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Surface Reconstruction via L1-Minimization

  • Veselin Dobrev
  • Jean-Luc Guermond
  • Bojan Popov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)

Abstract

A surface reconstruction technique based on the L 1- minimization of the variation of the gradient is introduced. This leads to a non-smooth convex programming problem. Well-posedness and convergence of the method is established and an interior point based algorithm is introduced. The L 1-surface reconstruction algorithm is illustrated on various test cases including natural and urban terrain data.

Keywords

Quadrature Rule Quadrature Point Preconditioned Conjugate Gradient Ematical Programming Gaussian Rule 
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.

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References

  1. 1.
    Lavery, J.E.: Shape-preserving interpolation of irregular data by bivariate curvature-based cubic \(L\sb 1\) splines in spherical coordinates. Comput. Aided Geom. Design 22(9), 818–837 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation. I. Fast and exact optimization. J. Math. Imaging Vision 26(3), 261–276 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D. 60, 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lavery, J.E.: Univariate cubic \(L\sb p\) splines and shape-preserving, multiscale interpolation by univariate cubic \(L\sb 1\) splines. Comput. Aided Geom. Design 17(4), 319–336 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Candes, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inform. Theory 51(12), 4203–4215 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lavery, J.E.: Solution of steady-state one-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 26(5), 1081–1089 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lavery, J.E.: Solution of steady-state, two-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 28(1), 141–155 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guermond, J.L.: A finite element technique for solving first-order PDEs in \(L\sp P\). SIAM J. Numer. Anal. 42(2), 714–737 (2004) (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Guermond, J.L., Popov, B.: Linear advection with ill-posed boundary conditions via L 1 minimization. Int. J. Numer. Anal. Model. 4(1), 39–47 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Guermond, J.L., Popov, B.: \(L\sp 1\)-minimization methods for Hamilton-Jacobi equations: the one-dimensional case. Numer. Math. 109(2), 269–284 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Guermond, J.L., Popov, B.: l1-minimization methods for Hamilton-Jacobi equations. SIAM J. Numer. Anal. (accepted)Google Scholar
  14. 14.
    Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  15. 15.
    Wright, S.J.: Primal-dual interior-point methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  16. 16.
    Wang, Y., Fang, S.C., Lavery, J.E.: A compressed primal-dual method for generating bivariate cubic \(L\sb 1\) splines. J. Comput. Appl. Math. 201(1), 69–87 (2007)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Veselin Dobrev
    • 1
  • Jean-Luc Guermond
    • 1
  • Bojan Popov
    • 1
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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