## Hilbert Space Filling Curves: Theory and Applications

### Abstract

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 $2^n \times 2^n$ 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.