Visual Form pp 451-467 | Cite as

A Review of Hierarchical Representations of Shape and Some Applications

  • Hanan Samet
Chapter

Abstract

A review is presented of the use of hierarchical data structures for the representation of shape. The presentation is primarily in terms of variants of a spatial data structure known as a quadtree. These data structures are based on a recursive decomposition of the interior of the shape or on its boundary. The focus is on the representation of two-dimensional shape used in applications in image processing and computer vision. The adaptation of the concepts of a distance, a skeleton, and a medial axis transform to a shape represented by a quadtree is described, as is a region expansion (i.e., image dilation) algorithm that executes in time independent of the radius of expansion.

Keywords

shape hierarchical data structures quadtrees medial axis transforms skeletons region expansion image dilation 
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References

  1. [Ang89]
    C. Ang, Analysis and applications of hierarchical data structures, Ph.D. dissertation, TR-2255, Computer Science Department, University of Maryland, College Park, MD, June 1989.Google Scholar
  2. [Ang90]
    C. H. Ang, H. Samet, and C. A. Shaffer, A new region expansion for quadtrees, IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 7(July 1990), 682–686.CrossRefGoogle Scholar
  3. [Ayal85]
    D. Ayala, P. Brunet, R. Juan, and I. Navazo, Object representation by means of nonminimal division quadtrees and octrees, ACM Transactions on Graphics 4, 1(January 1985), 41–59.CrossRefGoogle Scholar
  4. [Ball81]
    D. H. Ballard, Strip trees: A hierarchical representation for curves, Communications of the ACM 24, 5(May 1981), 310–321 (see also corrigendum, Communications of the ACM 25, 3(March 1982), 213).CrossRefGoogle Scholar
  5. [Blum67]
    H. Blum, A transformation for extracting new descriptors of shape, in Models for the Perception of Speech and Visual Form, W. Wathen-Dunn, ed., MIT Press, Cambridge, MA, 1967, 362–380.Google Scholar
  6. [Carl85]
    I. Carlbom, I. Chakravarty, and D. Vanderschel, A hierarchical data structure for representing the spatial decomposition of 3-D objects, IEEE Computer Graphics and Applications 5, 4(April 1985), 24–31.CrossRefGoogle Scholar
  7. [Chie84]
    C. H. Chien and J. K. Aggarwal, A normalized quadtree representation, Computer Vision, Graphics, and Image Processing 26, 3(June 1984), 331–346.CrossRefGoogle Scholar
  8. [Fink74]
    R. A. Finkel and J. L. Bentley, Quad trees: a data structure for retrieval on composite keys, Acta Informatica 4, 1(1974), 1-9.Google Scholar
  9. [Faug84]
    O. D. Faugeras, M. Hebert, P. Mussi, and J. D. Boissonnat, Polyhedral approximation of 3-d objects without holes, Computer Vision, Graphics, and Image Processing 25, 2(February 1984), 169–183.CrossRefGoogle Scholar
  10. [Free74]
    H. Freeman, Computer processing of line-drawing images, ACM Computing Surveys 6, 1(March 1974), 57–97.MATHCrossRefGoogle Scholar
  11. [Fuji85]
    K. Fujimura and T. L. Kunii, A hierarchical space indexing method, Proceedings of Computer Graphics’85, Tokyo, 1985, T1-4, 1-14.Google Scholar
  12. [Giint90]
    O. Günther and E. Wong, The arc tree: an approximation scheme to represent arbitrary curved shapes, Computer Vision, Graphics, and Image Processing 51, 3(September 1990), 313–337.CrossRefGoogle Scholar
  13. [Horo76]
    S. L. Horowitz and T. Pavlidis, Picture segmentation by a tree traversal algorithm, Journal of the ACM 23, 2(April 1976), 368–388.MATHCrossRefGoogle Scholar
  14. [Hunt78]
