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Embedded Hypercube Graph Applied to Image Analysis Problems

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Abstract

Hypercubes have interesting geometric and topological properties with applications in several different fields, such as computer networks, information retrieval, data fusion, social networks, coding theory and linguistics. In this work, we present and discuss the use of hypercubes in some image analysis problems. Hypercube graphs take advantage of high dimensional features to provide low-cost image transformations. The downsampling of an image is performed as a pixel permutation, with no need for interpolation and, consequently, addition and multiplication operations. The hypercube graph is employed on demand one edge at once, such that there is no memory usage to traverse the image. Experimental results demonstrate the effectiveness of hypercubes as a powerful space representation both in terms of computational time and memory requirements.

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References

  1. Amanatiadis, A., & Andreadis, I. (2009). A survey on evaluation methods for image interpolation. Measurement Science and Technology, 20(10), 1–9.

    Article  Google Scholar 

  2. Asratian, A.S., Denley, T.M., & Häggkvist, R. (1998). Bipartite Graphs and their Applications Vol. 131. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  3. Bhuyan, L., & Agrawal, D. (1984). Generalized hypercube and hyperbus structures for a computer network. IEEE Transactions on Computers, C-33(4), 323–333.

    Article  MATH  Google Scholar 

  4. Bollobás, B. (1998). Modern Graph Theory, vol. 184 Springer Science & Business Media.

  5. Boykov, Y., & Funka-Lea, G. (2006). Graph cuts and efficient ND image segmentation. International Journal of Computer Vision, 70(2), 109–131.

    Article  Google Scholar 

  6. Chan, T., & Saad, Y. (1986). Multigrid algorithms on the hypercube multiprocessor. IEEE Transactions on Computers, C-35(11), 969–977.

    Article  Google Scholar 

  7. Chen, C., Lee, S., & Cho, Z. (1990). A parallel implementation of 3-D CT image reconstruction on hypercube multiprocessor. IEEE Transactions on Nuclear Science, 37(3), 1333–1346.

    Article  Google Scholar 

  8. Chepoi, V., & Hagen, M.F. (2013). On Embeddings of CAT (0) Cube Complexes into Products of Trees via Colouring their Hyperplanes. Journal of Combinatorial Theory Series B, 103(4), 428–467.

    Article  MathSciNet  MATH  Google Scholar 

  9. Cover, T.M. (1965). Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Transactions on Electronic Computers (3), 326–334.

  10. Cypher, R., Sanz, J., & Snyder, L. (1989). Hypercube and Shuffle-Exchange algorithms for image component labeling. Journal of Algorithms, 10(1), 140–150.

    Article  MathSciNet  MATH  Google Scholar 

  11. Darbon, J., & Sigelle, M. (2006). Image restoration with discrete constrained total variation Part I: Fast and exact optimization. Journal of Mathematical Imaging and Vision, 26(3), 261–276.

    Article  MathSciNet  Google Scholar 

  12. Day, K., & Tripat, A. (1994). A comparative study of topological properties of hypercubes and star graphs. IEEE Transactions on Parallel and Distributed Systems, 5(1), 31–38.

    Article  MathSciNet  Google Scholar 

  13. Deshpande, S., & Jenevin, R.M. (1986). Scalability of a binary tree on a hypercube. International Parallel and Distributed Processing Symposium, 661–668.

  14. Elmoataz, A., Lezoray, O., & Bougleux, S. (2008). Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Transactions on Image Processing, 17(7), 1047–1060.

    Article  MathSciNet  MATH  Google Scholar 

  15. Eshera, M., & Fu, K.S. (1986). An Image Understanding System using Attributed Symbolic Representation and Inexact Graph-Matching. IEEE Transactions on Pattern Analysis and Machine Intelligence (5), 604–618.

  16. Firsov, V. (1965). Isometric embedding of a graph in a boolean cube. Cybernetics and Systems Analysis, 1 (6), 112–113.

