Shape Transformation of Multidimensional Density Functions using Distribution Interpolation of the Radon Transforms

Márton József Tóth, Balázs Csébfavi

2014

Abstract

In this paper, we extend 1D distribution interpolation to 2D and 3D by using the Radon transform. Our algorithm is fundamentally different from previous shape transformation techniques, since it considers the objects to be interpolated as density distributions rather than level sets of Implicit Functions (IF). First, we perform distribution interpolation on the precalculated Radon transforms of two different density functions, and then an intermediate density function is obtained by an inverse Radon transform. This approach guarantees a smooth transition along all the directions the Radon transform is calculated for. Unlike the IF methods, our technique is able to interpolate between features that do not even overlap and it does not require a one dimension higher object representation. We will demonstrate that these advantageous properties can be well exploited for 3D modeling and metamorphosis.

References

  1. Akmal Butt, M. and Maragos, P. (1998). Optimum design of chamfer distance transforms. IEEE Transactions on Image Processing, 7(10):1477-1484.
  2. Bajaj, C. L., Coyle, E. J., and nan Lin, K. (1996). Arbitrary topology shape reconstruction from planar cross sections. In Graphical Models and Image Processing, pages 524-543.
  3. Beier, T. and Neely, S. (1992). Feature-based image metamorphosis. SIGGRAPH Computer Graphics, 26(2):35-42.
  4. Borgefors, G. (1986). Distance transformations in digital images. Computer Vision, Graphics, and Image Processing, 34(3):344-371.
  5. Buhmann, M. (2009). Radial Basis Functions: Theory and Implementations. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press.
  6. Cheng, S.-W. and Dey, T. K. (1999). Improved constructions of delaunay based contour surfaces. In Proc. ACM Sympos. Solid Modeling and Applications 99, pages 322-323.
  7. Cohen-Or, D., Solomovic, A., and Levin, D. (1998). Threedimensional distance field metamorphosis. ACM Transactions on Graphics, 17(2):116-141.
  8. Csébfalvi, B., Neumann, L., Kanitsar, A., and Gr öller, E. (2002). Smooth shape-based interpolation using the conjugate gradient method. In Proceedings of Vision, Modeling, and Visualization, pages 123-130.
  9. Deans, S. R. (1983). The Radon Transform and Some of Its Applications. Krieger Publishing.
  10. Fang, S., Srinivasan, R., Raghavan, R., and Richtsmeier, J. T. (2000). Volume morphing and rendering - an integrated approach. Computer Aided Geometric Design, 17(1):59-81.
  11. Grevera, G. J. and Udupa, J. K. (1996). Shape-based interpolation of multidimensional grey-level images. IEEE Transactions on Medical Imaging, 15(6):881-92.
  12. Herman, G. T., Zheng, J., and Bucholtz, C. A. (1992). Shape-based interpolation. IEEE Computer Graphics and Applications, 12(3):69-79.
  13. Jones, M. W., Brentzen, J. A., and Sramek, M. (2006). 3D distance fields: A survey of techniques and applications. IEEE Transactions on Visualization and Computer Graphics, 12:581-599.
  14. Kak, A. C. and Slaney, M. (1988). Principles of Computerized Tomographic Imaging. IEEE Press.
  15. Kniss, J., Premoze, S., Hansen, C., and Ebert, D. (2002). Interactive Translucent Volume Rendering and Procedural Modeling. In Proceedings of IEEE Visualization Conference (VIS) 2002, pages 109-116.
  16. Lerios, A., Garfinkle, C. D., and Levoy, M. (1995). Featurebased volume metamorphosis. In Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, SIGGRAPH 7895, pages 449- 456.
  17. Liu, L., Bajaj, C., Deasy, J., Low, D. A., and Ju, T. (2008). Surface reconstruction from non-parallel curve networks. Computer Graphics Forum, 27(2):155-163.
  18. Lorensen, W. E. and Cline, H. E. (1987). Marching cubes: A high resolution 3d surface construction algorithm. Computer Graphics, 21(4):163-169.
  19. Neumann, L., Csébfalvi, B., Viola, I., Mlejnek, M., and Gröller, E. (2002). Feature-Preserving Volume Filtering . In VisSym 2002 : Joint Eurographics - IEEE TCVG Symposium on Visualization, pages 105-114.
  20. Raya, S. and Udupa, J. (1990). Shape-based interpolation of multidimensional objects. IEEE Transactions on Medical Imaging, 9(1):32-42.
  21. Read, A. L. (1999). Linear interpolation of histograms. Nuclear Instruments and Methods, A425:357-360.
  22. Schneider, J., Kraus, M., and Westermann, R. (2009). GPUbased real-time discrete euclidean distance transforms with precise error bounds. In International Conference on Computer Vision Theory and Applications (VISAPP), pages 435-442.
  23. Treece, G. M., Prager, R. W., Gee, A. H., and Berman, L. H. (2000). Surface interpolation from sparse crosssections using region correspondence. IEEE Transactions on Medical Imaging, 19(11):1106-1114.
  24. Turk, G. and O'Brien, J. F. (1999). Shape transformation using variational implicit functions. In Proceedings of ACM SIGGRAPH 1999, pages 335-342.
  25. Turk, G. and O'brien, J. F. (2002). Modelling with implicit surfaces that interpolate. ACM Transanctions on Graphics, 21(4):855-873.
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Paper Citation


in Harvard Style

Tóth M. and Csébfavi B. (2014). Shape Transformation of Multidimensional Density Functions using Distribution Interpolation of the Radon Transforms . In Proceedings of the 9th International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2014) ISBN 978-989-758-002-4, pages 5-12. DOI: 10.5220/0004640800050012


in Bibtex Style

@conference{grapp14,
author={Márton József Tóth and Balázs Csébfavi},
title={Shape Transformation of Multidimensional Density Functions using Distribution Interpolation of the Radon Transforms},
booktitle={Proceedings of the 9th International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2014)},
year={2014},
pages={5-12},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0004640800050012},
isbn={978-989-758-002-4},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 9th International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2014)
TI - Shape Transformation of Multidimensional Density Functions using Distribution Interpolation of the Radon Transforms
SN - 978-989-758-002-4
AU - Tóth M.
AU - Csébfavi B.
PY - 2014
SP - 5
EP - 12
DO - 10.5220/0004640800050012