Lower and Upper Bounds for Scaling Factors Used for Integer Approximation of 3D Anisotropic Chamfer Distance Operator

  • Didier Coquin
  • Philippe Bolon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)

Abstract

For 3D images composed of successive scanner slices (e.g. medical imaging, confocal microscopy or computed tomography), the sampling step may vary according to the axes, and specially according to the depth which can take values lower or higher than 1. Hence, the sampling grid turns out to be parallelepipedic. In this paper, 3D anisotropic local distance operators are introduced. The problem of coefficient optimization is addressed for arbitrary mask size. Lower and upper bounds of scaling factors used for integer approximation are given. This allows, first, to derive analytically the maximal normalized error with respect to Euclidean distance, in any 3D anisotropic lattice, and second, to compute optimal chamfer coefficients. As far as large images or volumes are concerned, 3D anisotropic operators are adapted to the measurement of distances between objects sampled on non-cubic grids as well as for quantitative comparison between grey level images.

Keywords

Distance transformation Chamfer distance Anisotropic lattice 

References

  1. 1.
    Borgefors, G.: Applications using distance transforms. In: Arcelli, C., Cordella, L.P., Sanniti di Baja., G. (eds.) Aspects of Visual Form Processing, pp. 83–108. World Scientific, Singapore (1994)Google Scholar
  2. 2.
    Borgefors, G.: Distance transformations in arbitrary dimensions. Computer Vision, Graphics and Image Processing 27, 312–345 (1984)CrossRefGoogle Scholar
  3. 3.
    Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics and Image Processing 34, 344–371 (1986)CrossRefGoogle Scholar
  4. 4.
    Cuisenaire, O., Macq, B.: Fast Euclidean distance transformation by propagation using multiple neighborhood. Computer Vision and Image Understanding 76, 163–172 (1999)CrossRefGoogle Scholar
  5. 5.
    Maurer Jr., 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)CrossRefGoogle Scholar
  6. 6.
    Svensson, S., Borgefors, G.: Digital distance transforms in 3D images using information from neighbourhoods up to 5x5x5. Computer Vision and Image Understanding 88, 24–53 (2002)CrossRefMATHGoogle Scholar
  7. 7.
    Fouard, C., Malandain, G.: 3-D chamfer distances and norms in anisotropic grids. Image and Vision Computing 23, 143–158 (2005)CrossRefGoogle Scholar
  8. 8.
    Fouard, C., Strand, R., Borgefors, G.: Weighted distance transforms generalize to modules and their computation on point lattices. Pattern Recognition 40, 2453–2474 (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Hulin, J., Thiel, E.: Chordal axis on weighted distance transforms. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 271–282. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Verwer, B.: Local distances for distance transformations in two and three dimensions. Pattern Recognition Letters 12, 671–682 (1991)CrossRefGoogle Scholar
  11. 11.
    Coquin, D., Bolon, P.: Discrete distance operator on rectangular grids. Pattern Recognition Letters 16, 911–923 (1995)CrossRefGoogle Scholar
  12. 12.
    Remy, E., Thiel, E.: Optimizing 3D chamfer mask with norm constraints. In: Proceedings of International Workshop on Combinatorial Image Analysis, Caen, France, pp. 39–56 (2000)Google Scholar
  13. 13.
    Strand, R.: Weighted distances based on neighbourhood sequences. Pattern Recognition Letters 28(15) (2007)Google Scholar
  14. 14.
    Borgefors, G.: On digital distance transforms in three dimensions. Computer Vision and Image Understanding 64, 368–376 (1996)CrossRefGoogle Scholar
  15. 15.
    Borgefors, G.: Weighted digital distance transforms in four dimensions. Discrete Applied Mathematics 125, 161–176 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kiselman, C.: Regularity properties of distance transformations in image analysis. Computer Vision and Image Understanding 64, 390–398 (1996)CrossRefGoogle Scholar
  17. 17.
    Sintorn, I.M., Borgefors, G.: Weighted distance transforms in rectangular grids. In: 11th International Conference on Image Analysis and Processing, Palermo, Italy, pp. 322–326 (2001)Google Scholar
  18. 18.
    Svensson, S., Borgefors, G.: Distance transforms in 3D using four different weights. Pattern Recognition Letters 23, 1407–1418 (2002)CrossRefMATHGoogle Scholar
  19. 19.
    Sintorn, I.M., Borgefors, G.: Weighted distance transforms for images using elongated voxel grids. In: Proc. 10th Discret Geometry for Computer Imagery, Bordeaux, France, pp. 244–254 (2002)Google Scholar
  20. 20.
    Chehadeh, Y., Coquin, D., Bolon, P.: A generalization to cubic and non cubic local distance operators on parallelepipedic grids. In: Proc. 5th Discret Geometry for Computer Imagery, pp. 27–36 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Didier Coquin
    • 1
  • Philippe Bolon
    • 1
  1. 1.LISTICDomaine UniversitaireAnnecy le Vieux CedexFrance

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