    MathSciNet  Google Scholar 

  17. Gonzalez, R.C., & Woods, R.E. (2006). Digital image processing. NJ: Prentice-Hall, Inc.

    Google Scholar 

  18. Harary, F., Hayes, J.P., & Wu, H.J. (1988). A survey of the theory of hypercube graphs. Computers & Mathematics with Applications, 15(4), 277–289.

    Article  MathSciNet  MATH  Google Scholar 

  19. Hayes, J.P. (1976). A graph model for Fault-Tolerant computing systems. IEEE Transactions on Computers, 100(9), 875–884.

    Article  MathSciNet  MATH  Google Scholar 

  20. Ho, C., & Johnsson, S. (1989). Spanning balanced trees in boolean cubes. SIAM Journal on Scientific and Statistical Computing, 10(4), 607–630.

    Article  MathSciNet  MATH  Google Scholar 

  21. Horowitz, E., Sahni, S., & Rajasekaran, S. (2007). Computer Algorithms/C++ silicon press.

  22. Hussain, Z., Bose, B., & Al-Dhelaan, A. (2014). Generalized hypercubes: Edge-disjoint hamiltonian cycles and gray codes. IEEE Transactions on Computers, 63(2), 375–382.

    Article  MathSciNet  MATH  Google Scholar 

  23. Intel 17455-03 (1985). IPSC User’s Guide. Portland, OR, USA.

  24. Jeulin, D. (2014). Random tessellations generated by boolean random functions. Pattern Recognition Letters, 47, 139–146.

    Article  Google Scholar 

  25. Johnsson, S.L. (1987). Communication effect basic linear algebra computations on hypercube architectures. Journal of Parallel and Distributed Computing, 4(2), 133–172.

    Article  Google Scholar 

  26. Kautz, W. (1958). Unit-Distance Error-Checking Codes. IRE Transactions on Electronic Computers (2), 179–180.

  27. Kolmogorov, V., & Zabin, R. (2004). What Energy Functions can be Minimized via Graph Cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(2), 147– 159.

    Article  Google Scholar 

  28. Kuo, C.N., Chou, H.H., Chang, N.W., & Hsieh, S.Y. (2013). Fault-Tolerant Path embedding in folded hypercubes with both node and edge faults. Theoretical Computer Science, 475, 82– 91.

    Article  MathSciNet  MATH  Google Scholar 

  29. Lai, T.H., & White, W. (1990). Mapping pyramid algorithms into hypercubes. Journal of Parallel and Distributed Computing, 9(1), 42–54.

    Article  Google Scholar 

  30. Leighton, F.T. (2014). Introduction to Parallel Algorithms and Architectures: Arrays·Trees· Hypercubes Elsevier.

  31. Martín-Herrero, J. (2007). Anisotropic diffusion in the hypercube. IEEE Transactions on Geoscience and Remote Sensing, 45(5), 1386–1398.

    Article  Google Scholar 

  32. Mathew, K.A., & Östergård, P.R. (2015). On Hypercube Packings, Blocking Sets and a Covering Problem. Information Processing Letters, 115(2), 141–145.

    Article  MathSciNet  MATH  Google Scholar 

  33. Mendez-Rial, R., & Martin-Herrero, J. (2012). Efficiency of Semi-Implicit schemes for anisotropic diffusion in the hypercube. IEEE Transactions on Image Processing, 21(5), 2389–2398.

    Article  MathSciNet  Google Scholar 

  34. Mudge, T. (1987). An analysis of hypercube architectures for image pattern recognition algorithms. In Proc. SPIE 0755, image pattern recognition: Algorithm implementations, techniques, and technology, pp. 71–83. International society for optics and photonics.

  35. nCUBE V1.0. (1990). NCUBE Vol. 6400. Beaverton: Processor Manual.

    Google Scholar 

  36. Pease, M.C. (1977). The Indirect Binary n-Cube Microprocessor Array. IEEE Transactions on Computers, C-26(5), 458– 473.

    Article  MATH  Google Scholar 

  37. Raju, S.V., & Govardhan, A. (2013). Overlapped text partition algorithm for pattern matching on hypercube networked model global journal of computer science and technology 13(4).

