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TVL\({}_1\) Planarity Regularization for 3D Shape Approximation

  • Eugen FunkEmail author
  • Laurence S. Dooley
  • Anko Börner
Part of the Communications in Computer and Information Science book series (CCIS, volume 598)

Abstract

The modern emergence of automation in many industries has given impetus to extensive research into mobile robotics. Novel perception technologies now enable cars to drive autonomously, tractors to till a field automatically and underwater robots to construct pipelines. An essential requirement to facilitate both perception and autonomous navigation is the analysis of the 3D environment using sensors like laser scanners or stereo cameras. 3D sensors generate a very large number of 3D data points when sampling object shapes within an environment, but crucially do not provide any intrinsic information about the environment which the robots operate within.

This work focuses on the fundamental task of 3D shape reconstruction and modelling from 3D point clouds. The novelty lies in the representation of surfaces by algebraic functions having limited support, which enables the extraction of smooth consistent implicit shapes from noisy samples with a heterogeneous density. The minimization of total variation of second differential degree makes it possible to enforce planar surfaces which often occur in man-made environments. Applying the new technique means that less accurate, low-cost 3D sensors can be employed without sacrificing the 3D shape reconstruction accuracy.

Keywords

Radial Basis Function Move Little Square Radial Basis Function Shape Approximation Total Variation Regularization 
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.
    Agoston, M.: Computer Graphics and Geometric Modelling: Implementation & Algorithms, 1st edn. Springer, London (2005)zbMATHGoogle Scholar
  2. 2.
    Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.T.: Point set surfaces. In: Proceedings of the Conference on Visualization 2001, VIS 2001, pp. 21–28. IEEE Computer Society, Washington, DC, USA (2001)Google Scholar
  3. 3.
    Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.T.: Computing and rendering point set surfaces. IEEE Trans. Vis. Comput. Graph. 9(1), 3–15 (2003)CrossRefGoogle Scholar
  4. 4.
    Alizadeh, F., Alizadeh, F., Goldfarb, D., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bach, F.R., Jenatton, R., Mairal, J., Obozinski, G.: Optimization with sparsity-inducing penalties. Found. Trends Mach. Learn. 4(1), 1–106 (2012)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bernardini, F., Mittleman, J., Rushmeier, H., Silva, C., Taubin, G.: The ball-pivoting algorithm for surface reconstruction. IEEE Trans. Vis. Comput. Graph. 5(4), 349–359 (1999)CrossRefGoogle Scholar
  7. 7.
    Bodenmüller, T.: Streaming surface reconstruction from real time 3D-measurements. Ph.D. thesis, Technical University Munich (2009)Google Scholar
  8. 8.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Img. Sci. 3(3), 492–526 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bregman, L.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7(3), 200–217 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Calakli, F., Taubin, G.: SSD: smooth signed distance surface reconstruction. Comput. Graph. Forum 30(7), 1993–2002 (2011)CrossRefGoogle Scholar
  12. 12.
    Canelhas, D.R.: Scene representation, registration and object detection in a truncated signed distance function representation of 3D space. Ph.D. thesis, Örebro University (2012)Google Scholar
  13. 13.
    Canelhas, D.R., Stoyanov, T., Lilienthal, A.J.: SDF tracker: a parallel algorithm for on-line pose estimation and scene reconstruction from depth images. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3671–3676. IEEE (2013)Google Scholar
  14. 14.
    Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3D objects with radial basis functions. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 2001, pp. 67–76. ACM, New York, NY, USA (2001)Google Scholar
  15. 15.
    Chen, X., Lin, Q., Kim, S., Peña, J., Carbonell, J.G., Xing, E.P.: An efficient proximal-gradient method for single and multi-task regression with structured sparsity. CoRR, abs/1005.4717 (2010)
  16. 16.
    Duchon, J.: Splines minimizing rotation-invariant semi-norms in sobolev spaces. In: Schempp, W., Zeller, K. (eds.) Constructive Theory of Functions of Several Variables. Lecture Notes in Mathematics, vol. 571, pp. 85–100. Springer, Berlin (1977)CrossRefGoogle Scholar
