A generalized 3-D Hilbert scan using look-up tables

Abstract

The Hilbert curve is a one-to-one mapping between multidimensional space and one-dimensional (1-D) space. Due to the advantage of preserving high correlation of multidimensional points, it receives much attention in many areas. Especially in image processing, Hilbert curve is studied actively as a scan technique (Hilbert scan). Currently there have been several Hilbert scan algorithms, but they usually have strict implementation conditions. For example, they use recursive functions to generate scans, which makes the algorithms complex and difficult to implement in real-time systems. Moreover the length of each side in a scanned region should be same and equal to the power of two, which limits the application of Hilbert scan greatly. In this paper, to remove the constraints and improve the Hilbert scan for a general application, an effective generalized three-dimensional (3-D) Hilbert scan algorithm is proposed. The proposed algorithm uses two simple look-up tables instead of recursive functions to generate a scan, which greatly reduces the computational complexity and saves storage memory. Furthermore, the experimental results show that the proposed generalized Hilbert scan can also take advantage of the high correlation between neighboring lattice points in an arbitrarily-sized cuboid region, and give competitive performance in comparison with some common scan techniques.