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Volumetric Analysis of Digital Objects Using Distance Transformation: Performance Issues and Extensions

  • Conference paper
Applications of Discrete Geometry and Mathematical Morphology (WADGMM 2010)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 7346))

Abstract

In binary images, the distance transformation (DT) and the geometrical medial axis are classic tools for shape analysis. In the digital geometry literature, recent articles have demonstrated that fast algorithms can be designed without any approximation of the Euclidean metric. The aim of the paper is to first give an overview of separable techniques to compute the distance transformation, the reverse distance transformation and a discrete medial axis extraction with the Euclidean metric. Then we will focus on performance issues and different extensions of these techniques.

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References

  1. Aurenhammer, F.: Power Diagrams: Properties, Algorithms, and Applications. SIAM Journal on Computing 16, 78–96 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics, and Image Processing 34(3), 344–371 (1986)

    Article  Google Scholar 

  3. Borgefors, G.: Hierarchical chamfer matching: a parametric edge matching algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence 10(6), 849–865 (1988), http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=9107

    Article  Google Scholar 

  4. Breu, H., Gil, J., Kirkpatrick, D., Werman, M.: Linear time euclidean distance transform algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(5), 529–533 (1995)

    Article  Google Scholar 

  5. Cai, J., Chu, J., Recine, D., Sharam, M., Nguyeb, C., Rodebaugh, R., Saxena, V., Ali, A.: CT and PET lung image registration and fusion in radiotherapy treatment planning using the chamfer-matching method. International Journal of Radiation Oncology Biology Physics 43(4), 883–891 (1999), http://linkinghub.elsevier.com/retrieve/pii/S036030169800399X

    Article  Google Scholar 

  6. Cao, T.T., Tang, K., Mohamed, A., Tan, T.: Parallel Banding Algorithm to compute exact distance transform with the GPU. In: Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games, vol. (2), pp. 83–90. ACM, New York (2010), http://portal.acm.org/citation.cfm?id=1730804.1730818

    Chapter  Google Scholar 

  7. Chaussard, J., Bertrand, G., Couprie, M.: Characterizing and Detecting Toric Loops in n-Dimensional Discrete Toric Spaces. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 129–140. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  8. Ciesielski, K.C., Chen, X., Udupa, J.K., Grevera, G.J.: Linear Time Algorithms for Exact Distance Transform. Journal of Mathematical Imaging and Vision 39(3), 193–209 (2010), http://www.springerlink.com/index/10.1007/s10851-010-0232-4

    Article  MathSciNet  Google Scholar 

  9. Coeurjolly, D.: Distance Transformation, Reverse Distance Transformation and Discrete Medial Axis on Toric Spaces. In: 19th International Conference on Pattern Recognition, ICPR 2008, pp. 1–4. IEEE Computer Society (December 2008), http://scholar.google.com/scholar?hl=en&btnG=Search&q=intitle:Distance+transformation,+reverse+distance+transformation+and+discrete+medial+axis+on+toric+spaces#0

  10. Coeurjolly, D., Montanvert, A.: Optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(3), 437–448 (2007)

    Article  Google Scholar 

  11. Couprie, M., Coeurjolly, D., Zrour, R.: Discrete bisector function and Euclidean skeleton in 2D and 3D. Image and Vision Computing 25, 1543–1556 (2007), http://linkinghub.elsevier.com/retrieve/pii/S0262885606003064

    Article  Google Scholar 

  12. Cuisenaire, O., Macq, B.: Fast Euclidean distance transformations by propagation using multiple neighbourhoods. Computer Vision and Image Understanding 76, 163–172 (1999)

    Article  Google Scholar 

  13. Culver, T., Keyser, J., Lin, M., Manocha, D.: Fast Computation of Generalized Voronoi Diagrams Using Graphics Hardware. In: International Conference on Computer Graphics and Interactive Techniques, pp. 277–286 (1999)

    Google Scholar 

  14. Danielsson, P.E.: Euclidean distance mapping. Computer Graphics and Image Processing 14, 227–248 (1980)

    Article  Google Scholar 

  15. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry. Springer (2000)

