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

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

Tensor

Example Eq 2.5 Eq 2.4

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

Example Curvature calculation

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)

Example

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

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

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

PCA

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

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

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

Average Diagonal Entries Average diffusion

Visualizing Anisotropy

Visualizing Anisotropy

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

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

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

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

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.

Hyperstreamline

End of Chap. 7 Note: