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Tensor Algorithms & Visualization By Arun Rao. Presentation Outline Introduction to Tensors Boussinesq Problem Visualizations Recent Uses.

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Presentation on theme: "Tensor Algorithms & Visualization By Arun Rao. Presentation Outline Introduction to Tensors Boussinesq Problem Visualizations Recent Uses."— Presentation transcript:

1 Tensor Algorithms & Visualization By Arun Rao

2 Presentation Outline Introduction to Tensors Boussinesq Problem Visualizations Recent Uses

3 Tensor Rank RankAliasElement notation Common transformation * Geometrical interpretation 0ScalaraS'=|a|S· 1Vectoraiai V' i =|a|a ij V j 2Matrixa ij M' ij =|a|a ik a jl M kl 3?a ijk M' ijk =|a|a il a js a km M lsm

4 “So, tensors are…?” –Multidimensional arrays –N dimensional generalizations of scalars 1 dimensional vectors 2 dimensional matrices – Elements: Numbers Functions Differentials

5 Boussinesq’s Problem…Saada To The Rescue Given: –Surface on Semi-infinite Domain –Point Load Need: –Stress felt on all points in the material

6 Saada’s Solution – Stress Tensor Solve for the stress tensor on the left.

7 Visualizing Tensors Ellipsoids –Eigenvectors and Eigenvalues

8 Visualizing Tensors Hyperstreamline Generation

9 Visualizing Tensors Results

10 Other uses of Tensors… Human brain pathways recovered from DT-MRI data using an oriented tensor reconstruction algorithm (below)

11 References Wikipedia - http://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Tensor Mathworld - http://mathworld.wolfram.com/Tensor.htmlhttp://mathworld.wolfram.com/Tensor.html Oriented Tensor Reconstruction - http://www.gg.caltech.edu/~zhukov/research/fiber_tracking/vis02/vi s02.html http://www.gg.caltech.edu/~zhukov/research/fiber_tracking/vis02/vi s02.html

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