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Exploring Space-Filling Curves in Information Visualization

Learn about the concept and applications of space-filling curves for mapping multi-dimensional data to one-dimensional sequences. Discover various non-recursive and recursive categories such as Z-Scan, Hilbert, and Peano curves, with properties like coherence and direction preservation. Explore the use of SFCs in information visualization through examples with color mapping and data representation.

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Exploring Space-Filling Curves in Information Visualization

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  1. Space-Filling Curves in Information Visualization Jiwen Huo jhuo@cgl.uwaterloo.ca

  2. Space-filling curves • Definition: • A continuous curve which passes through every point of a closed n-cell in Euclidean n-space En is called a space filling curve (SFC).

  3. Application of SFC • Mapping multi-dimensional space to one dimensional sequence • Applications in computer science: • Database multi-attribute access • Image compression • Information visualization • ……

  4. Categories of SFC • Non-recursive • Z-Scan Curve • Snake Scan Curve • Recursive • Hilbert Curve • Peano Curve • Gray Code Curve

  5. Non-recursive Space Filling Curves Z-Scan Curve Snake Scan Curve

  6. Recursive Space Filling Curves Hilbert Curve Gray Code Curve Peano Curve

  7. 3D SFC Hilbert Curve Z Curve Peano Curve

  8. Hilbert Curve Z Curve Z Curve Hilbert Curve Good Good Properties of SFCs • Coherent in Continuity • Clustering Property • Direction Preserving

  9. Coherent in Continuity

  10. Clustering property Z H

  11. Direction Preserving P1: (X1, Y1) X1 > X2> X3 P2: (X2, Y2) Y1 > Y2> Y3 P3: (X3, Y3) Z(P1) > Z(P2) > Z(P3) H(P3) < H(P1) < H(P2)

  12. SFC in Information Visualization • Example Data • Color (R, G, B: [0, 255]) • Data with obvious geometric pattern • 4D Hyper Sphere • Data without obvious geometric pattern • Iris flowers (5 attributes, 3 classes) • Example SFC • Z-Curve • Hilbert Curve

  13. (160, 100) (160, 100, 100) ? 100 Visualizing Color (RGB)

  14. Visualizing Color (RGB) Z-Curve Mapping

  15. Visualizing Color (RGB) Hilbert Curve Mapping

  16. Visualizing Color (RGB) Green & Blue Z Red H

  17. Z-Curve Visualizing 4D Hyper-Sphere Surface • Hilbert Curve

  18. Z-Curve Hilbert Curve The Visualization of Iris Flowers

  19. Extension to Hilbert Curve

  20. The Visualization of Iris Flowers • Extended Hilbert Curve

  21. Questions? Suggestions?

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