<title>N-dimensional Hilbert scanning and its application to data compression</title>

Abstract

Hilbert scanning defines a mapping, hn : R yields Un, that maps the unit interval onto the n-dimensional unit hypercube continuously. In the discrete case the mapping can be described in terms of Reflected Binary Gray Codes (RBGC). In order to extend the quantized mapping to arbitrary precision it is necessary to define induction rules. Induction rules are defined in terms of a single canonical sequence and a set of rotations. In general, in an n-dimensional hypercube there are n2n possible orientations of a canonical form. Beyond two dimensions, it is possible to have nontrivially different paths between two possible orientations and it is better to define the induction rule in terms of the end points of the RBGC subsequences. Hilbert coding is used for n- dimensional binary data compression. The effectiveness of this method to data compression is confirmed. Experimental evaluation shows Hilbert- Wyle coding to be consistently better than other standard compression methods.