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Alternative patterns of the multidimensional Hilbert curve

Application in image retrieval

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Abstract

Locality-preserving (distance preserving-mapping) is a useful property to manage multidimensional data. Close points in space remain -as much as possible- close after mapping on curve. That is why Hilbert space-filling curve is used in many domains and applications. Hilbert curve preserves well locality because from a construction aspect, it is guided by adajacency constraint on points ordering : the curve connects all points of a D-dimensional discrete space, without favoring any direction, under the constrainst that two successive points are separated by an unit distance. Originally defined in 2-D, all existing multidimensional extensions of the Hilbert curve satisfy adjacency by using the RBG pattern (based on Reflected Binary Gray code). The RBG pattern is then duplicated and arranged (geometrical transformations) to build the multidimensional Hilbert curve at a given order. In this paper, we emphasize that there are other patterns that can satisfy the adjacency. A formulation is given, an algorithm to find out solutions is provided and their respective level of locality preservation is estimated through a standard criterion. Results show that some new patterns can carry a comparable levels of locality and sometimes better than RBG. Moreover, selecting the best locality preserving pattern allows one to design, through orders, a new curve with a comparable overall locality preserving refer to Hilbert curve. The contribution of new patterns is experimented through a CBIR (Content-Based Image Retrieval) application. Large-scale image retrieval tests show that exploring the image feature space with an alternative way to the classical Hilbert curve can lead to improved image searching performances.

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Notes

  1. http://grec2013.loria.fr/GREC2013/node/16

  2. http://mpeg.chiariglione.org/standards/mpeg-7

  3. http://www.lems.brown.edu/~dmc/

  4. The RBG pattern is founded on the Reflected Binary Gray code. The RBG code generates binary words of the given size D (i.e. each word contains D bits) satisfying the condition that two successive words have D − 1 identical bits.

  5. This is not surprising because by satisfying the proposed multidimensional generalization of the Hilbert rule, the patterns solution belongs to the same family of curves i.e. the family of Hilbert-like space-filling curve.

  6. The sample test was composed of 400 images randomly selected from the database.

  7. Our tests are realized on a standard laptop PC (Intel Core 2 Duo T9800 2.93Ghz x 2, 8GB RAM) running Ubuntu 12.10. The B+tree data structure is used for efficient implementation of the large-scale database.

  8. Practically, the considered most-locality preserving pattern is the best solution off-line obtained from Fig. 7 after twenty-four hours of calculation.

  9. The multidimensional Hilbert curve, is the RBG based-curve.

References

  1. Amory A, Sammouda R, Mathkour H, Jomaa R (2012) A content based image retrieval using k-means algorithm. In: IEEE 7th International Conference on Digital Information Management (ICDIM), pp 221–225

  2. Armstrong J, Ahmed M, Chau S (2009) A rotation-invariant approach to 2d shape representation using the hilbert curve. In: Springer (ed) Proceedings of the 6th International Conference on Image Analysis and Recognition, pp 594–603

  3. Bader M (2012) Space-Filling Curves: An Introduction with Applications in Scientific Computing. Springer Science and Business Media. doi:10.1007/978-3-642-31046-1

  4. Bilenko M, Basu S, Mooney R (2004) Integrating constraints and metric learning in semi-supervised clustering. In: Proceedings of the 21st International Conference on Machine Learning (ICML), pp 81–88

  5. Biswas S (2000) Hilbert scan and image compression. In: Proceedings of the 15th International Conference on Pattern Recognition, vol 3, pp 207–210

  6. Butz A (1969) Convergence with Hilbert’s space filling curve. J Comput Syst Sci 3(2):128–146

    Article  MathSciNet  MATH  Google Scholar 

  7. Butz A (1971) Alternative algorithm for Hilbert’s space-filling curve. IEEE Trans Comput 20(4):424–426

    Article  MATH  Google Scholar 

  8. Chatzichristofis SA, Zagoris K, Boutalis YS, Papamarkos N (2010) Accurate image retrieval based on compact composite descriptors and relevance feedback information. Int J Pattern Recognit Artif Intell 24:207–244

    Article  Google Scholar 

  9. Chierichetti F, Panconesi A, Raghavan P, Sozio M, Tiberi A, Upfal E (2007) Finding near neighbors through cluster pruning. In: Proceedings of the 26th ACM SIGMOD- SIGACT-SIGART symposium on Principles of Database Systems (PODS), pp 103–112

  10. Cox T, Cox M (1994) Multidimensional scalling. Chapman & Hal, London

    Google Scholar 

  11. Doulamis N, Doulamis A (2006) Evaluation of relevance feedback schemes in content-based in retrieval systems. Signal Process Image Commun 21:334–357

