Analysis of Multi-Dimensional Space-Filling Curves

  • Authors:

  • Mohamed F. Mokbel;Walid G. Aref;Ibrahim Kamel

  • Affiliations:

  • Department of Computer Sciences, Purdue University, West Lafayette, IN 47907-1398 mokbel@cs.purdue.edu;Department of Computer Sciences, Purdue University, West Lafayette, IN 47907-1398 aref@cs.purdue.edu;Panasonic Information and Networking Technologies Laboratory, Two Research Way, Princeton, NJ 08540 ibrahim@research.panasonic.com

  • Venue:
  • Geoinformatica
  • Year:
  • 2003

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Abstract

A space-filling curve is a way of mapping the multi-dimensional space into the 1-D space. It acts like a thread that passes through every cell element (or pixel) in the D-dimensional space so that every cell is visited exactly once. There are numerous kinds of space-filling curves. The difference between such curves is in their way of mapping to the 1-D space. Selecting the appropriate curve for any application requires knowledge of the mapping scheme provided by each space-filling curve. A space-filling curve consists of a set of segments. Each segment connects two consecutive multi-dimensional points. Five different types of segments are distinguished, namely, Jump, Contiguity, Reverse, Forward, and Still. A description vector V=(J, C, R, F, S), where J, C, R, F, and S are the percentages of Jump, Contiguity, Reverse, Forward, and Still segments in the space-filling curve, encapsulates all the properties of a space-filling curve. The knowledge of V facilitates the process of selecting the appropriate space-filling curve for different applications. Closed formulas are developed to compute the description vector V for any D-dimensional space and grid size N for different space-filling curves. A comparative study of different space-filling curves with respect to the description vector is conducted and results are presented and discussed.