TY - JOUR

T1 - Analysis of multi-dimensional space-filling curves

AU - Mokbel, Mohamed F.

AU - Aref, Walid G.

AU - Kamel, Ibrahim

PY - 2003/9

Y1 - 2003/9

N2 - 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.

AB - 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.

KW - Fractals

KW - Locality-preserving mapping

KW - Performance analysis

KW - Space-filling curves

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U2 - 10.1023/A:1025196714293

DO - 10.1023/A:1025196714293

M3 - Article

AN - SCOPUS:0141682542

VL - 7

SP - 179

EP - 209

JO - GeoInformatica

JF - GeoInformatica

SN - 1384-6175

IS - 3

ER -