    Article  MATH  Google Scholar 

  12. Faloutsos C, Roseman S (1989) Fractals for secondary key retrieval. In: Proceedings of the 8th ACM SIGACT-SIGMOD-SIGART symposium on Principles of Database Systems (PODS), pp 247–252

  13. Guimarães Pedronette DC, S Torres R (2013) Image re-ranking and rank aggregation based on similarity of ranked lists. Pattern Recogn 46(8):2350–2360

    Article  Google Scholar 

  14. Haji-Hashemi M, Mir-Mohammad Sadeghi H, Moghtadai V (2006) Space-filling patch antennas with cpw feed. In: Progress in Electromagnetics Research Symposium. Cambridge, USA, pp 69–73

  15. Hamilton CH, Rau-Chaplin A (2008) Compact Hilbert indices: Space-filling curves for domains with unequal side lengths. Inf Process Lett 105(5):155–163

    Article  MathSciNet  MATH  Google Scholar 

  16. Hartigan J, Wong M (1979) Algorithm as136: a k-means clustering algorithm. R Stat Soc Ser C 28:100–108

    MATH  Google Scholar 

  17. He X, Niyogi P (2003) Locality preserving projection. In: Advances in Neural Information Processing Systems, vol 16. MIT Press, pp 153–160

  18. Hilbert D (1891) Ueber die stetige abbildung einer line auf ein flächenstück. Math Ann 38(3):459–460

    Article  MathSciNet  MATH  Google Scholar 

  19. Ho J, Lin S, Fann C, Wang Y, Chang R (2012) A novel content based image retrieval system using k-means with feature extraction. In: IEEE International Conference on Systems and Informatics (ICSAI), pp 785–790

  20. Hosny KM (2008) Fast computation of accurate zernike moments. J Real-Time Image Proc 3(1-2):97–107

    Article  Google Scholar 

  21. Hue-Ling C, Ye-In C (2005) Neighbor-finding based on space-filling curves. Inf Syst 30:205–226

    Article  Google Scholar 

  22. Jin G, Mellor-Crummey J (2005) Sfcgen: a framework for efficient generation of multi-dimensional space-filling curves by recursion. ACM Trans Math Softw (TOMS) 31:120–148

    Article  MathSciNet  MATH  Google Scholar 

  23. Jolliffe I (2002) Principal component analysis, 2nd edn. Springer Series in Statistics, Springer

  24. Kanungo T, Haralick R, Baird H, Stuezle W (2000) A statistical, nonparametric methodology for document degradation model validation. IEEE Trans Pattern Anal Mach Intell 22(11):1209–1223

    Article  Google Scholar 

  25. Khotanzad A, Hong Y (1990) Invariant image recognition by zernike moments. IEEE Trans Pattern Anal Mach Intell 12(5):489–497

    Article  Google Scholar 

  26. Kim D, Chung C, Barnard K (2005) Relevance feedback using adaptive clustering for image similarity retrieval. J Syst Softw 78(1):9–23

    Article  Google Scholar 

  27. Kim WY, Kim YS (2000) A region-based shape descriptor using zernike moments. Signal Process Image Commun 16(1-2):95–102

    Article  Google Scholar 

  28. Lawder J, King P (2000) Using space-filling curves for multi-dimensional indexing. In: Advances in databases, vol 1832, pp 20–35

  29. Lawder J, King P (2001) Using state diagrams for hilbert curve mappings. Int J Comput Math 78(3):327–342

    Article  MathSciNet  MATH  Google Scholar 

  30. Lebesgue H (1904) Leċons sur l’intégration. Gauthier-Villars, Paris

    MATH  Google Scholar 

  31. Liu P, Jia K, Lv Z (2008) An effective and fast retrieval algorithm for content-based image retrieval. In: IEEE Congress on Image and Signal Processing (CISP’08), vol 2, pp 471–474

  32. Liu X (2004) Four alternative patterns of the hilbert curve. Appl Math Comput 147(3):741–752

    MathSciNet  MATH  Google Scholar 

  33. McVay J, Hoorfar A, Engheta N (2005) Thin absorbers using space-filling-curve high-impedance surfaces. In: IEEE International Symposium on Antennas and Propagation, vol 2A, pp 22–25

  34. Mehtre BM, Kankanhalli MS, Lee WF (1997) Shape measures for content based image retrieval: a comparison. Inf Process Manag 33(3):319–337

    Article  Google Scholar 

  35. Mitchison G, Durbin R (1986) Optimal numberings of an n*n array. SIAM Journal on Algebraic Discrete Methods 7(4):571–582

