)

  23. Oliensis, J.: Uniqueness in shape from shading. Int. J. Comput. Vis. 6, 75–104 (1991)

    Article  MATH  Google Scholar 

  24. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rouy, E., Tourin, A.: A viscosity solution approach to shape from shading. SIAM J. Numer. Anal. 29, 867–884 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Szelisi, R.: Fast shape from shading. Comput. Vis. Graph. Image Proc. Image Underst. 53, 129–153 (1991)

    Google Scholar 

  27. Tozza, S., Falcone, M., Analysis and approximation of some shape-from-shading models for non-Lambertian surfaces. J. Math. Imaging Vis. 55, 153–178 (2016)

  28. Wu, T., Sun, J., Tang, C., Shum, H.: Interactive normal reconstruction from a single image. ACM Trans. Graph. 27, 1–9 (2008). Article 119

  29. Worthington, P.L., Hancock, E.R.: New constraints on data-closeness and needle map consistency for shape from shading. IEEE Trans. Pattern Anal. Mach. Intell. 21, 1250–1267 (1999)

    Article  Google Scholar 

  30. Zhang, R., Tsai, P.-S., Cryer, J.E., Shah, M.: Shape-from-shading: a survey. IEEE Trans. Pattern Anal. Mach. Intell. 21, 690–706 (1999)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We thank the associate editor for the care taken with this paper and the two anonymous referees for their insightful comments, which led us to improve our work. Work partially supported by INdAM-GNCS.

Author information

Authors and Affiliations

  1. Department of Industrial Engineering, University of Florence, 50134, Florence, Italy

    Stefania Bellavia, Lapo Governi, Alessandra Papini & Luca Puggelli

Authors
  1. Stefania Bellavia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lapo Governi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alessandra Papini
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Luca Puggelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandra Papini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bellavia, S., Governi, L., Papini, A. et al. Regularized Quadratic Penalty Methods for Shape from Shading. Mediterr. J. Math. 14, 145 (2017). https://doi.org/10.1007/s00009-017-0946-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-017-0946-2

Keywords

Mathematics Subject Classification

Search

Navigation