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.
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References
Abada, L., Aouat, S.: Tabu search to solve the shape from shading ambiguity. Int. J. Artif. Intell. Tools 24(5) (2015)
Abada, L., Aouat, S.: Improved shape from shading without initial information. Front. Comput. Sci. 11(2), 320–331 (2017)
Bollobás, B.: Modern Graph Theory. Graduate Texts in Mathematics, vol. 184. Springer, Heidelberg (1998). https://doi.org/10.1007/978-1-4612-0619-4
Bruss, A.R.: The eikonal equation: some results applicable to computer vision. J. Math. Phys. 23(5), 890–896 (1982)
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
Chang, J.Y., Lee, K.M., Lee, S.U.: Shape from shading using graph cuts. Pattern Recogn. 41(12), 3749–3757 (2008)
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
Durou, J.D., Piau, D.: Ambiguous shape from shading with critical points. J. Math. Imaging Vis. 12(2), 99–108 (2000)
Hopcroft, J., Tarjan, R.: Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM 16(6), 372–378 (1973)
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)
Horn, B.K.P., Brooks, M.J. (eds.): Shape from Shading. MIT Press, Cambridge (1989)
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)
Kimmel, R., Bruckstein, A.M.: Global shape from shading. Comput. Vis. Image Underst. 62(3), 360–369 (1995)
Köhler, E.: Recognizing graphs without asteroidal triples. J. Discret. Algorithms 2(4), 439–452 (2004)
Oliensis, J.: Uniqueness in shape from shading. Int. J. Comput. Vis. 6(2), 75–104 (1991)
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
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
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)
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)
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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
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DOI: https://doi.org/10.1007/978-3-319-78199-0_22
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