    Article  MathSciNet  MATH  Google Scholar 

  36. Mokbel MF, Aref WG (2003) Analysis of multi-dimensional space-filling curves. GeoInformatiqua 7(3):179–209

    Article  Google Scholar 

  37. Moon B, Jagadish H, Faloutsos C, Saltz J (2001) Analysis of the clustering properties of the hilbert space-filling curve. IEEE Trans Knowl Data Eng 13(1):124–141

    Article  Google Scholar 

  38. Müller H, Müller W, McG Squire D, Marchand-Maillet S, Pun T (2001) Performance evaluation in content-based image retrieval: overview and proposal. Pattern Recogn Lett 22:593–601

    Article  MATH  Google Scholar 

  39. Murthy V, Vamsidhar E, Swarup Kumar J, Sankara Rao P (2010) Content based image retrieval using hierarchical and k-means and clustering techniques. Int J Eng Sci Technol 2(3):209–212

    Google Scholar 

  40. Nagthane D (2013) Content based image retrieval system using k-means clustering technique. Int J Comput Appl Inf Technol 3(1):22–29

    Google Scholar 

  41. Nguyen G (2013) Space-filling curves and their application in image processing. PhD thesis, University of La Rochelle (France), Laboratoire Informatique Image et Interactions (L3I). HAL Id: tel-01174960

  42. Nguyen G, Franco P, Mullot R, Ogier JM (2012) Mapping high dimensional image features onto hilbert curve: applying to fast image retrieval. In: Proceedings of the 21st International Conference on Pattern Recognition. IEEE Computer Society, pp 425–428

  43. Novotni M, Klein R (2004) Shape retrieval using 3d zernike descriptors. Comput Aided Des 36(11):1047–1062

    Article  Google Scholar 

  44. Park G, Baek Y, Lee HK (2005) Re-ranking algorithm using post-retrieval clustering for content-based image retrieval. Inf Process Manag 41(2):177–194

    Article  MATH  Google Scholar 

  45. Peano G (1890) Sur une courbe, qui remplit toute une aire plane. Math Ann 36(1):157–160

    Article  MathSciNet  MATH  Google Scholar 

  46. Perez A, Kamata S, Kawaguchi E (1992) Peano scanning of arbitrary size images. In: Proceedings of the International Conference on Pattern Recognition, pp 565–568

  47. Sagan H (2012) Space-Filling Curves. Springer Science and Business Media. doi:10.1007/978-1-4612-0871-6

  48. Sergeyev YD, Strongin RG, Lera D (2013) Introduction to Global Optimization Exploiting Space-Filling Curves. Springer Science & Business Media

  49. Teague M (1980) Image analysis via the general theory of moments. J Opt Soc Am 70:920–930

    Article  MathSciNet  Google Scholar 

  50. Tsai C (2012) Bag-of-words representation in image annotation: a review ISRN Artif Intell, 2012

  51. Velho L, Gomes JdM (1991) Digital halftoning with space filling curves. ACM SIGGRAPH Computer Graphics 25(4):81–90

    Article  Google Scholar 

  52. Yasser E, Maher A, Siu-Cheung C, Wegdan A (2007) A view-based 3d object shape representation technique. In: Image Analysis and Recognition, vol 4633. Springer, Berlin Heidelberg, pp 411–422

  53. Yasser E, Maher A, Wegdan A, Siu-Cheung C (2009) Shape representation and description using the hilbert curve. Pattern Recogn Lett 30(4):348–358

    Article  Google Scholar 

  54. Ying L, Dengsheng Z, Guojun L, Wei-Ying M (2007) A survey of content-based image retrieval with high-level semantics. Pattern Recogn 40(1):262–282

    Article  MATH  Google Scholar 

  55. Zhang D, Lu G (2002) Shape-based image retrieval using generic fourier descriptor. Signal Process Image Commun 17(10):825–848

    Article  Google Scholar 

  56. Zhang D, Lu G (2004) Review of shape representation and description techniques. Pattern Recogn 37(1):1–19

    Article  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Informatique, Image, Interaction (L3i), EA 2118- University of La Rochelle (France), Avenue M. Crepeau, 17000, La Rochelle, France

    Patrick Franco, Giap Nguyen, Remy Mullot & Jean-Marc Ogier

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  1. Patrick Franco
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  2. Giap Nguyen
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Correspondence to Patrick Franco.

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Franco, P., Nguyen, G., Mullot, R. et al. Alternative patterns of the multidimensional Hilbert curve. Multimed Tools Appl 77, 8419–8440 (2018). https://doi.org/10.1007/s11042-017-4744-4

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