  38. Ranka, S., & Sahni, S. (2012). Hypercube Algorithms with Applications to Image Processing and Pattern Recognition Springer Science & Business Media.

  39. Saad, Y., & Schultz, M.H. (1988). Topological properties of hypercubes. IEEE Transactions on Computers, 37(7), 867– 872.

    Article  Google Scholar 

  40. Schuld, M., Sinayskiy, I., & Petruccione, F. (2014). Quantum walks on graphs representing the firing patterns of a quantum neural network. Physical Review A, 89(3), 032,333.

    Article  MATH  Google Scholar 

  41. Shankar, R.V., & Ranka, S. (1992). Hypercube algorithms for operations on quadtrees. Pattern Recognition, 25(7), 741– 747.

    Article  Google Scholar 

  42. Silva, E., Guedes, A., & Todt, E. (2013). Independent Spanning Trees on Systems-on-chip Hypercubes Routing. In International conference on electronics, circuits and systems, vol. 75, pp. 93–96.

  43. Skiena, S. (1990). Hypercubes. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica 148–150.

  44. Trémeau, A., & Colantoni, P. (2000). Regions adjacency graph applied to color image segmentation. IEEE Transactions on Image Processing, 9(4), 735–744.

    Article  Google Scholar 

  45. Uyttendaele, M., Eden, A., & Skeliski, R. (2001). Eliminating ghosting and exposure artifacts in image mosaics. In IEEE Computer society conference on computer vision and pattern recognition, vol. 2, pp. II–509. IEEE.

  46. Wang, E., Wang, H., Chen, J., Gong, W., & Ni, F. (2014). A Novel Node-to-Set Node-Disjoint Fault-Tolerant Routing Strategy in Hypercube. In Advanced computer architecture, pp. 58–67. Springer.

  47. Wang, X., Li, H., Bichot, C.E., Masnou, S., & Chen, L. (2013). A Graph-Cut Approach to Image Segmentation using an Affinity Graph based on l 0- Sparse Representation of Features. In 20Th IEEE international conference on image processing, pp. 4019–4023. IEEE.

  48. Wu, R.Y., Chen, G.H., Fu, J.S., & Chang, G.J. (2008). Finding cycles in hierarchical hypercube networks. Information Processing Letters, 109(2), 112–115.

    Article  MathSciNet  MATH  Google Scholar 

  49. Wu, Z., & Leahy, R. (1993). An Optimal Graph Theoretic Approach to Data Clustering: Theory and its Application to Image Segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(11), 1101– 1113.

    Article  Google Scholar 

  50. Yang, J.S., & Chang, J.M. (2014). Optimal independent spanning trees on cartesian product of hybrid graphs. The Computer Journal, 57(1), 93–99.

    Article  Google Scholar 

  51. Ziavras, S.G., & Siddiqui, M.A. (1993). Pyramid mappings onto hypercubes for computer vision: Connection machine comparative study. Concurrency: Practice and Experience, 5(6), 471– 489.

    Article  Google Scholar 

  52. Ziavras, S.G., & Sideras, M.A. (1995). Facilitating High-Performance image analysis on reduced hypercube (RH) parallel computers. International Journal of Pattern Recognition and Artificial Intelligence, 9(4), 679–698.

    Article  Google Scholar 

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Acknowledgments

The authors are grateful to FAPESP - São Paulo Research Foundation (Grant 2011/22749-8), CNPq (Grant 307113/2012-4) and CAPES for their financial support.

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  1. Institute of Computing, University of Campinas, Campinas, São Paulo, 13083-852, Brazil

    Eduardo Sant’Ana da Silva & Helio Pedrini

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  1. Eduardo Sant’Ana da Silva
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  2. Helio Pedrini
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Correspondence to Helio Pedrini.

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da Silva, E.S., Pedrini, H. Embedded Hypercube Graph Applied to Image Analysis Problems. J Sign Process Syst 88, 453–462 (2017). https://doi.org/10.1007/s11265-016-1182-x

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