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Digital Topology on Adaptive Octree Grids

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Abstract

The theory of digital topology is used in many different image processing and computer graphics algorithms. Most of the existing theories apply to uniform cartesian grids, and they are not readily extensible to new algorithms targeting at adaptive cartesian grids. This article provides a rigorous extension of the classical digital topology framework for adaptive octree grids, including the characterization of adjacency, connected components, and simple points. Motivating examples, proofs of the major propositions, and algorithm pseudocodes are provided.

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References

  1. Ayala, R., Dominguez, E., Frances, A., Quintero, A.: Homotopy in digital spaces. Discrete Appl. Math. 125, 3–24 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bai, Y., Han, X., Prince, J.L.: Octree-based topology-preserving isosurface simplification. In: IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, New York, June 2006

  3. Bai, Y., Han, X., Prince, J.L.: Octree grid topology preserving geometric deformable models for 3D medical image segmentation. In: Information Processing in Medical Imaging (2007)

  4. Bazin, P., Pham, D.L.: Topology-preserving tissue classification of magnetic resonance brain images. IEEE Trans. Med. Imag. 26(4), 487–496 (2007)

    Article  Google Scholar 

  5. Bazin, P., Ellingsen, L., Pham, D.L.: Digital homeomorphisms in deformable registration. In: Information Processing in Medical Imaging, pp. 211–222 (2007)

  6. Bertrand, G.: A new characterization of three-dimensional simple points. Pattern Recogn. Lett. 15, 169–175 (1994)

    Article  MATH  Google Scholar 

  7. Bertrand, G.: Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recogn. Lett. 15, 1003–1011 (1994)

    Article  Google Scholar 

  8. Bertrand, G., Couprie, M.: A model for digital topology. Discrete Geom. Comput. Imag. 1568, 227–239 (1999)

    Article  Google Scholar 

  9. Brimkov, V., Klette, R.: Border and surface tracing–theoretical foundations. IEEE Trans. Pattern Anal. Mach. Intell. 30(4), 577–590 (2008)

    Article  Google Scholar 

  10. de Andrade, M.C., Bertrand, G., de Albuquerde Araujo, A.: Segmentation of microscopic images by flooding simulation: a catchment-basins merging algorithm. In: Proceedings of SPIE: Nonlinear Image Processing VIII (1997)

  11. Droske, M., Meyer, B., Schaller, C., Rumpf, M.: An adaptive level set method for medical image segmentation. In: Insana, M.F., Leahy, R.M. (eds.) Information Processing in Medical Imaging. Lecture Notes in Computer Science, vol. 2082, pp. 416–422. Springer, Berlin (2001)

    Google Scholar 

  12. Fourey, S., Malgouyres, R.: A concise characterization of 3D simple points. Discrete Appl. Math. 125, 59–80 (2005)

    Article  MathSciNet  Google Scholar 

  13. Frisken, S.F., Perry, R.N., Rockwood, A.P., Jones, T.R.: Adaptively sampled distance fields: A general representation of shape for computer graphics. In: SIGGRAH, pp. 249–254 (2000)

  14. Han, X., Xu, C., Braga-Neto, U., Prince, J.L.: Topology correction in brain cortex segmentation using a multiscale graph-based approach. IEEE Trans. Med. Imag. 21, 109–121 (2002)

    Article  Google Scholar 

  15. Han, X., Xu, C., Prince, J.L.: A topology preserving level set method for geometric deformable models. IEEE Trans. Pattern Anal. Mach. Intell. 25, 755–768 (2003)

    Article  Google Scholar 

  16. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  17. Herman, G.T. (ed.): Geometry of Digital Spaces. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  18. Ju, T., Baker, M.L., Chiu, W.: Computing a family of skeletons of volumetric models for shape description. Comput. Aided Des. 39(5), 352–360 (2007)

    Article  Google Scholar 

  19. Kenmochi, Y., Kotani, K., Imiya, A.: Marching cubes method with connectivity. In: International Conference on Image Processing (1999)

  20. Khalimsky, E., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topol. Its Appl. 36, 1–17 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kong, T.Y.: A digital fundamental group. Comput. Graph. 13, 159–166 (1989)

    Article  Google Scholar 

  22. Kong, T.Y.: On topology preservation in 2-D and 3-D thinning. Int. J. Pattern Recogn. Artif. Intell. 9, 813–844 (1995)

    Article  Google Scholar 

  23. Kong, T.Y.: Foundation of Image Understanding, pp. 73–93. Kluwer Academic, Dordrecht (2001). Chap. Digital topology

    Google Scholar 

  24. Kong, T.Y.: Topological adjacency relations on z n. Theor. Comput. Sci. 283, 3–28 (2002)

    Article  MATH  Google Scholar 

  25. Kong, T.Y., Rosenfeld, A.: Digital topology: Introduction and survey. CVGIP, Image Underst. 48, 357–393 (1989)

    Google Scholar 

  26. Kong, T.Y., Rosenfeld, A.: If we use 4- or 8-connectedness for both the objects and the background, the Euler characteristics is not locally computable. Pattern Recogn. Lett. 11, 231–232 (1990)

    Article  MATH  Google Scholar 

  27. Kong, T.Y., Udupa, J.K.: A justification of a fast surface tracking algorithm. CVGIP, Graph. Models Image Process. 54, 162–170 (1992)

