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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.

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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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  3. Remy Mullot
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  4. Jean-Marc Ogier
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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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