Strategies for Direct Volume Rendering of Diffusion Tensor Fields Gordon 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 lookup 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 littensors 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 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 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 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
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