    G. M. Hunter, Efficient computation and data structures for graphics, Ph.D. dissertation, Department of Electrical Engineering and Computer Science, Princeton University, Princeton, NJ, 1978.Google Scholar
  15. [Hunt8l]
    G. M. Hunter, Geometrees for interactive visualization of geology: an evaluation, System Science Department, Schlumberger-Doll Research, Ridgefield, CT, 1981.Google Scholar
  16. [Hunt79]
    G. M. Hunter and K. Steiglitz, Operations on images using quad trees, IEEE Transactions on Pattern Analysis and Machine Intelligence 1, 2(April 1979), 145–153.CrossRefGoogle Scholar
  17. [Jack80]
    C. L. Jackins and S. L. Tanimoto, Oct-trees and their use in representing three-dimensional objects, Computer Graphics and Image Processing 14, 3(November 1980), 249–270.CrossRefGoogle Scholar
  18. [Kim86]
    Y. C. Kim and J. K. Aggarwal, Rectangular parallelepiped coding: a volumetric representation of three-dimensional objects, IEEE Journal of Robotics and Automation 2, 3(September 1986), 127–134.Google Scholar
  19. [Klin71]
    A. Klinger, Patterns and search statistics, in Optimizing Methods in Statistics, J. S. Rustagi, ed., Academic Press, New York, 1971, 303–337.Google Scholar
  20. [Klin76]
    A. Klinger and C. R. Dyer, Experiments in picture representation using regular decomposition, Computer Graphics and Image Processing 5, 1(March 1976), 68–105.CrossRefGoogle Scholar
  21. [Li82]
    M. Li, W. I. Grosky, and R. Jain, Normalized quadtrees with respect to translations, Computer Graphics and Image Processing 20, 1(September 1982), 72–81.MATHCrossRefGoogle Scholar
  22. [Meag80]
    D. Meagher, Octree encoding: a new technique for the representation, the manipulation, and display of arbitrary 3-d objects by computer, Electrical and Systems Engineering Technical Report IPL-TR-80-111, Rensselaer Polytechnic Institute, Troy, NY, October 1980.Google Scholar
  23. [Meag82]
    D. Meagher, Geometric modeling using octree encoding, Computer Graphics and Image Processing 19, 2(June 1982), 129–147.CrossRefGoogle Scholar
  24. [Mort66]
    G. M. Morton, A computer oriented geodetic data base and a new technique in file sequencing, IBM Ltd., Ottawa, Canada, 1966.Google Scholar
  25. [Nava86]
    I. Navazo, Contribució a les tècniques de modelat geomètric d’objectés polièdrics usant la codificació amb arbres octals, Ph.D. dissertation, Escola Tecnica Superior d’Enginyers Industrials, Department de Metodes Informatics, Universität Politècnica de Catalunya, Barcelona, Spain, January 1986.Google Scholar
  26. [Nels86]
    R. C. Nelson and H. Samet, A consistent hierarchical representation for vector data, Computer Graphics 20, 4(August 1986), 197–206 (also Proceedings of the SIGGRAPW’86 Conference, Dallas, August 1986).CrossRefGoogle Scholar
  27. [Ponc87]
    J. Ponce and O. Faugeras, An object centered hierarchical representation for 3d objects: the prism tree, Computer Vision, Graphics, and Image Processing 88, 1(April 1987), 1–28.CrossRefGoogle Scholar
  28. [Quin82]
    K. M. Quinlan and J. R. Woodwark, A spatially-segmented solids database—justification and design, Proceedings of CAD’82 Conference, Butterworth, Guildford, Great Britain, 1982, 126-132.Google Scholar
  29. [Redd78]
    D. R. Reddy and S. Rubin, Representation of three-dimensional objects, CMU-CS-78-113, Computer Science Department, Carnegie-Mellon University, Pittsburgh, April 1978.Google Scholar
  30. [Rose82]
    A. Rosenfeld and A. C. Kak, Digital Picture Processing, Second Edition, Academic Press, New York, 1982.Google Scholar
  31. [Rose66]
    A. Rosenfeld and J. L. Pfaltz, Sequential operations in digital image processing, Journal of the ACM 18, 4(October 1966), 471–494.Google Scholar
  32. [Ruto68]