  17. 17.
    Dykstra, R.: An algorithm for restricted least squares regression. Technical report, Mathematical Sciences. University of Missouri-Columbia, Department of Statistics (1982)Google Scholar
  18. 18.
    Edelsbrunner, H., Mücke, E.P.: Three-dimensional alpha shapes. ACM Trans. Graph. 13(1), 43–72 (1994)CrossRefzbMATHGoogle Scholar
  19. 19.
    Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32, 407–499 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Friedman, J.H., Hastie, T., Tibshirani, R.: Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33(1), 1–22 (2010)CrossRefGoogle Scholar
  21. 21.
    Getreuer, P.: Rudin-Osher-Fatemi total variation denoising using split bregman. Image Process. On Line 2, 74–95 (2012)CrossRefGoogle Scholar
  22. 22.
    Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM J. Img. Sci. 2(2), 323–343 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 1st edn. Springer Publishing Company, London (2009)CrossRefzbMATHGoogle Scholar
  24. 24.
    Guennebaud, G., Gross, M.: Algebraic point set surfaces. ACM Trans. Graph. 26(3), July 2007Google Scholar
  25. 25.
    Hägele, M.: Wirtschaftlichkeitsanalysen neuartiger Servicerobotik-Anwendungen und ihre Bedeutung für die Robotik-Entwicklung (2011)Google Scholar
  26. 26.
    Hirschmüller, H.: Semi-global matching - motivation, developments and applications. In: Fritsch, D. (ed.) Photogrammetric Week, pp. 173–184. Wichmann, Heidelberg (2011)Google Scholar
  27. 27.
    Hughes, J., Foley, J., van Dam, A., Feiner, S.: Computer Graphics: Principles and Practice. The Systems Programming Series. Addison-Wesley, Boston (2014)zbMATHGoogle Scholar
  28. 28.
    Kazhdan, M., Hoppe, H.: Screened poisson surface reconstruction. ACM Trans. Graph. 32(3), 29:1–29:13 (2013)CrossRefzbMATHGoogle Scholar
  29. 29.
    Ohtake, Y., Belyaev, A., Alexa, M., Turk, G., Seidel, H.-P.: Multi-level partition of unity implicits. ACM Trans. Graph. 22(3), 463–470 (2003)CrossRefGoogle Scholar
  30. 30.
    Open Source Community. Blender, open source film production software (2014). http://blender.org. Accessed on 6 June 2014
  31. 31.
    Oztireli, C., Guennebaud, G., Gross, M.: Feature preserving point set surfaces based on non-linear kernel regression. Comput. Graph. Forum 28(2), 493–501 (2009)CrossRefGoogle Scholar
  32. 32.
    Piegl, L., Tiller, W.: The NURBS Book. Monographs in Visual Communication. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  33. 33.
    Rogers, D.F.: Preface. In: Rogers, D.F. (ed.) An Introduction to NURBS. The Morgan Kaufmann Series in Computer Graphics, pp. 105–107. Morgan Kaufmann, San Francisco (2001)Google Scholar
  34. 34.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  36. 36.
    Schölkopf, B., Smola, A.J.: Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge (2001)Google Scholar
  37. 37.
    Tennakoon, R., Bab-Hadiashar, A., Suter, D., Cao, Z.: Robust data modelling using thin plate splines. In: 2013 International Conference on Digital Image Computing, Techniques and Applications (DICTA), pp. 1–8. November 2013Google Scholar
  38. 38.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. Ser. B 58, 267–288 (1994)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Wahba, G.: Spline models for observational data, vol. 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1990)Google Scholar
  40. 40.
    Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4(1), 389–396 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  42. 42.
    Wolff, D.: OpenGL 4 Shading Language Cookbook, 2nd edn. Packt Publishing, Birmingham (2013)Google Scholar
  43. 43.
    Zach, C., Pock, T., Bischof, H.: A globally optimal algorithm for robust TV-L1 range image integration. In: IEEE 11th International Conference on Computer Vision, 2007, ICCV 2007, pp. 1–8, October 2007Google Scholar
  44. 44.
    Zhao, H., Oshery, S., Fedkiwz, R.: Fast surface reconstruction using the level set method. In: Proceedings of the IEEE Workshop on Variational and Level Set Methods, VLSM 2001 (2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Eugen Funk
    • 1
    • 2
    Email author
  • Laurence S. Dooley
    • 1
  • Anko Börner
    • 2
  1. 1.Department of Computing and CommunicationsThe Open UniversityMilton KeynesUK
  2. 2.Department of Information Processing for Optical Systems, Institute of Optical Sensor SystemsGerman Aerospace Center (DLR)BerlinGermany

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