Visualization of Anatomic Covariance Tensor Fields Gordon L. Kindlmann, David M. Weinstein, Agatha D. Lee, Arthur W. Toga, and Paul M. Thompson Presented by: Eugene (Austin) Stoudenmire 31 Jan 2007

Problem Current methods of visualizing brain variability –Not sufficiently informative and interactive Current methods involve computing –for each vertex of brain surface model –distribution of displacement vectors between individual brains and average brain –summarizing as covariance tensor tensor – akin to vector field –And displaying as ellipsoid tensor glyphs glyph – icon that maps features onto primitive –shape, size, orientation, appearance

Ellipsoidal Tensor Glyphs But – feature values are hard to discern

We Care Understanding anatomic variability of the brain is important Visualization is an important component of understanding anatomic variability

Anatomic Variability Importance Functional organization differs among people –measures are required to represent and visualize systematic patterns Determine patterns of altered brain structure in diseases –based on databases of brain scans Statistical info on anatomical variance to facilitate computer vision algorithms that automatically identify brain structures

Visualization Importance Important component of understanding anatomic variability –Provide feedback for variability algorithm verification –Means of mucking with the data to form/refine hypotheses about variability

Approach Map values onto superquadric glyphs MRI brain scans of 40 subjects –Aligned and converted to 3D models Interest: Deformation that would transform average onto each individual Represented as 3D covariance tensor

Superquadric Glyphs

Superquadric Ellipsoid

Superquadric Ellipsoid Kindlmann, “Superquadric Tensor Glyphs”, Joint Eurographics – IEEE TCVG Symposium on Visualization 2004

Subject Mapping MRI brain scans of 40 subjects –Aligned to standardized brain –Converted aligned brains to 3D models –Matched up major fissures Interest: Minimum-energy 3D nonlinear elastic deformation that transforms the landmarks of average onto those of each individual Represented at each point as 3D covariance tensor of the displacement vectors induced by the deformations from the average to all individuals

Tensor Creation 3 x 3 covariance tensor T Diagonalized T = R  R -1 Where  = Diagonal matrix of eigenvalues R = Rotational matrix that transforms standard basis onto eigenvector basis

Tensor Creation Tensor orientation – Eigenvectors Tensor shape – Eigenvalues Kindlmann, “Superquadric Tensor Glyphs”, Joint Eurographics – IEEE TCVG Symposium on Visualization 2004

What Makes Superquadric So Good Edges of superquadric tensor glyph –indicate eigenvalue differences Eigenvalue differences –imply lack of rotational symmetry Therefore need to emphasize glyph edges –which will emphasize eigenvalue differences

Space of Superquadratics

Tensor Creation Create superquadric shape Modify standard parameterization of sphere p(  ) = ( ) cos   sin   sin   sin   cos  , 0 <=  <= 2  0 <=  <=  where x  = sgn(x)|x| 

And More Colormaps were used –To make large-scale patterns stand out Depicted two tensor attributes simulataneously –Different map on mesh surface and glyph –Frobenius norm (overall variability) –Fractional anisotropy (extent to which variability extends more in some directions than others (e.g. not rotationally symmetrical)) –Skew (precise manner of difference from a sphere)

Evaluation Back to the beginning, importance was –Understanding anatomic variability of the brain –Visualizing in order to understand anatomic variability Comparative images were used for verification that the visualization was more discriminatory than elliptical tensors –These pictures, along with those of other references, appeared to more clearly differentiate among attribute values than did ellipsoid glyphs No objective verification or validation But –Their previous work referenced studies that did quantitatively measure an increased discrimination when superquadratic tensors were used.

Superquadric Ellipsoid Kindlmann, “Superquadric Tensor Glyphs”, Joint Eurographics – IEEE TCVG Symposium on Visualization 2004

Superquadric Ellipsoid Kindlmann, “Superquadric Tensor Glyphs”, Joint Eurographics – IEEE TCVG Symposium on Visualization 2004

Superquadric Ellipsoid Scientific Computing and Imaging Institute website

Alternative Methods Spinor mentioned in another paper Other methods from “Tensorlines: Advection-Diffusion based Propagation through Diffusion Tensor Fields”, David Weinstein, Gordon Kindlmann, Eric Lundberg –Brush strokes (stroke shape, color, texture) –Glyphs –Ellipsoids (one kind of glyph) –Stream-polygons (show info along a path) –Hyperstreamlines (show info along a path) Other work of theirs did discuss ray-tracing the resulting superquadratic tensors with the caveat that it wouldn’t be real time

Conclusion Computational efficiency – could be interactive, depending on problem size Not designed to definitively depict attribute value Certainly not designed as a physician’s diagnostic tool Could be applicable to many other uses Example calculation sure would have been helpful!

Next Steps Other applications Efficiency Additional attributes (e.g. their colormapping scheme)

Question What are the advantages of superquadric tensor fields (or are there any)

Question Do the shapes really convey the info they are supposed to (i.e. differences)