    Google Scholar 

  16. Fabbri, R., da Fontoura Costa, L., Torelli, J.C., Bruno, O.M.: 2D euclidean distance transform algorithms: A comparative survey. ACM Computing Surveys 40(1), 1–44 (2008), http://doi.acm.org/10.1145/1322432.1322434

    Article  Google Scholar 

  17. Fouard, C., Malandain, G.: 3-D chamfer distances and norms in anisotropic grids. Image and Vision Computing 23, 143–158 (2005)

    Article  Google Scholar 

  18. Gotsman, C., Lindenbaum, M.: Euclidean Voronoi Labelling on the Multidimensional Grid. Pattern Recognition Letters 16, 409–415 (1995)

    Article  Google Scholar 

  19. Guan, W., Ma, S.: A list-processing approach to compute voronoi diagrams and the euclidean distance transform. IEEE Transactions on Pattern Analysis and Machine Intelligence 20(7), 757–761 (1998)

    Article  Google Scholar 

  20. Herman, G., Zheng, J., Bucholtz, C.: Shape-based interpolation. IEEE Computer Graphics and Applications 12(3), 69–79 (1992), http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=135915

    Article  Google Scholar 

  21. Hesselink, W.: A linear-time algorithm for Euclidean feature transform sets. Information Processing Letters 102, 181–186 (2007), http://linkinghub.elsevier.com/retrieve/pii/S0020019006003681

    Article  MathSciNet  Google Scholar 

  22. Hildebrand, T., Laib, A., Müller, R., Dequeker, J., Rüegsegger, P.: Direct three-dimensional morphometric analysis of human cancellous bone: microstructural data from spine, femur, iliac crest, and calcaneus. Journal of Bone and Mineral Research: The Official Journal of the American Society for Bone and Mineral Research 14(7), 1167–1174 (1999), http://www.ncbi.nlm.nih.gov/pubmed/10404017

    Article  Google Scholar 

  23. Hildebrand, T., Rüegsegger, P.: A new method for the model-independent assessment of thickness in three-dimensional images. Journal of Microscopy 185(1), 67–75 (1997), http://www.blackwell-synergy.com/links/doi/10.1046%2Fj.1365-2818.1997.1340694.x

    Article  Google Scholar 

  24. Hirata, T.: A unified linear-time algorithm for computing distance maps. Information Processing Letters 58(3), 129–133 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lucet, Y.: A Linear Euclidean Distance Transform Algorithm Based on the Linear-Time Legendre Transform. In: The 2nd Canadian Conference on Computer and Robot Vision (CRV 2005), pp. 262–267 (2005), http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=1443139

  26. Lucet, Y.: New sequential exact Euclidean distance transform algorithms based on convex analysis. Image and Vision Computing 27(1-2), 37–44 (2009), http://linkinghub.elsevier.com/retrieve/pii/S0262885606003647

    Article  Google Scholar 

  27. Maurer, C.R., Qi, R., Raghavan, V.: A linear time algorithm for computing exact euclidean distance transforms of binary images in arbitrary dimensions. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(2), 265–270 (2003)

    Article  Google Scholar 

  28. Meijster, A., Roerdink, J.B.T.M., Hesselink, W.H.: A general algorithm for computing distance transforms in linear time. In: Mathematical Morphology and its Applications to Image and Signal Processing, pp. 331–340. Kluwer (2000)

    Google Scholar 

  29. Mukherjee, J., Das, P.P., Kumarb, M.A., Chatterjib, B.N.: On approximating euclidean metrics by digital distances in 2D and 3D. Pattern Recognition Letters 21(6–7), 573–582 (2000)

    Article  Google Scholar 

  30. Mullikin, J.C.: The vector distance transform in two and three dimensions. Computer Vision, Graphics, and Image Processing. Graphical Models and Image Processing 54(6), 526–535 (1992)

