Advertisement

Sparse-Data Based 3D Surface Reconstruction for Cartoon and Map

  • Bin WuEmail author
  • Talal Rahman
  • Xue-Cheng Tai
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

A model combining the first-order and the second-order variational regularizations for the purpose of 3D surface reconstruction based on 2D sparse data is proposed. The model includes a hybrid fidelity constraint which allows the initial conditions to be switched flexibly between vectors and elevations. A numerical algorithm based on the augmented Lagrangian method is also proposed. The numerical experiments are presented, showing its excellent performance both in designing cartoon characters, as well as in recovering oriented three dimensional maps from contours or points with elevation information.

Notes

Acknowledgements

XC Tai acknowledges the support from Norwegian Research Council through ISP-Matematikk (Project no. 239033/F20). The authors also thank Dr. Jie Qiu for providing us example strokes.

References

  1. 1.
    R.C. Zeleznik, K.P. Herndon, J.F. Hughes, Sketch: an interface for sketching 3d scenes, in The 23rd Annual Conference on Computer Graphics and Interactive Techniques (1996), pp. 163–170Google Scholar
  2. 2.
    T. Igarashi, S. Matsuoka, H. Tanaka, Teddy: a sketching interface for 3d freeform design, in The 26th Annual Conference on Computer Graphics and Interactive Techniques (1999), pp. 409–416Google Scholar
  3. 3.
    O.A. Karpenko, J.F. Hughes, SmoothSketch: 3D free-form shapes from complex sketches, in The 33th Annual Conference on Computer Graphics and Interactive Techniques (2006), pp. 589–598Google Scholar
  4. 4.
    A. Nealen, T. Igarashi, O. Sorkine, M. Alexa, FiberMesh: designing freeform surfaces with 3D curves. ACM Trans. Graph. 26(3), Article No. 41 (2007)CrossRefGoogle Scholar
  5. 5.
    L. Olsen, F.F. Samavati, M.C. Sousa, J.A. Jorge, Sketch-based modeling: a survey. Comput. Graph. 33(1), 88–103 (2009)CrossRefGoogle Scholar
  6. 6.
    Y. Gingold, T. Igarashi, D. Zorin, Structured annotations for 2d-to-3d modeling. ACM Trans. Graph. 28(5), Article No. 148 (2009)CrossRefGoogle Scholar
  7. 7.
    L. Olsen, F.F. Samavati, M.C. Sousa, J.A. Jorge, Sketch-based mesh augmentation, in The 2nd Eurographics Workshop on Sketch-Based Interfaces and Modeling (2005)Google Scholar
  8. 8.
    J. Hahn, J. Qiu, E. Sugisaki, L. Jia, X.-C. Tai, H. Seah, Stroke-based surface reconstruction. CAM Report 12–18, UCLA (2012)Google Scholar
  9. 9.
    J. Hahn, J. Qiu, E. Sugisaki, L. Jia, X.-C. Tai, H. Seah, Stroke-based surface reconstruction. Numer. Math. Theory Meth. Appl. 6(1), 297–324 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. Agrawal, R. Raskar, R. Chellappa, What is the range of surface reconstructions from a gradient field?, in Computer Vision C ECCV (2006), pp. 578–591Google Scholar
  11. 11.
    R.T. Frankot, R. Chellappa, S. Member, A method for enforcing integrability in shape from shading algorithms. IEEE Trans. Pattern Anal. Mach. Intell. 10, 118–128 (1987)Google Scholar
  12. 12.
    N. Petrovic, I. Cohen, B.J. Frey, R. Koetter, T.S. Huang, Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models, in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1 (2001), p. 743Google Scholar
  13. 13.
    T. Simchony, R. Chellappa, M. Shao, Direct analytical methods for solving Poisson equations in computer vision problems. IEEE Trans. Pattern Anal. Mach. Intell. 12, 435–446 (1990)CrossRefGoogle Scholar
  14. 14.
    L. Zhang, G. Dugas-Phocion, J.-S. Samson, Single-view modeling of free-form scenes. J. Vis. Comput. Anim. 13, 225–235 (2002)CrossRefGoogle Scholar
  15. 15.
    T.-P. Wu, C.-K. Tang, M. Brown, H.-Y. Shum, Shapepalettes: interactive normal transfer visa sketching. ACM Trans. Graph. 26(3), 07, Article No. 44CrossRefGoogle Scholar
  16. 16.
    M. Prasad, A. Fitzgibbon, Single view reconstruction of curved surfaces, in The 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2 (2006), pp. 1345–1354Google Scholar
  17. 17.
    H.-S. Ng, T.-P. Wu, C.-K. Tang, Surface-from-gradients without discrete integrability enforcement: a Gaussian kernel approach. IEEE Trans. Pattern Anal. Mach. Intell. 32(11), 2085–2099 (2010)CrossRefGoogle Scholar
