TVL1 Shape Approximation from Scattered 3D Data

Eugen Funk, Laurence S. Dooley, Anko Boerner

2015

Abstract

With the emergence in 3D sensors such as laser scanners and 3D cameras, large 3D point clouds can now be sampled from physical objects within a scene. The raw 3D samples delivered by these sensors however, do not contain any information about the environment the objects exist in, which means that further geometrical high-level modelling is essential. In addition, issues like sparse data measurements, noise, missing samples due to occlusion, and the inherently huge datasets involved in such representations makes this task extremely challenging. This paper addresses these issues by presenting a new 3D shape modelling framework for samples acquired from 3D sensor. Motivated by the success of nonlinear kernel-based approximation techniques in the statistics domain, existing methods using radial basis functions are applied to 3D object shape approximation. The task is framed as an optimization problem and is extended using non-smooth L1 total variation regularization. Appropriate convex energy functionals are constructed and solved by applying the Alternating Direction Method of Multipliers approach, which is then extended using Gauss-Seidel iterations. This significantly lowers the computational complexity involved in generating 3D shape from 3D samples, while both numerical and qualitative analysis confirms the superior shape modelling performance of this new framework compared with existing 3D shape reconstruction techniques.

References

  1. Agoston, M. (2005). Computer Graphics and Geometric Modelling: Implementation & Algorithms. Computer Graphics and Geometric Modeling. Springer.
  2. Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., and Silva, C. T. (2001). Point set surfaces. In Proceedings of the Conference on Visualization 7801, VIS 7801, pages 21-28, Washington, DC, USA. IEEE Computer Society.
  3. Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., and Silva, C. T. (2003). Computing and rendering point set surfaces. Visualization and Computer Graphics, IEEE Transactions on, 9(1):3-15.
  4. Alizadeh, F., Alizadeh, F., Goldfarb, D., and Goldfarb, D. (2003). Second-order cone programming. Mathematical Programming, 95:3-51.
  5. Bach, F. R., Jenatton, R., Mairal, J., and Obozinski, G. (2012). Optimization with sparsity-inducing penalties. Foundations and Trends in Machine Learning, 4(1):1-106.
  6. Bernardini, F., Mittleman, J., Rushmeier, H., Silva, C., and Taubin, G. (1999). The ball-pivoting algorithm for surface reconstruction. IEEE Transactions on Visualization and Computer Graphics, 5(4):349-359.
  7. Bodenmüller, T. (2009). Streaming surface reconstruction from real time 3D-measurements. PhD thesis, Technical University Munich.
  8. Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn., 3(1):1-122.
  9. Bredies, K., Kunisch, K., and Pock, T. (2010). Total generalized variation. SIAM J. Img. Sci., 3(3):492-526.
  10. Bregman, L. (1967). The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. fUSSRg Computational Mathematics and Mathematical Physics, 7(3):200 - 217.
  11. Calakli, F. and Taubin, G. (2011). SSD: Smooth signed distance surface reconstruction. Computer Graphics Forum, 30(7):1993-2002.
  12. Canelhas, D. R. (2012). Scene Representation, Registration and Object Detection in a Truncated Signed Distance Function Representation of 3D Space. PhD thesis, O rebro University.
  13. Canelhas, D. R., Stoyanov, T., and Lilienthal, A. J. (2013). Sdf tracker: A parallel algorithm for on-line pose estimation and scene reconstruction from depth images. In Intelligent Robots and Systems (IROS), 2013 IEEE/RSJ International Conference on, pages 3671-3676. IEEE.
  14. Carr, J. C., Beatson, R. K., Cherrie, J. B., Mitchell, T. J., Fright, W. R., McCallum, B. C., and Evans, T. R. (2001). Reconstruction and representation of 3d objects with radial basis functions. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 7801, pages 67- 76, New York, NY, USA. ACM.
  15. Chen, X., Lin, Q., Kim, S., Pen˜a, J., Carbonell, J. G., and Xing, E. P. (2010). An efficient proximalgradient method for single and multi-task regression with structured sparsity. CoRR, abs/1005.4717.
  16. Duchon, J. (1977). Splines minimizing rotation-invariant semi-norms in sobolev spaces. In Schempp, W. and Zeller, K., editors, Constructive Theory of Functions of Several Variables, volume 571 of Lecture Notes in Mathematics, pages 85-100. Springer Berlin Heidelberg.
  17. Dykstra, R. (1982). An Algorithm for Restricted Least Squares Regression. Technical report, mathematical sciences. University of Missouri-Columbia, Department of Statistics.
