Strategies for Direct Volume Rendering of Diffusion Tensor

FieldsGordon Kindlmann, David Weinstein, and David Hart

Presented by Chris Kuck

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 MRI’s.

Diffusion Tensor fields

• The direction and magnitude are stored as the systems orthogonal eigenvectors and eigenvalues

• Each tensor, and it’s corresponding eigensystem, can be represented in a concise and elegant way as an ellipsoid.

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.

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.

Strategies

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

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.

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.

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.

Anisotropy Index

Which gives way to:

Ca is defined as the anisotropy index.

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 .

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.

Barycentric Mapping

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.

Barycentric Mapping

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.

• They’re solution is a shading technique termed lit-tensors, which can indicate the type and orientation of anisotropy.

Lit-Tensors

• They follow these constraints to do so:1) With linear anisotropy lighting should be identical to

illuminated streamlines

2) In planar anisotropy lighting should be identical to standard surface rendering.

3) Every where else the surface normals must be smoothly interpolated

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.

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. λ 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.

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.

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

Lit-Tensor calculation

Where are the eigenvalues sorted as lambda1 >= lambda2 >= lambda3.

• Ctheta ranges from completely linear ( 0 ) to completely planar ( π/2 ). Then the dot product can be rearranged as

Lit-Tensors

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.

Lit-Tensors

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.

Hue-Balls and Deflection Mapping

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.

Hue-Balls and Deflection Mapping

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.

Reaction Diffusion Textures

Where at t=0, a=b=4. 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 β is a pattern of uniformly distributed random values in a small interval centered around 0.

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.

Reaction Diffusion calculation

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

Reaction Diffusion calculation

• To get the texture to be representative of the tensor data, they use Fick’s second law to determine diffusion.

• The resulting equations change the discrete grid once again.

Reaction Diffusion visualization

• To visualize the data in a more convenient fashion, they remove all area’s that are not anisotropic.

• They do this by calculating the average anisotropy, and any spots below .5 * average anisotropy are removed.

Diffusion Tensor Interpolation

• 3 different ways to interpolate the tensors:

1) MRI channels

2) Tensor matrices

3) Eigensystems

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

MRI channel Interpolation

Then the diffusion tensor is:

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

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.

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.

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.

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.