  18. 18.
    T. Rahman, X.C. Tai, S. Osher, A TV-stokes denoising algorithm, in Scale Space and Variational Methods in Computer Vision (Springer, Berlin, 2007), pp. 473–483CrossRefGoogle Scholar
  19. 19.
    C.A. Elo, A. Malyshev, T. Rahman, A dual formulation of the TV-stokes algorithm for image denoising, in Scale Space and Variational Methods in Computer Vision (Springer, Berlin, 2009), pp. 307–318CrossRefGoogle Scholar
  20. 20.
    X.C. Tai, S. Borok, J. Hahn, Image denoising using TV-Stokes equation with an orientation-matching minimization, in International Conference on Scale Space and Variational Methods in Computer Vision (Springer, Berlin, 2009), pp. 490–501zbMATHGoogle Scholar
  21. 21.
    J. Hahn, X.C. Tai, S. Borok, A.M. Bruckstein, Orientation-matching minimization for image denoising and inpainting. Int. J. Comput. Vis. 92(3), 308–324 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    W.G. Litvinov, T. Rahman, X.C. Tai, A modified TV-stokes model for image processing. SIAM Sci. Comput. 33(4), 1574–1597 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    S.F. Johnston, Lumo: illumination for CEL animation, in The 2nd International Symposium on Non-photorealistic Animation and Rendering (ACM, New York, 2002), pp. 45–52Google Scholar
  24. 24.
    D. Meyers, S. Skinner, K. Sloan, Surfaces from contours. Trans. Graph. 11(3), 228–258 (1992)CrossRefGoogle Scholar
  25. 25.
    S. Masnou, J. Morel, Level lines based disocclusion, in 5th IEEE International Conference on Image Processing, Chicago, Oct 4–7 (1998), pp. 259–263Google Scholar
  26. 26.
    T. Meyer, Coastal elevation from sparse level curves. Summer project under the guidance of T. Wittman, A. Bertozzi, and A. Chen, UCLA (2011)Google Scholar
  27. 27.
    L. Alvarez, F. Guichard, P.L. Lions, J.M. Morel, Axioms and fundamental equations of image processing. Arch. Ration. Mech. 123, 199–257 (1993)MathSciNetCrossRefGoogle Scholar
  28. 28.
    V. Caselles, J.-M. Morel, C. Sbert, An axiomatic approach to image interpolation. Trans. Image Proc. 7(3), 376–386 (1998)MathSciNetCrossRefGoogle Scholar
  29. 29.
    R. Franke, Scattered data interpolation: test of some methods. Math. Comput. 38, 181–200 (1982)MathSciNetzbMATHGoogle Scholar
  30. 30.
    J. Meinguet, Approximation theory and spline functions, in Surface Spline Interpolation: Basic Theory and Computational Aspects (Holland, Dordrecht, 1984), pp. 124–142zbMATHGoogle Scholar
  31. 31.
    J.C. Carr, W.R. Fright, R.K. Beatson, Surface interpolation with radial basis functions for medical imaging. Trans. Med. Imaging 16(1), 96–107 (1997)CrossRefGoogle Scholar
  32. 32.
    L. Mitas, H. Mitasova, Spatial Interpolation (Wiley, New York, 1999)zbMATHGoogle Scholar
  33. 33.
    J. Lellmann, J.M. Morel, C.-B. Schönlieb, Anisotropic Third-Order Regularization for Sparse Digital Elevation Models (Springer, Berlin, 2013)CrossRefGoogle Scholar
  34. 34.
    R. Glowinski, P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (Society for Industrial and Applied Mathematics, Philadelphia, 1989)CrossRefGoogle Scholar
  35. 35.
    X.-C. Tai, C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, in SSVM, ed. by X.-C. Tai, K. Mrken, M. Lysaker, K.-A. Lie. Lecture Notes in Computer Science, vol. 5567 (Springer, Berlin, 2009), pp. 502–513Google Scholar
  36. 36.
    C.L. Wu, J.Y. Zhang, X.C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Prob. Imaging 5(1), 237–261 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    C.D. Meyer, Matrix Analysis and Applied Linear Algebra, vol. 2 (SIAM, Philadelphia, 2000)CrossRefGoogle Scholar
  38. 38.
    Y. Wang, J. Yang, W. Yin, Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    C. Van Loan, Computational Frameworks for the Fast Fourier Transform (Society for Industrial and Applied Mathematics, Philadelphia, 1992)CrossRefGoogle Scholar
  40. 40.
    C.A. Elo, Image denoising algorithms based on the dual formulation of total variation, Master thesis. University of Bergen, 2009Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computing, Mathematics and PhysicsWestern Norway University of Applied SciencesBergenNorway
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

Personalised recommendations