  18. Edelsbrunner, H. and Mücke, E. P. (1994). Threedimensional alpha shapes. ACM Trans. Graph., 13(1):43-72.
  19. Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004). Least angle regression. Annals of Statistics, 32:407-499.
  20. Friedman, J. H., Hastie, T., and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1):1-22.
  21. Getreuer, P. (2012). Rudin-Osher-Fatemi Total Variation Denoising using Split Bregman. Image Processing On Line, 2:74-95.
  22. Goldstein, T. and Osher, S. (2009). The split bregman method for l1-regularized problems. SIAM J. Img. Sci., 2(2):323-343.
  23. Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., and Galbraith, C. (2009). Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms. Springer Publishing Company, Incorporated, 1st edition.
  24. Guennebaud, G. and Gross, M. (2007). Algebraic point set surfaces. ACM Trans. Graph., 26(3).
  25. Hägele, M. (2011). Wirtschaftlichkeitsanalysen neuartiger Servicerobotik-Anwendungen und ihre Bedeutung für die Robotik-Entwicklung.
  26. Hirschmüller, H. (2011). Semi-global matching - motivation, developments and applications. In Fritsch, D., editor, Photogrammetric Week, pages 173-184. Wichmann.
  27. Hughes, J., Foley, J., van Dam, A., and Feiner, S. (2014). Computer Graphics: Principles and Practice. The systems programming series. Addison-Wesley.
  28. Kazhdan, M. and Hoppe, H. (2013). Screened poisson surface reconstruction. ACM Trans. Graph., 32(3):29:1- 29:13.
  29. Ohtake, Y., Belyaev, A., Alexa, M., Turk, G., and Seidel, H.-P. (2003). Multi-level partition of unity implicits. ACM Trans. Graph., 22(3):463-470.
  30. Open Source Community (2014). Blender, open source film production software. http://blender.org. Accessed: 2014-06-6.
  31. Oztireli, C., Guennebaud, G., and Gross, M. (2009). Feature Preserving Point Set Surfaces based on NonLinear Kernel Regression. Computer Graphics Forum, 28(2):493-501.
  32. Piegl, L. and Tiller, W. (1997). The NURBS Book. Monographs in Visual Communication. U.S. Government Printing Office.
  33. Rogers, D. F. (2001). Preface. In Rogers, D. F., editor, An Introduction to NURBS, The Morgan Kaufmann Series in Computer Graphics, pages xv - xvii. Morgan Kaufmann, San Francisco.
  34. Rudin, L. I., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Phys. D, 60(1-4):259-268.
  35. Saad, Y. (2003). Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2nd edition.
  36. Schölkopf, B. and Smola, A. J. (2001). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MA, USA.
  37. Tennakoon, R., Bab-Hadiashar, A., Suter, D., and Cao, Z. (2013). Robust data modelling using thin plate splines. In Digital Image Computing: Techniques and Applications (DICTA), 2013 International Conference on, pages 1-8.
  38. Tibshirani, R. (1994). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58:267-288.
  39. Wahba, G. (1990). Spline models for observational data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  40. Wendland, H. (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, 4(1):389-396.
  41. Wendland, H. (2004). Scattered Data Approximation. Cambridge University Press.
  42. Wolff, D. (2013). OpenGL 4 Shading Language Cookbook, Second Edition. EBL-Schweitzer. Packt Publishing.
  43. Zach, C., Pock, T., and Bischof, H. (2007). A globally optimal algorithm for robust tv-l1 range image integration. In Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference on, pages 1-8.
  44. Zhao, H., Oshery, S., and Fedkiwz, R. (2001). Fast surface reconstruction using the level set method. In In VLSM 01: Proceedings of the IEEE Workshop on Variational and Level Set Methods.
Download


Paper Citation


in Harvard Style

Funk E., Dooley L. and Boerner A. (2015). TVL1 Shape Approximation from Scattered 3D Data . In Proceedings of the 10th International Conference on Computer Vision Theory and Applications - Volume 3: VISAPP, (VISIGRAPP 2015) ISBN 978-989-758-091-8, pages 294-304. DOI: 10.5220/0005301802940304


in Bibtex Style

@conference{visapp15,
author={Eugen Funk and Laurence S. Dooley and Anko Boerner},
title={TVL1 Shape Approximation from Scattered 3D Data},
booktitle={Proceedings of the 10th International Conference on Computer Vision Theory and Applications - Volume 3: VISAPP, (VISIGRAPP 2015)},
year={2015},
pages={294-304},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005301802940304},
isbn={978-989-758-091-8},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 10th International Conference on Computer Vision Theory and Applications - Volume 3: VISAPP, (VISIGRAPP 2015)
TI - TVL1 Shape Approximation from Scattered 3D Data
SN - 978-989-758-091-8
AU - Funk E.
AU - Dooley L.
AU - Boerner A.
PY - 2015
SP - 294
EP - 304
DO - 10.5220/0005301802940304