    D. Rutovitz, Data structures for operations on digital images, in Pictorial Pattern Recognition, G. C. Cheng et al., eds., Thompson Book Co., Washington, DC, 1968, 105–133.Google Scholar
  33. [Same82]
    H. Samet, Distance transform for images represented by quadtrees, IEEE Transactions on Pattern Analysis and Machine Intelligence 4, 3(May 1982), 298–303.CrossRefGoogle Scholar
  34. [Same83]
    H. Samet, A quadtree medial axis transform, Communications of the ACM 26, 9(September 1983), 680–693 (also corrigendum, Communications of the ACM 27, 2(February 1984), 151).CrossRefGoogle Scholar
  35. [Same84a]
    H. Samet, The quadtree and related hierarchical data structures, ACM Computing Surveys 16, 2(June 1984), 187–260.MathSciNetCrossRefGoogle Scholar
  36. [Same85a]
    H. Samet, Reconstruction of quadtrees from quadtree medial axis transforms, Computer Vision, Graphics, and Image Processing 29, 3(March 1985), 311–328.CrossRefGoogle Scholar
  37. [Same90a]
    H. Samet, The Design and Analysis of Spatial Data Structures, Addison-Wesley, Reading, MA, 1990.Google Scholar
  38. [Same90b]
    H. Samet, Applications of Spatial Data Structures: Computer Graphics, Image Processing, and GIS, Addison-Wesley, Reading, MA, 1990.Google Scholar
  39. [Same84b]
    H. Samet, A. Rosenfeld, C. A. Shaffer, R. C. Nelson, and Y. G. Huang, Application of hierarchical data structures to geographical information systems: phase III, Computer Science TR-1457, University of Maryland, College Park, MD, November 1984.Google Scholar
  40. [Same85b]
    H. Samet and R. E. Webber, Storing a collection of polygons using quadtrees, ACM Transactions on Graphics 4, 3(July 1985), 182–222 (also Proceedings of Computer Vision and Pattern Recognition 88, Washington, DC, June 1983, 127-132).CrossRefGoogle Scholar
  41. [Scot86]
    D. S. Scott and S. S. Iyengar, TID—a translation invariant data structure for storing images, Communications of the ACM 29, 5(May 1985), 418–429.CrossRefGoogle Scholar
  42. [Shaf90]
    C. A. Shaffer, H. Samet, and R. C. Nelson, QUILT: a geographic information system based on quadtrees, International Journal of Geographical Information Systems 4, 2(April-June 1990), 103–131.CrossRefGoogle Scholar
  43. [Shaf88]
    C. A. Shaffer and H. Samet, An algorithm to expand regions represented by linear quadtrees, Image and Vision Computing 6, 3(August 1988), 162–168.CrossRefGoogle Scholar
  44. [Shne81]
    M. Shneier, Path-length distances for quadtrees, Information Sciences 28, 1(February 1981), 49–67.MathSciNetCrossRefGoogle Scholar
  45. [Tamm8l]
    M. Tamminen, The EXCELL method for efficient geometric access to data, Acta Polytechnica Scandinavica, Mathematics and Computer Science Series No. 34, Helsinki, Finland, 1981.Google Scholar
  46. [Tani75]
    S. Tanimoto and T. Pavlidis, A hierarchical data structure for picture processing, Computer Graphics and Image Processing 4, 2(June 1975), 104–119.CrossRefGoogle Scholar
  47. [Vand84]
    D. J. Vanderschel, Divided leaf octal trees, Research Note, Schlumberger-Doll Research, Ridgefield, CT, March 1984.Google Scholar
  48. [Warn69]
    J. E. Warnock, A hidden surface algorithm for computer generated half tone pictures, Computer Science Department TR 4-15, University of Utah, Salt Lake City, June 1969.Google Scholar
  49. [Wyvi85]
    G. Wyvill and T. L. Kunii, A functional model for constructive solid geometry, Visual Computer 1, 1(July 1985), 3–14.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Hanan Samet
    • 1
  1. 1.Computer Science Department, Center for Automation Research, and Institute for Advanced Computer StudiesUniversity of MarylandCollege ParkUSA

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