Tree Triangular Coding Image Compression Algorithms

  • K. K. ShuklaEmail author
  • M. V. Prasad
Part of the SpringerBriefs in Computer Science book series (BRIEFSCOMPUTER)


This chapter presents four new image compression algorithms namely, Three-triangle decomposition scheme, Six-triangle decomposition scheme, Nine-triangle decomposition scheme and the Delaunay Triangulation Scheme. Performance of these algorithms is evaluated using standard test images. The asymptotic time complexity of Three-, Six-, and Nine-triangle decomposition algorithms is the same: O(nlogn) for coding and θ(n), for decoding. The time complexity of the Delaunay triangulation algorithm is O(n 2 logn) for coding and O(nlogn) for decoding, where n is the number of pixels in the image.


Triangulation Decomposition schemes Delaunay triangulation Algorithm complexity 


  1. 1.
    Distasi R, Nappi M, Vitulano S (1997) Image compression by B-tree triangular coding. IEEE Trans Commun 45(9):1095–1100CrossRefGoogle Scholar
  2. 2.
    Dimento LJ, Brekovich SY (1990) The compression effects of the binary tree overlapping method on digital imagery. IEEE Trans Commun 38:1260–1265CrossRefGoogle Scholar
  3. 3.
    Radha H, Leonadi R, Vetterli M (1991) A multiresolution approach to binary tree representation of Images. Proc ICASSP 91:2653–2656Google Scholar
  4. 4.
    Strobach P (1989) Image coding based on quadtree–structured recursive least squares approximation. Proc IEEE ICAASP 89:1961–1964Google Scholar
  5. 5.
    Wu X (1992) Image coding by adaptive tree-structured segmentation. IEEE Trans Inf Theory 38(6):1755–1767zbMATHCrossRefGoogle Scholar
  6. 6.
    Davoine F, Svensson J, Chassery J-M (1995) A mixed triangular and quadrilateral partition for fractal image coding. In: Proceedings of the international conference on image processing (ICIP ‘95)Google Scholar
  7. 7.
    VC Da Silva, JM De Carvalho (2000) Image compression via TRITREE decomposition. In: Proceedings of the XIII Brazilian symposium on computer graphics and image processing (SIBGRAPI)Google Scholar
  8. 8.
    Li X, Knipe J, Cheng H (1997) Image compression and encryption using tree structures. Pattern Recogn Lett 18:1253–1259CrossRefGoogle Scholar
  9. 9.
    Kotlov A (2001) Note: tree width and regular triangulations. Discret Math 237:187–191MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Wu X, Fang Y (1995) A segmentation-based predictive multiresolution image coder. IEEE Trans Image Process 4:34–47CrossRefGoogle Scholar
  11. 11.
    Radha H, Vetterli M, Leonardi R (1996) Image compression using binary space partitioning trees. IEEE Trans image process 5(12):1610–1623CrossRefGoogle Scholar
  12. 12.
    Samet H (1984) The quadtree and related Hierarchical data structures. ACM Comput Surv 16(2):187–260MathSciNetCrossRefGoogle Scholar
  13. 13.
    Babuska I, Aziz AK (1976) On the angle condition in the finite element method. Siam J Numer Anal 13(2):214–226MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Rippa S (1992) Long and thin triangles can be good for linear interpolation. Siam J Numer Anal 29(1):257–270MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Plaza A, Rivara M-C (2002) On the adjacencies of triangular meshes based on skeleton-regular partitions. J Comput Appl Math 140:673–693MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Mitchell SA (1997) Approximating the maximum-angle covering triangulation. Comput Geom 7:93–111MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Szelinski R (1990) Fast surface interpolation using hierarchical basis functions. IEEE Trans Pattern Anal Mach Intell 12:513–528CrossRefGoogle Scholar
  18. 18.
    Vreelj B, Vaidyanathan PP (2001) Efficient implementation of all digital interpolation. IEEE Trans image process 10(11):1639–1646CrossRefGoogle Scholar
  19. 19.
    Li J, Chen CS (2002) A simple efficient algorithm for interpolation between different grids in both 2D and 3D. Math Comput Sim 58:125–132zbMATHCrossRefGoogle Scholar
  20. 20.
    Chuah C-S, Leou J-J (2001) An adaptive image interpolation algorithm for image/video processing. Pattern Recogn 34:2383–2393zbMATHCrossRefGoogle Scholar
  21. 21.
    Boissonnant JD, Cazals F (2002) Smooth surface reconstruction via natural neighbour interpolation of distance functions. Comput Geom 22:185–203MathSciNetCrossRefGoogle Scholar
  22. 22.
    Antonini M, Barlaud M, Mathieu P, Daubechies I (1992) Image coding using the wavelet transform. IEEE Trans Image Process 1(2):205–220CrossRefGoogle Scholar
  23. 23.
    Vore BAD, Jawerth B, Lucien BJ (1992) Image compression through wavelet transform coding. IEEE Trans Inf Theory 38:719–746CrossRefGoogle Scholar
  24. 24.
    Jacquin AE (1992) Image coding based on a fractal theory of iterated contractive image transformations. IEEE Trans Image Process 1:18–30CrossRefGoogle Scholar
  25. 25.
    Helsingius M, Kuosmanen P, Astola J (2000) Image compression using multiple transforms. Signal Process Image Commun 15:513–529CrossRefGoogle Scholar
  26. 26.
    Prasad MVNK, Mishra VN, Shukla KK (2003) Space partitioning based image compression using quality measures. Appl Soft Comput Elsevier Sci 3:273–282CrossRefGoogle Scholar
  27. 27.
    Bramble JH, Zlamal M (1970) Triangular elements in the finite elements method. Math Comput 24(112):809–820MathSciNetCrossRefGoogle Scholar
  28. 28.
    Waldron S (1998) The error in linear interpolation at the vertices of a simplex. Siam J Numer Anal 35:1191–1200MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Aurenhammer F (1991) Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput Surv 23(3):345–405CrossRefGoogle Scholar
  30. 30.
    Guibas L, Stolfi J (1985) Primitives for the manipulation of general subdivisions and the computation of voronoi diagrams. ACM Trans Graph 4(2):74–123zbMATHCrossRefGoogle Scholar
  31. 31.
    Su P, Drysdale RLS (1995) A comparison of sequential delaunay triangulation algorithms, 1lth Symposium computational geometry, Vancouver, B.C. Canada, pp 61–70Google Scholar
  32. 32.
    Shewchuk JR (1999) Lecture notes on delaunay mesh generation. University of California, Berkeley, p 20Google Scholar
  33. 33.
    Letrattanapanich S, Bose NK (2002) High resolution image formation from low resolution frames using delaunay triangulation. IEEE Trans Image Process 11(12):1427–1441MathSciNetCrossRefGoogle Scholar
  34. 34.
    Davoine F, Antonini M, Chassery J-M, Barlaud M (1996) Fractal image compression based on delaunay triangulation and vector quantization. IEEE Trans Image Process 5(2):338–346CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited  2011

Authors and Affiliations

  1. 1.Department of Computer EngineeringIndian Institute of Technology, Banaras Hindu UniversityVaranasiIndia
  2. 2.Institute for Development and Research in Banking TechnologyHyderabadIndia

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