Distance Transform Algorithms And Their Implementation And Evaluation

  • George J. Grevera
Part of the Topics in Biomedical Engineering. International Book Series book series (ITBE)

Consider an n-dimensional binary image consisting of one or more objects. A value of 1 indicates a point within some object and a value of 0 indicates that that point is part of the background (i.e., is not part of any object). For every point in some object, a distance transform assigns a value indicating the distance from that point within the object to the nearest background point. Similarly for every point in the background, a distance transform assigns a value indicating the minimum distance from that background point to the nearest point in any object. By convention, positive values indicate points within some object and negative values indicate background points. A number of elegant and efficient distance transform algorithms have been proposed, with Danielsson being one of the earliest in 1980 and Borgefors in 1986 being a notable yet simple improvement. In 2004 Grevera proposed a further improvement of this family of distance transform algorithms that maintains their elegance but increases accuracy and extends them to n-dimensional space as well. In this paper, we describe this family of algorithms and compare and contrast them with other distance transform algorithms. We also present a novel framework for evaluating distance transform algorithms and discuss applications of distance transforms to other areas of image processing and analysis such as interpolation and skeletonization.

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References

  1. 1.
    Udupa JK. 1994. Multidimensional digital boundaries. Comput Vision Graphics Image Process: Graphical Models Image Process 56(4):311-323.Google Scholar
  2. 2.
    Cuisenaire O. 1999. Distance transformations: fast algorithms and applications to medical image processing. PhD thesis. Universit é Catholique de Louvian.Google Scholar
  3. 3.
    Rosenfeld A, Pfaltz JL. 1968. Distance functions on digital pictures. Pattern Recognit 1(1):33-61.CrossRefMathSciNetGoogle Scholar
  4. 4.
    Montanari U. 1968. A method for obtaining skeletons using a quasi-Euclidean distance. J Assoc Comput Machin 15:600-624.Google Scholar
  5. 5.
    Danielsson P-E. 1980. Euclidean distance mapping. Comput Graphics Image Process 14:227-248.CrossRefGoogle Scholar
  6. 6.
    Borgefors G. 1986. Distance transformations in digital images. Comput Vision Graphics Image Process 34:344-371.CrossRefGoogle Scholar
  7. 7.
    Borgefors G. 1996. On digital distance transforms in three dimensions. Comput Vision Image Understand 64(3):368-376.CrossRefGoogle Scholar
  8. 8.
    Butt MA, Maragos P. 1998. Optimum design of chamfer distance transforms. IEEE Trans Image Process 7(10):1477-1484.CrossRefGoogle Scholar
  9. 9.
    Marchand-Maillet S, Sharaiha YM. 1999. Euclidean ordering via Chamfer distance calculations. Comput Vision Image Understand 73(3):404-413.MATHCrossRefGoogle Scholar
  10. 10.
    Nilsson NJ. Artificial intelligence: a new synthesis. San Francisco: Morgan Kaufmann, 1998.MATHGoogle Scholar
  11. 11.
    Verwer BJH, Verbeek PW, Dekker ST. 1989.An efficient uniform cost algorithm applied to distance transforms. IEEE Trans Pattern Anal Machine Intell 11(4):425-429.CrossRefGoogle Scholar
  12. 12.
    Leymarie F, Levine MD. 1992. Fast raster scan distance propagation on the discrete rectangular lattice. Comput Vision Graphics Image Process: Image Understand 55(1):84-94.MATHGoogle Scholar
  13. 13.
    Satherley R, Jones MW. 2001. Vector-city vector distance transform. Comput Vision Image Un- derstand 82:238-254.MATHCrossRefGoogle Scholar
  14. 14.
    Ragnemalm I. 1992. Neighborhoods for distance transformations using ordered propagation. Com-put Vision Graphics Image Process: Image Understand 56(3):399-409.MATHGoogle Scholar
  15. 15.
    Guan W, Ma S. 1998. A list-processing approach to compute Voronoi diagrams and the Euclidean distance transform. IEEE Trans Pattern Anal Machine Intell 20(7):757-761.CrossRefGoogle Scholar
  16. 16.
    Eggers H. 1998. Two fast Euclidean distance transformations in Z2 based on sufficient propagation. Comput Vision Image Understand 69(1):106-116.CrossRefGoogle Scholar
  17. 17.
