Skip to main content

A Graph Theoretic Approach for Shape from Shading

Part of the Lecture Notes in Computer Science book series (LNIP,volume 10746)

Abstract

Resolving ambiguities is a fundamental problem in shape from shading (SFS). The classic SFS approach allows to reconstruct the surface locally around singular points up to an ambiguity of convex, concave or saddle point type.

In this paper we follow a recent approach that seeks to resolve the local ambiguities in a global graph-based setting so that the complete surface reconstruction is consistent. To this end, we introduce a novel graph theoretic formulation for the underlying problem that allows to prove for the first time in the literature that the underlying surface orientation problem is \(\mathcal {NP}\)-complete. Moreover, we show that our novel framework allows to define an algorithmic framework that solves the disambiguation problem. It makes use of cycle bases for dealing with the graph construction and enables an easy embedding into an optimization method that amounts here to a linear program.

Keywords

  • Shape from shading
  • Ambiguity
  • Configuration graph
  • Cycle basis

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-78199-0_22
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-78199-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   107.00
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

References

  1. Abada, L., Aouat, S.: Tabu search to solve the shape from shading ambiguity. Int. J. Artif. Intell. Tools 24(5) (2015)

    Google Scholar 

  2. Abada, L., Aouat, S.: Improved shape from shading without initial information. Front. Comput. Sci. 11(2), 320–331 (2017)

    CrossRef  Google Scholar 

  3. Bollobás, B.: Modern Graph Theory. Graduate Texts in Mathematics, vol. 184. Springer, Heidelberg (1998). https://doi.org/10.1007/978-1-4612-0619-4

    MATH  Google Scholar 

  4. Bruss, A.R.: The eikonal equation: some results applicable to computer vision. J. Math. Phys. 23(5), 890–896 (1982)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Bruss, A.R.: Is what you see what you get? In: Proceedings of the International Joint Conference on Artificial Intelligence, pp. 1053–1056, August 1983

    Google Scholar 

  6. Chang, J.Y., Lee, K.M., Lee, S.U.: Shape from shading using graph cuts. Pattern Recogn. 41(12), 3749–3757 (2008)

    CrossRef  MATH  Google Scholar 

  7. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-662-53622-3

    MATH  Google Scholar 

  8. Durou, J.D., Piau, D.: Ambiguous shape from shading with critical points. J. Math. Imaging Vis. 12(2), 99–108 (2000)

    MathSciNet  CrossRef  MATH  Google Scholar 

  9. Hopcroft, J., Tarjan, R.: Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM 16(6), 372–378 (1973)

    CrossRef  Google Scholar 

  10. Horn, B.K.P.: Shape from shading: a method for obtaining the shape of a smooth opaque object from one view. Ph.D. thesis, Massachusetts Institute of Technology (1970)

    Google Scholar 

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

    Google Scholar 

  12. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    CrossRef  Google Scholar 

  13. Kimmel, R., Bruckstein, A.M.: Global shape from shading. Comput. Vis. Image Underst. 62(3), 360–369 (1995)

    CrossRef  Google Scholar 

  14. Köhler, E.: Recognizing graphs without asteroidal triples. J. Discret. Algorithms 2(4), 439–452 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Oliensis, J.: Uniqueness in shape from shading. Int. J. Comput. Vis. 6(2), 75–104 (1991)

    CrossRef  MATH  Google Scholar 

  16. Prados, E., Soatto, S.: Fast marching method for generic shape from shading. In: Paragios, N., Faugeras, O., Chan, T., Schnörr, C. (eds.) VLSM 2005. LNCS, vol. 3752, pp. 320–331. Springer, Heidelberg (2005). https://doi.org/10.1007/11567646_27

    CrossRef  Google Scholar 

  17. Quéau, Y., Durou, J.-D.: Edge-preserving integration of a normal field: weighted least-squares, TV and \(L^1\) approaches. In: Aujol, J.-F., Nikolova, M., Papadakis, N. (eds.) SSVM 2015. LNCS, vol. 9087, pp. 576–588. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18461-6_46

    Google Scholar 

  18. Sethian, J.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  19. Zhu, Q., Shi, J.: Shape from shading: recognizing the mountains through a global view. In: 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1839–1846. IEEE, New York (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Robert Scheffler .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Scheffler, R., Mansouri Yarahmadi, A., Breuß, M., Köhler, E. (2018). A Graph Theoretic Approach for Shape from Shading. In: Pelillo, M., Hancock, E. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2017. Lecture Notes in Computer Science(), vol 10746. Springer, Cham. https://doi.org/10.1007/978-3-319-78199-0_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-78199-0_22

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78198-3

  • Online ISBN: 978-3-319-78199-0

  • eBook Packages: Computer ScienceComputer Science (R0)