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Journal of Information Science and Engineering, Vol. 24 No. 1, pp. 261-275


Tensor Product Formulation for Hilbert Space-Filling Curves

Shen-Yi Lin, Chih-Shen Chen, Li Liu+ and Chua-Huang Huang
Department of Information Engineering and Computer Science 
Feng Chia University 
Taichung, 407 Taiwan 
+Graduate Institute of Medical Informatics 
Taipei Medical University 
Taipei, 110 Taiwan

    We present a tensor product formulation for Hilbert space-filling curves. Both recursive and iterative formulas are expressed in the paper. We view a Hilbert space-filling curve as a permutation which maps two-dimensional 2n × 2n data elements stored in the row major or column major order to the order of traversing a Hilbert curve. The tensor product formula of Hilbert space-filling curves uses several permutation operations: stride permutation, radix-2 Gray permutation, transposition, and anti-diagonal transposition. The iterative tensor product formula can be manipulated to obtain the inverse Hilbert permutation. Also, the formulas are directly translated into computer programs which can be used in various applications including image processing, VLSI component layout, and R-tree indexing, etc.

Keywords: tensor product, block recursive algorithm, Hilbert space-filling curve, stride permutation, gray permutation, transposition, anti-diagonal transposition, data allocation

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