    Saito T, Toriwaki J-I. 1994. New algorithms for euclidean distance transformation of an n-dimensional digitized picture with application. Pattern Recognit 27(11):1551-1565.CrossRefGoogle Scholar
  18. 18.
    Boxer L, Miller R. 2000. Efficient computation of the Euclidean distance transform. Comput Vision Image Understand 80:379-383.MATHCrossRefGoogle Scholar
  19. 19.
    Meijster A, Roerdink JBTM, Hesselink WH. 2000. A general algorithm for computing distance transforms in linear time. In Mathematical morphology and its applications to image and signal processing, pp. 331-340. Ed. J Goutsias, L Vincent, DS Bloombers. New York: Kluwer.Google Scholar
  20. 20.
    Lotufo RA, Falcao AA, Zampirolli FA. 2000. Fast Euclidean distance transform using a graph- search algorithm. SIBGRAPI 2000:269-275.Google Scholar
  21. 21.
    da Fontoura Costa L. 2000. Robust skeletonization through exact Euclidean distance transform and its application to neuromorphometry. Real-Time Imaging 6:415-431.MATHCrossRefGoogle Scholar
  22. 22.
    Pudney C. 1998. Distance-ordered homotopic thinning: a skeletonization algorithm for 3D digital images. Comput Vision Image Understand 72(3):404-413.CrossRefGoogle Scholar
  23. 23.
    Sanniti di Baja G. 1994. Well-shaped, stable, and reversible skeletons from the (3,4)-distance transform. J Visual Commun Image Represent 5(1):107-115.CrossRefGoogle Scholar
  24. 24.
    Svensson S, Borgefors G. 1999. On reversible skeletonization using anchor-points from distance transforms. J Visual Commun Image Represent 10:379-397.CrossRefGoogle Scholar
  25. 25.
    Herman GT, Zheng J, Bucholtz CA. 1992. Shape-based interpolation, IEEE Comput Graphics Appl 12(3):69-79.CrossRefGoogle Scholar
  26. 26.
    Raya SP, Udupa JK. 1990. Shape-based interpolation of multidimensional objects. IEEE Trans Med Imaging 9(1):32-42.CrossRefGoogle Scholar
  27. 27.
    Grevera GJ, Udupa JK. 1996. Shape-based interpolation of multidimensional grey-level images. IEEE Trans Med Imaging 15(6):881-892.CrossRefGoogle Scholar
  28. 28.
    Kozinska D. 1997. Multidimensional alignment using the Euclidean distance transform. Graphical Models Image Process 59(6):373-387.CrossRefGoogle Scholar
  29. 29.
    Paglieroni DW. 1997. Directional distance transforms and height field preprocessing for efficient ray tracing. Graphical Models Image Process 59(4):253-264.CrossRefGoogle Scholar
  30. 30.
    Remy E, Thiel E. 2000. Computing 3D medial axis for Chamfer distances. Discrete Geom Comput Imagery pp. 418-430.Google Scholar
  31. 31.
    Remy E, Thiel E. 2002. Medial axis for chamfer distances: computing look-up tables and neigh- bourhoods in 2D or 3D. Pattern Recognit Lett 23(6):649-662.MATHCrossRefGoogle Scholar
  32. 32.
    Grevera GJ, Udupa JK. 1998. An objective comparison of 3D image interpolation methods. IEEE Trans Med Imaging 17(4):642-652.CrossRefGoogle Scholar
  33. 33.
    Grevera GJ, Udupa JK, Miki Y. 1999. A task-specific evaluation of three-dimensional image inter-polation techniques. IEEE Trans Med Imaging 18(2):137-143.CrossRefGoogle Scholar
  34. 34.
    Travis AJ, Hirst DJ, Chesson A. 1996. Automatic classification of plant cells according to tissue type using anatomical features obtained by the distance transform. Ann Botany 78:325-331.CrossRefGoogle Scholar
  35. 35.
    Van Der Heijden GWAM, Van De Vooren JG, Van De Wiel CCM. 1995. Measuring cell wall dimensions using the distance transform. Ann Botany 75:545-552.CrossRefGoogle Scholar
  36. 36.
    Schnabel JA, Wang L, Arridge SR. 1996. Shape description of spinal cord atrophy in patients with MS. Comput Assist Radiol ICS 1124:286-291.Google Scholar
  37. 37.
    Grevera GJ. 2004. The “dead reckoning” signed distance transform. Comput Vision Image Under- stand 95:317-333.CrossRefGoogle Scholar
  38. 38.
    Oppenheim AV, Schafer RW, Buck JR. 1999. Discrete-time signal processing, 2d ed. Englewood Cliffs: Prentice Hall.Google Scholar
  39. 39.
    Cormen TH, Leiserson CE, Rivest RL, Stein C. 2001. Introduction to algorithms, 2d ed. Cambridge: MIT Press.MATHGoogle Scholar
  40. 40.
    Svensson S, Borgefors G. 2002. Digital distance transforms in 3D images using information from neighbourhoods up to 5x5x5. Comput Vision Image Understand 88:24-53.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • George J. Grevera
    • 1
  1. 1.Mathematics and Computer Science DepartmentSt. Joseph's UniversityPhiladelphiaUSA

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