    Article  Google Scholar 

  28. Kovalevsky, V.: Axiomatic digital topology. J. Math. Imaging Vis. 26, 41–58 (2006)

    Article  MathSciNet  Google Scholar 

  29. Lachaud, J.: Topologically Defined Iso-surfaces. Lecture Notes in Computer Science, vol. 1176, pp. 245–256. Springer, Berlin (1996)

    Google Scholar 

  30. Lachaud, J.-O., Montanvert, A.: Continuous analogs of digital boundaries: A topological approach to isosurfaces. Graph. Models 62, 129–164 (2000)

    Article  Google Scholar 

  31. Ma, C.M., Sonka, M.: A fully parallel 3D thinning algorithm and its applications. Comput. Vis. Image Underst. 64, 420–433 (1996)

    Article  Google Scholar 

  32. Malandain, G., Bertrand, G., Ayache, N.: Topological segmentation of discrete surfaces. Int. J. Comput. Vis. 10(2), 183–197 (1993)

    Article  Google Scholar 

  33. Mangin, J., Frouin, V., Bloch, I., Regis, J., Lopez-Krahe, J.: From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations. J. Math. Imaging Vis. 5, 297–318 (1995)

    Article  Google Scholar 

  34. Milne, R.B.: Adaptive level sets methods interfaces. Ph.D. Thesis, Dept. Math., UC Berkely (1995)

  35. Morgenthaler, D.G.: Three-dimensional simple points: Serial erosion, parallel thinning and skeletonization. Technical Report, Computer Vision Lab., Univ. of Maryland (1981)

  36. Natarajan, B.: On generating topologically consistent isosurfaces from uniform samples. Vis. Comput. 11(1), 52–62 (1994)

    Article  Google Scholar 

  37. Nielson, G.M., Hamann, B.: The asymptotic decider: Resolving the ambiguity in marching cubes. In: IEEE Visualization, pp. 83–91, Los Alamitos (1991)

  38. Niethammer, M., Kalies, W.D., Mischaikow, K., Tannenbaum, A.: On the detection of simple points in higher dimensions using cubical homology. IEEE Trans. Image Process. 15, 2462–2469 (2006)

    Article  MathSciNet  Google Scholar 

  39. Pudney, C.: Distance-ordered homotopic thinning: A skeletonization algorithm for 3d digital images. Comput. Vis. Image Underst. 72, 404–413 (1998)

    Article  Google Scholar 

  40. Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach. 17, 146–160 (1970)

    MATH  MathSciNet  Google Scholar 

  41. Rosenfeld, A.: Arcs and curves in digital pictures. J. Assoc. Comput. Mach. 20, 81–87 (1973)

    MATH  MathSciNet  Google Scholar 

  42. Rosenfeld, A., Kak, A.C.: Digital Picture Processing. Academic Press, San Diego (1982)

    Google Scholar 

  43. Saha, P., Chaudhuri, B., Majumder, D.: A new shape preserving parallel thinning algorithm for 3D digital images. Pattern Recogn. 30, 1939–1955 (1997)

    Article  Google Scholar 

  44. Saha, P.K., Chaudhuri, B.B.: Detection of 3D simple points for topology preserving transformation with application to thinning. IEEE Trans. Pattern Anal. Mach. Intell. 16, 1028–1032 (1994)

    Article  Google Scholar 

  45. Saha, P.K., Chaudhuri, B.B.: 3D digital topology under binary transformation with applications. Comput. Vis. Image Underst. 63, 418–429 (1996)

    Article  Google Scholar 

  46. Segonne, F., Pons, J.-P., Grimson, E., Fischl, B.: Active contours under topology control genus preserving level sets. In: Computer Vision for Biomedical Image Applications, pp. 135–145 (2005)

  47. Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93, 1591–1595 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  48. Stelldinger, P., Latecki, L., Siqueira, M.: Topological equivalence between a 3D object and the reconstruction of its digital image. IEEE Trans. Pattern Anal. Mach. Intell. 29(1), 126–140 (2007)

    Article  Google Scholar 

  49. Teo, P., Sapiro, G., Wandell, B.: Creating connected representations of cortical gray matter for functional MRI visualization. IEEE Trans. Med. Imag. 16, 852–863 (1997)

    Article  Google Scholar 

  50. Tsitsiklis, J.N.: Efficient algorithm for globally optimal trajectories. IEEE Trans. Automat. Contr. 40(9), 1528–1538 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  51. Udupa, J.K.: Multidimensional digital boundaries. CVGIP, Graph. Models Image Process. 56, 311–323 (1994)

    Article  Google Scholar 

  52. Xie, W., Thompson, R.P., Perucchio, R.: A topology-preserving parallel 3d thinning algorithm for extracting the curve skeleton. Pattern Recogn. 36, 1529–1544 (2003)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD, 21218, USA

    Ying Bai & Jerry L. Prince

  2. CMS Software, Elekta Inc., St. Louis, MO, 63043, USA

    Xiao Han

Authors
  1. Ying Bai
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  2. Xiao Han
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  3. Jerry L. Prince
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Correspondence to Ying Bai.

Additional information

This work was supported in part by NIH/NINDS Grant R01NS37747.

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Bai, Y., Han, X. & Prince, J.L. Digital Topology on Adaptive Octree Grids. J Math Imaging Vis 34, 165–184 (2009). https://doi.org/10.1007/s10851-009-0140-7

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  • DOI: https://doi.org/10.1007/s10851-009-0140-7

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