Strategies for Direct Volume Rendering of Diffusion Tensor Fields

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Strategies for Direct Volume Rendering of
Diffusion Tensor Fields

• Gordon Kindlmann, David Weinstein, and David Hart
• Presented by Chris Kuck

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Diffusion Tensor fields

• In the living tissue water molecules exhibit
  Brownian motion. This water motion, or
  diffusion, can be isotropic or anisotropic.
• The diffusion can be described by a 3x3 symmetric
  real valued matrix and is used as a good
  approximation to the diffusion process.
• These matrices are calculated from a sequence of
  diffusion-weighted MRIs.

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Diffusion Tensor fields

• The direction and magnitude are stored as the
  systems orthogonal eigenvectors and eigenvalues
• Each tensor, and its corresponding eigensystem,
  can be represented in a concise and elegant way
  as an ellipsoid.

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Why use diffusion tensor fields?

• Currently we have no way of visualizing the
  structure of the white matter contained in the
  brain.
• However, these intricate structures can be
  differentiated from the surrounding material by
  observing that white matter generally exhibits
  the property of anisotropy.
• However, the cases where highly concentrated
  white matter are involved, this property could be
  false.

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White Matter Visualization

• Creating a detailed understanding of the white
  matter tracts in the brain by visualizing the
  structure of these white matter tracts, could
  lead to advances in neuroanatomy, surgical
  planning, and cognitive sciences.

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Strategies

• Barycentric Mapping
• Lit-Tensors
• Hue-Balls and Deflection Mapping
• Reaction-Diffusion Textures
• Diffusion Interpolation

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Barycentric Mapping

• Motivation Displaying 6 dimensions of data at
  once is not a useful visualization. We need some
  way to reduce the diffusion-tensor field.
• Ideally we would like the resulting visualization
  to be opaque where there are regions of interest
  and transparent elsewhere.

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Barycentric Mapping

• Before we can adequately remove isotropic areas
  of diffusion from the brain visualization, we
  must first define what anisotropy means
• They used Westen et al.s formulas to derive the
  amount of anisotropy in each tensor.

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Isotropic/anisotropic coefficients

• Where cl is the amount of linear anisotropy, cp
  is the amount of planar anisotropy and cs is the
  amount of isotropy and cl cp cs 1.

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Anisotropy Index

• Which gives way to

Ca is defined as the anisotropy index.
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Barycentric space

• Now we define a space that is called barycentric
  space. This space is the combination of all
  different types of anisotropy and isotropy.
• With this space we can mark each tensor with a
  value of ca, cl, cp , and cs .

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Barycentric Mapping

• After marking each element, we can create a
  look-up table in Barycentric space to see what
  value for opacity we should use, thus reducing
  the entire dataset down to one dimension.

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Barycentric Mapping
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Barycentric Mapping

• As well as a look-up table for opacity, it is
  possible to create a similar look-up table in
  Barycentric space for color such that the user
  can see the different types of anisotropy in the
  brain.

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Barycentric Mapping
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Lit-Tensors

• Now that we have reduced the data set to a
  reasonably amount of data we need to somehow
  depict accurate lighting as well.
• Theyre solution is a shading technique termed
  lit-tensors, which can indicate the type and
  orientation of anisotropy.

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Lit-Tensors

• They follow these constraints to do so
• With linear anisotropy lighting should be
  identical to illuminated streamlines
• In planar anisotropy lighting should be identical
  to standard surface rendering.
• Every where else the surface normals must be
  smoothly interpolated

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Lit-Tensors

• This problem can be viewed as a codimension
  problem. The ellipsoid that is represents
  linear anisotropy has a codimension of 2, and 1
  with planar anisotropy.
• D.C. Banks, Illumination in Diverse
  Codimension
• This paper, unlike Banks, however makes no
  claims of its physical accuracy or plausibility.

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Lit-Tensor calculation

• First, start with Blinn-Phong Shading model

Where ka, kd, and ks are the respective intensity
coefficients, A is the amount of ambient light, O
is the Object color. ? is replaced with either
r, g, or b for color. L is the vector pointing
to the directional light source, N is the normal
of the surface, and H is the half-way vector.
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Lit-Tensor calculation

• You can view linear anisotropic tensors as
  streamlines and because of this, there are an
  infinite set of normals.
• By using the Pythagorean theorem the dot product
  can by expressing in terms of a T tangent to the
  surface.

