1. Tensor Visualization Chap. 7 October 21, 2008 Jie Zhang Copyright ©
CDS 301
Fall, 2008
Jie Zhang
Copyright ©

2. Outline 7.1. Principle Component Analysis (PCA)
7.2. Visualizing Components
7.3. Visualizing Scalar PCA Information
7.4. Visualizing Vector PCA Information
7.5. Tensor Glyphs
7.6. Fiber Tracking
7.7. Hyperstreamlines

3. Tensor

4. Example
Eq 2.5
Eq 2.4

5. Example Curvature of a 3-D surface is a tensor.
Curvature is different along different direction

6. Example
Curvature calculation

7. Example
Question: for the following Gaussian function that defines a 3-D surface,
Calculate its normal direction at point (0,0),
Calculate the curvatures along α=0, 45 and 90 degrees at point (0,0)
Calculate the gradient of the corresponding 2-D function at point (0,0)

8. Example

9. (continued) Tensor Visualization
Chap. 7
October 23, 2008

10. Visualizing Components
Data: (3 X 3) Diffusion tensor data from DT-MRI scan
Show only one slice of the 3-D data
Nine Components.

11. Principal Component Analysis (PCA)
A tensor has principal directions, e.g., along which the curvatures are extremal (maximum or minimum)
s: eigenvector
λ: eigenvalue

12. PCA

13. PCA: example
Question: calculate the eigenvalue and eigenvector of the following tensor

14. PCA If we order the eigenvalue in decreasing order:
In case of curvature:
e1 the direction of maximal curvature
e2 the direction of minimum curvature
e3: the direction of surface normal

15. PCA maximum curvature: along red line  maximum eigenvector
Minimum curvature: along the yellow line  medium eigenvector

16. Average Diagonal Entries
Average diffusion

17. Visualizing Anisotropy

18. Visualizing Anisotropy

19. Visualizing Vector PCA
Show major eigenvector direction using color coding
Shaded color sphere:
R: horizontal (X)
G: vertical (Y)
B: depth (Z)

20. Tensor Glyphs Ellipsoid Tensor Glyphs In triangle “glyph space”
Three corners:
Linear,
Planar,
Spherical

21. Tensor Glyphs DT-MRI Ellipsoid Tensor Glyphs
Color: direction on shaded sphere

22. Fiber Tracking Similar to vector streamline
Tracing along the major eigenvector
Start from a seed region

23. Hyperstreamline Similar to stream tube
The stream line is not convolved with a circular cross section
But an elliptic cross section, whose axes are along the directions of the medium and minor eigenvectors.

24. Hyperstreamline

25. End of Chap. 7
Note: