Tensor Data Visualization Mengxia Zhu

Tensor data A tensor is a multivariate quantity  Scalar is a tensor of rank zero s = s(x,y,z)  Vector is a tensor of rank one  For a symmetric tensor of rank 2, its nine components are related byfor i,j = 1,2,3. A tensor field is a field which associates a tensor with each point in space Examples include Stress tensor (compose of forces acting on a surface) Diffusion tensor

Diffusion Tensor Diffusion tensor describes the molecular mobility along each direction and the correlation between these directions Diffusion MRI measures such diffusion anisotropy and provides information on the structure of the tissues at microscopic scale. Diffusion MRI does not interfere the diffusion physical process itself Symmetric tensor  zz  zy  zx

Ellipsoid The diffusion tensor can be visualized using an ellipsoid. Principle axes correspond to the direction of the eigenvector. Ellipsoids  The principal axes can be taken as minor, medium and major axes of an ellipsoid  The shape and orientation of ellipsoid represent the size of the eigenvalues and orientation of the eigenvectors

Background Eigenvalues and eigenvectors are based upon a common behavior in linear systems. We now generalize this concept of when a matrix/vector product is the same as a product by a scalar as above: essentially if we have a n×n matrix A, we seek solutions in v to find the eigenvectors, and solutions in λ to find the eigenvalues for the equation

Eigenvalues and Eigenvectors Any n-dimensional symmetric tensor A always has n eigenvaluesand n mutually perpendicular eigenvectors The eigenvectors and eigenvalues of the diffusion tensor give the direction and the length of the principle axes of the ellipsoid.

Eigenvalue Equation The eigenvectors and eigenvalues of tensor (matrix) A are obtained as follows  Cramer’s rule, a linear system of equations has nontrivial solution iff the determinant vanishes.  Eigenvectors form a 3D orthogonal coordinate system; axes are called the principal axes of the tensor  For order 1 ≥ 2 ≥ 3, the vectors v1, v2 and v3 are referred to as the major, medium and minor eigenvectors

Comparison of ellipsoid and a composite shape Linear, planar and spherical components Radius of sphere represents the smallest eigenvalue, radius of the disk represents the second largest eigenvlaue, the length of the rod is twice the value of the largest eigenvalue

Geometric measures Divide diffusion tensor into three geometric cases  Linear case: Diffusion is mainly along the direction with the largest eigenvector  Planar case: Diffusion mainly occurs on the plane spanned by two largest eigenvectors  Spherical case: Isotropic diffusion

Quantitive shape measures

Coloring scheme