Data Visualization
Fall 2015
Tensor Data Visualization
• 0th order tensors – scalars (magnitude)
• 1st order tensors – vectors (magnitude + direction)
• 2nd order tensors – quantities that have magnitude and two
directions, represents variation of magnitude
• Tensors (math.) are independent of the coordinate systems
• Components are tensor representation in a coordinate system
• Tensor field T: R3xR3 → R
Fall 2015
(vector field v: R3 → R3)
Data Visualization
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Tensor Data Visualization
• A tensor with rank (order) r in a n-dimensional space has r indices and
nr components
• r … number of simultaneous directions
• Scalars (r = 0) … n0 component with no index (value)
• Vectors (r = 1) … n1 components with only one index (vector[i])
• Matrices (r = 2) … n2 components with two indices (matrix[i][j])
• Tensors (r ≥ 3)
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Data Visualization
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Tensor Data Visualization
• Cauchy stress tensor – 2nd order tensor
• Completely define the state of stress at a point inside a material
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Data Visualization
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Tensor Data Visualization
• Tensor fields are common quantity in engineering and physical
sciences:
• Stress, strain, diffusion, velocity gradients, etc.
• Mostly second-order tensors – interpreted as a linear transformation
between vectors (represented in 3D by 3x3 matrices):
• Stress to strain, force to deformation
• Special case – symmetric second-order tensors:
• Can be viewed as anisotropic ellipsoids (eigenvectors and
eigenvalues are principal axes of the diffusion ellipsoids)
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Data Visualization
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Tensor Data Visualization
• Tensor attributes are high-dimensional generalization of vectors and
matrices
• In other words, tensor data encode some spatial property that varies
as a function of position and direction
• Example: curvature as a tensor
• For every point x0 on planar curve consider local coordinate
system (x, y) such that:
• x – tangent to the curve
• y – normal to the curve
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Data Visualization
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Tensor Data Visualization
• Example: curvature as a tensor
• For every point x0 on planar curve consider local coordinate
system (x, y) such that:
• x – tangent to the curve
• y – normal to the curve
• In this system, curve can be described as y = f(x) in the
neighborhood of x0, where f(x0) = 0
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Data Visualization
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Tensor Data Visualization
• Example: curvature as a tensor
• Curve is described as y = f(x) in the neighborhood of x0,
where f(x0) = 0
2 f
• Curvature in 1D: C ( x)  2 ( x)
x
• C(x) = 1 / radius of circle tangent to f at x
• How quickly normal n changes around x
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Data Visualization
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Tensor Data Visualization
• Example: curvature as a tensor
• What if we deal with surfaces?
• Analogy with 1D case but…
• In which direction to look for changes?
2 f
• We have to compute C (x, s)  2 (x)
s
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Data Visualization
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Tensor Data Visualization
• Example: curvature as a tensor
2 f
• We have to compute C (x, s)  2 (x)  sT Hs
s
where H is Hessian matrix as follows
 2 f
 2
x
H 2
 f

 yx
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2 f 

xy 
2 f 
2 
y 
Data Visualization
The problem is that we need
to construct local coordinate
systems at every point on
surface and it is not obvious
how to do that
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Tensor Data Visualization
• Example: curvature as a tensor
• Let the surface be describe as an implicit function (i.e. the zerolevel isosurface of a function)
2 f
• We have to compute C (x, s)  2 (x)  sT Hs
 2 f
 2
 x
 2 f
H
 2yx
 f
 zx

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s
2 f
xy
2 f
y 2
2 f
zy
Data Visualization
2 f 

xz 
2 f 

yz 
2 f 
z 2 
A curvature tensor is fully
described by 3x3 matrix of
2nd order derivatives
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Tensor Data Visualization
• Example: diffusion tensor
• Consider an anisotropic material
• We have to compute diffusivity at a point x in a direction s
2 f
D(x, s)  2 (x)
s
f … speed of water motion in tissue
• Application: Diffusion of water in the human brain tissue
• Strong along neural fibers
• Weak across fibers
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Data Visualization
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Tensor Data Visualization
• Example: diffusion tensor
• Compute Hessian in R3
• Select some slice of interest
• Visualize all components of H using
color mapping
• We get 9 images, some are same
due to the symmetry
• But we do not really care about
diffusion along x, y, z axes
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Data Visualization
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Tensor Data Visualization
• Principal component analysis
α … angle of s with local coordinate axis x0
2 f
C (x, s)  2 (x)  sT Hs  h11 cos 2   (h12  h21 ) sin  cos   h22 cos 2 
s
C
( x, s )  0

We are looking for extremal curvature
• This is equivalent to a system of equations in matrix form
Hs  s
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( H  I )s  0
Data Visualization
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Tensor Data Visualization
• Principal component analysis

1
1  2  3 
3
• White = strong mean diff.
• Black = weak mean diff.
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Data Visualization
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Tensor Data Visualization
Fall 2015
Data Visualization
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Tensor Data Visualization
Fall 2015
Data Visualization
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Tensor Data Visualization
• Tensor visualization:
• Component visualization
• Anisotropy visualization
• Major eigenvector visualization
• Fiber tracking
• Basic fiber tracking
• Stream tubes
• Hyperstreamlines
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Data Visualization
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