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Semidefinite Programming Heuristics for Surface Reconstruction Ambiguities

  • Ady Ecker
  • Allan D. Jepson
  • Kiriakos N. Kutulakos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5302)

Abstract

We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based on semidefinite programming may reveal the shape of the surface.

Keywords

Singular Vector Photometric Stereo Quadratic Cost Function Sweep Line Compute Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Helmberg, C.: Semidefinite programming for combinatorial optimization. Technical Report ZIB-Report ZR-00-34, TU Berlin (2000)Google Scholar
  2. 2.
    Laurent, M., Rendl, F.: Semidefinite programming and integer programming. In: Handbook on Discrete Optimization, pp. 393–514. Elsevier, Amsterdam (2005)CrossRefGoogle Scholar
  3. 3.
    Todd, M.J.: Semidefinite optimization. Acta Numerica 10, 515–560 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Keuchel, J., Schnorr, C., Schellewald, C., Cremers, D.: Binary partitioning, perceptual grouping, and restoration with semidefinite programming. PAMI 25(11), 1364–1379 (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Keuchel, J.: Multiclass image labeling with semidefinite programming. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3954, pp. 454–467. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Carl Olsson, A.E., Kahl, F.: Solving large scale binary quadratic problems: Spectral methods vs. semidefinite programming. In: CVPR 2007, pp. 1–8 (2007)Google Scholar
  7. 7.
    Bai, X., Yu, H., Hancock, E.: Graph matching using spectral embedding and semidefinite programming. In: BMVC 2004, pp. 297–307 (2004)Google Scholar
  8. 8.
    Yu, H., Hancock, E.R.: Graph seriation using semi-definite programming. In: Brun, L., Vento, M. (eds.) GbRPR 2005. LNCS, vol. 3434, pp. 63–71. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Schellewald, C., Schnörr, C.: Probabilistic subgraph matching based on convex relaxation. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 171–186. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Torr, P.: Solving markov random fields using semi definite programming. In: Proc. Ninth International Workshop on Artificial Intelligence and Statistics (2003)Google Scholar
  11. 11.
    Zhu, Q., Shi, J.: Shape from shading: Recognizing the mountains through a global view. In: CVPR 2006, pp. 1839–1846 (2006)Google Scholar
  12. 12.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Forsyth, D.: Shape from texture and integrability. In: ICCV 2001, pp. 447–452 (2001)Google Scholar
  14. 14.
    Forsyth, D.: Shape from texture without boundaries. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 225–239. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Onn, R., Bruckstein, A.: Integrability disambiguates surface recovery in two-image photometric stereo. Int. J. Comput. Vision 5(1), 105–113 (1990)CrossRefGoogle Scholar
  16. 16.
    Naito, S., Rosenfeld, A.: Shape from random planar features. CVGIP 42(3), 345–370 (1988)Google Scholar
  17. 17.
    Koenderink, J., van Doorn, A.: Shape from chebyshev nets. In: Burkhardt, H., Neumann, B. (eds.) ECCV 1998. LNCS, vol. 1407, pp. 215–225. Springer, Heidelberg (1998)Google Scholar
  18. 18.
    Pentland, A.P.: A new sense for depth of field. PAMI 9(4), 523–531 (1987)CrossRefGoogle Scholar
  19. 19.
    Arora, S., Berger, E., Hazan, E., Kindler, G., Safra, M.: On non-approximability for quadratic programs. In: FOCS 2005, pp. 206–215 (2005)Google Scholar
  20. 20.
    Chan, T.F., Gilbert, J.R., Teng, S.H.: Geometric spectral partitioning. Technical Report Tech. Report CSL-94-15, Xerox PARC (1995)Google Scholar
  21. 21.
    Fjallstrom, P.O.: Algorithms for graph partitioning: A survey. Linkoping Electronic Articles in Computer and Information Science 3(10) (1998)Google Scholar
  22. 22.
    Burer, S., Monteiro, R.D.C., Zhang, Y.: Rank-two relaxation heuristics for max-cut and other binary quadratic programs. SIAM J. on Optimization 12(2), 503–521 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. The Bell system technical journal 49(1), 291–307 (1970)CrossRefzbMATHGoogle Scholar
  24. 24.
    Fiduccia, C., Mattheyses, R.: A linear-time heuristic for improving network partitions. In: Proc. 19th Design Automation Conference, pp. 175–181 (1982)Google Scholar
  25. 25.
    Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Krishnan, K., Mitchell, J.E.: A semidefinite programming based polyhedral cut and price approach for the maxcut problem. Comp. Optim. Appl. 33(1), 51–71 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Benson, S.J., Ye, Y.: Algorithm 875: DSDP5: Software for semidefinite programming. ACM Trans. Math. Software 34(3) (2008)Google Scholar
  28. 28.
    Frieze, A., Jerrum, M.: Improved approximation algorithms for maxk-cut and max bisection. Algorithmica 18(1), 67–81 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    de Klerk, E., Pasechnik, D.V., Warners, J.P.: On approximate graph colouring and max-k-cut algorithms based on the theta-function. J. Comb. Optim. 8(3), 267–294 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Technical report, Alpen-Adria-Universität Klagenfurt, Inst. f. Mathematik (2007)Google Scholar
  31. 31.
    Atick, J.J., Griffin, P.A., Redlich, A.N.: Statistical approach to shape from shading: Reconstruction of three-dimensional face surfaces from single two-dimensional images. Neural Computation 8(6), 1321–1340 (1996)CrossRefGoogle Scholar
  32. 32.
    Zhang, L., Dugas-Phocion, G., Samson, J.S., Seitz, S.M.: Single view modeling of free-form scenes. In: CVPR 2001, pp. 990–997 (2001)Google Scholar
  33. 33.
    White, R., Forsyth, D.: Combining cues: Shape from shading and texture. In: CVPR 2006, pp. 1809–1816 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ady Ecker
    • 1
  • Allan D. Jepson
    • 1
  • Kiriakos N. Kutulakos
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoCanada

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