Strategies for Direct Volume Rendering of Diffusion Tensor FieldsDiffusion Tensor fieldsSlide 3Why use diffusion tensor fields?White Matter VisualizationStrategiesBarycentric MappingSlide 8Isotropic/anisotropic coefficientsAnisotropy IndexBarycentric spaceSlide 12Slide 13Slide 14Slide 15Lit-TensorsSlide 17Slide 18Lit-Tensor calculationSlide 20Slide 21Slide 22Slide 23Slide 24Slide 25Hue-Balls and Deflection MappingSlide 27Slide 28Slide 29Reaction Diffusion TexturesSlide 31Reaction Diffusion calculationSlide 33Slide 34Reaction Diffusion visualizationPowerPoint PresentationSlide 37Diffusion Tensor InterpolationMRI channel interpolationMRI channel InterpolationMatrix InterpolationEigensystem interpolationSlide 43EvaluationSlide 45Strategies for Direct Volume Rendering of Diffusion Tensor FieldsGordon Kindlmann, David Weinstein, and David HartPresented by Chris KuckDiffusion 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 InterpolationBarycentric MappingMotivation: 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 coefficientsWhere 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 IndexWhich 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 MappingBarycentric 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 MappingLit-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 streamlines2) In planar anisotropy lighting should be identical to standard surface rendering.3) Every where else the surface normals must be smoothly interpolatedLit-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 modelWhere 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 calculationWhere 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 neededLit-Tensor calculationWhere 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 asLit-TensorsLit-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-TensorsHue-Balls and Deflection MappingThe 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 MappingHue-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
View Full Document
Unlocking...