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 E n 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 CurveSnake Scan Curve

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

7. 3D SFC Z CurvePeano Curve Hilbert Curve

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

9. Coherent in Continuity

10. Clustering property ZHZH

11. Direction Preserving P1: (X 1, Y 1 ) X 1 > X 2 > X 3 P2: (X 2, Y 2 ) Y 1 > Y 2 > Y 3 P3: (X 3, Y 3 ) 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. Visualizing Color (RGB) (160, 100) (160, 100, 100) 100 ?

14. Visualizing Color (RGB) Z-Curve Mapping

15. Visualizing Color (RGB) Hilbert Curve Mapping

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

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

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

19. Extension to Hilbert Curve

20. The Visualization of Iris Flowers Extended Hilbert Curve

21. Questions? Suggestions?