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Lit-Tensor calculation

• Where U is either L or H depending on whether you
  are doing specular or diffusion lighting
  respectively.
• T could be represented with either 1 vector in
  the planar case, or 2 vectors in the linear case,
  thus a new parameter is needed

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Lit-Tensor calculation

• Where are the eigenvalues sorted as lambda1 gt
  lambda2 gt lambda3.
• Ctheta ranges from completely linear ( 0 ) to
  completely planar ( p/2 ). Then the dot product
  can be rearranged as

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Lit-Tensors
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Lit-Tensors

• Lit-Tensors accomplish their goal of computing
  lighting conditions via the direction and
  magnitude of each tensor.
• However this does not provide a very intuitive
  way to view the lighting conditions.
• Their solution was to mix, or use completely,
  opacity gradient shading.

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Lit-Tensors
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Hue-Balls and Deflection Mapping

• The idea Take a tensor and reduce it to a
  vector. Then map from this vector to a point on
  a color unit sphere.
• How they accomplish this is they pick some input
  vector, and multiply it by the tensor.
• This output vector is then mapped onto the color
  unit sphere.

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Hue-Balls and Deflection Mapping
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Hue-Balls and Deflection Mapping

• In Addition to finding the color, they compute
  deflection by finding the difference between
  the input vector and output vector.
• This vector, when there are high levels of
  anisotropy, will be deflected a great amount.
  Since they are trying to make all anisotropic
  areas opaque, they use this value to assign an
  opacity.

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Hue-Balls and Deflection Mapping
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Reaction Diffusion Textures

• Reaction Diffusion textures are an idea that was
  introduced by Turing that was looking for a
  mathematical model to describe pigmentation
  patterns in the animal kingdom.
• The equations that Turing proposed are simple in
  nature.

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Reaction Diffusion Textures
Where at t0, ab4. k is the controlling factor
in the growth of the patterns. da and db control
how fast the two chemicals can spread throughout
the medium. The overall convergence speed is
controlled by s. Finally ß is a pattern of
uniformly distributed random values in a small
interval centered around 0.
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Reaction Diffusion calculation

• In practice, a regular two-dimensional grid is
  used to simulate the Laplacian.
• This could be extended to three dimensions pretty
  easily.

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Reaction Diffusion calculation

• In three dimensions the discrete grid to convolve
  to approximate the Laplacian is similar.

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Reaction Diffusion calculation

• To get the texture to be representative of the
  tensor data, they use Ficks second law to
  determine diffusion.
• The resulting equations change the discrete grid
  once again.

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Reaction Diffusion visualization

• To visualize the data in a more convenient
  fashion, they remove all areas that are not
  anisotropic.
• They do this by calculating the average
  anisotropy, and any spots below .5 average
  anisotropy are removed.

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Diffusion Tensor Interpolation

• 3 different ways to interpolate the tensors
• MRI channels
• Tensor matrices
• Eigensystems

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MRI channel interpolation

• It would be more accurate to interpolate straight
  from the MRI channels, to do this first you must
  calculate a set of log image values via

Where Ai is the channel and b is the direction
independent diffusion weight
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MRI channel Interpolation

• Then the diffusion tensor is

While this is more accurate it is not
computationally feasible to do this.
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Matrix Interpolation

• Another way to do this is component wise
  interpolation in the matrices.
• If the process of getting the log images were
  linear, interpolation between the MRI channels
  and the interpolation between matrices, would be
  identical. However, it is not linear.

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Eigensystem interpolation

• Another approach is to solve the eigensystems for
  each sample point, store this volume instead of
  the tensor volume, and then interpolate as
  needed.
• This holds an extreme advantage over MRI channel
  interpolation and Matrix interpolation since each
  eigensystem needs to be solved only once.

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Eigensystem interpolation

• However, the orientation of the principal
  eigenvector, is undetermined at several times.
• During planar anisotropy the direction of the two
  principle eigenvectors degenerates to every
  vector perpendicular to the minor axis.
• Also between any two points, how are we
  guaranteed that the direction is the same as when
  we started.

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Evaluation

• The results that they found were computed using
  Barycentric maps.
• They came to the conclusion that even though
  eigensystem interpolation is dramatically
  cheaper, if the density of points are not close
  enough, it loses too much information and ends up
  being too inaccurate.
• They chose matrix interpolation for every
  calculation they used in this paper.

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