    Google Scholar 

  31. Nagy, B.: A Comparison Among Distances Based on Neighborhood Sequences in Regular Grids. In: Kalviainen, H., Parkkinen, J., Kaarna, A. (eds.) SCIA 2005. LNCS, vol. 3540, pp. 1027–1036. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  32. Pottmann, H., Wallner, J., Huang, Q., Yang, Y.: Integral invariants for robust geometry processing. Computer Aided Geometric Design 26(1), 37–60 (2009), http://linkinghub.elsevier.com/retrieve/pii/S0167839608000095

    Article  MathSciNet  MATH  Google Scholar 

  33. Ragnemalm, I.: Contour processing distance transforms, pp. 204–211. World Scientific (1990)

    Google Scholar 

  34. Remy, E., Thiel, E.: Optimizing 3D chamfer masks with norm constraints. In: International Workshop on Combinatorial Image Analysis, Caen, pp. 39–56 (July 2000)

    Google Scholar 

  35. Rong, G., Tan, T.S.: Jump flooding in GPU with applications to Voronoi diagram and distance transform. In: Proceedings of the 2006 Symposium on Interactive 3D Graphics and Games, SI3D 2006, p. 109 (2006), http://portal.acm.org/citation.cfm?doid=1111411.1111431

  36. Rong, G., Tan, T.S.: Variants of Jump Flooding Algorithm for Computing Discrete Voronoi Diagrams. In: 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007), pp. 176–181 (July 2007), http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4276119

  37. Rosenfeld, A., Pfaltz, J.L.: Sequential operations in digital picture processing. Journal of the ACM 13(4), 471–494 (1966)

    Article  MATH  Google Scholar 

  38. Rosenfeld, A., Pfaltz, J.L.: Distance functions on digital pictures. Pattern Recognition 1, 33–61 (1968)

    Article  MathSciNet  Google Scholar 

  39. Saito, T., Toriwaki, J.I.: New algorithms for Euclidean distance transformations of an $n$-dimensional digitized picture with applications. Pattern Recognition 27, 1551–1565 (1994)

    Article  Google Scholar 

  40. Strand, R.: Distance Functions and Image Processing on Point-Lattices With Focus on the 3D Face- and Body-centered Cubic Grids. Phd thesis, Uppsala Universitet (2008)

    Google Scholar 

  41. Sud, A., Otaduy, M.A., Manocha, D.: DiFi: Fast 3D Distance Field Computation Using Graphics Hardware. Computer Graphics Forum 23(3), 557–566 (2004), http://www.blackwell-synergy.com/links/doi/10.1111%2Fj.1467-8659.2004.00787.x

    Article  Google Scholar 

  42. Vacavant, A., Coeurjolly, D.: First Results on Medial Axis Extraction on Two-Dimensional Irregular Isothetic Grids. In: 13th International Workshop on Combinatorial Image Analysis. Resarch Publishing Services (November 2009), http://liris.cnrs.fr/publis/?id=4333

  43. Vacavant, A., Coeurjolly, D., Tougne, L.: A Novel Algorithm for Distance Transformation on Irregular Isothetic Grids. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 469–480. Springer, Heidelberg (2009), http://liris.cnrs.fr/publis/?id=4166

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Université de Lyon, CNRS, LIRIS, UMR5205, F-69622, France

    David Coeurjolly

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  1. David Coeurjolly
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Editors and Affiliations

  1. HCI, University of Heidelberg, Germany, Speyerer Strasse 6, D-69115, Heidelberg, Germany

    Ullrich Köthe

  2. GIPSA-lab, 961 rue de la Houille Blanche, 38402, Saint Martin d’Hères cedex, France

    Annick Montanvert

  3. EC JRC Space Application Institute, T.P. 262, 21020, Ispra, Italy

    Pierre Soille

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Coeurjolly, D. (2012). Volumetric Analysis of Digital Objects Using Distance Transformation: Performance Issues and Extensions. In: Köthe, U., Montanvert, A., Soille, P. (eds) Applications of Discrete Geometry and Mathematical Morphology. WADGMM 2010. Lecture Notes in Computer Science, vol 7346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32313-3_6

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  • DOI: https://doi.org/10.1007/978-3-642-32313-3_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32312-6

  • Online ISBN: 978-3-642-32313-3

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