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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
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)
Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D. 60, 259–268 (1992)
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)
Candes, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inform. Theory 51(12), 4203–4215 (2005)
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)
Lavery, J.E.: Solution of steady-state one-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 26(5), 1081–1089 (1989)
Lavery, J.E.: Solution of steady-state, two-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 28(1), 141–155 (1991)
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)
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)
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)
Guermond, J.L., Popov, B.: l1-minimization methods for Hamilton-Jacobi equations. SIAM J. Numer. Anal. (accepted)
Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)
Wright, S.J.: Primal-dual interior-point methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1997)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dobrev, V., Guermond, JL., Popov, B. (2009). Surface Reconstruction via L 1-Minimization. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2008. Lecture Notes in Computer Science, vol 5434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00464-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-00464-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00463-6
Online ISBN: 978-3-642-00464-3
eBook Packages: Computer ScienceComputer Science (R0)