Fabric-like Line Integral Convolution

The physical interpretation of mathematical features of tensor fields is highly application-specific. Our technique is a global method that represents the physical meaning of these tensor fields with their central features: regions of compression and expansion. The method is based on two steps: first we define a positive definite metric, with the same topological structure as the tensor field; second, the resulting metric is visualized. The eigenvector fields are represented using a texture-based approach resembling LIC methods. The eigenvalues of the metric are encoded in free parameters of the texture definition. Our method supports an intuitive distinction between positive and negative eigenvalues.

Publications

- Ingrid Hotz, Louis Feng, Hans Hagen, Bernd Hamann, and Kenneth I. Joy, “Tensor Fields Visualization Using a Metric Interpretation”, Joachim Weickert and Hans Hagen, eds., Visualization and Image Processing of Tensor Fields. Heidelberg, Germany, 2005, pp. 269-281.

- Ingrid Hotz, Louis Feng, Hans Hagen, Bernd Hamann, Boris Jeremic, and Kenneth I. Joy, “Physically Based Methods for Tensor Field Visualization”, Proceedings of IEEE Visualization Conference 2004, Austin, Texas, USA, October 2004, pp. 123-130.

Love and Tensor Algebra

from "The Cyberiad" by Stanislaw Lem

Come, let us hasten to a higher plane,
Where dyads tread the fairy fields of Venn,
Their indices bedecked from one to n,
Commingled in an endless Markov chain!

Come, every frustum longs to be a cone,
And every vector dreams of matrices.
Hark to the gentle gradient of the breeze:
It whispers of a more ergodic zone.

In Riemann, Hilbert or in Banach space
Let superscripts and subscripts go their ways.
Our asymptotes no longer out of phase,
We shall encounter, counting, face to face.

I'll grant thee random access to my heart,
Thou'lt tell me all the constants of thy love;
And so we two shall all love's lemmas prove,
And in our bound partition never part.

For what did Cauchy know, or Christoffel,
Or Fourier, or any Boole or Euler,
Wielding their compasses, their pens and rulers,
Of thy supernal sinusoidal spell?

Cancel me not--for what then shall remain?
Abscissas, some mantissas, modules, modes,
A root or two, a torus and a node:
The inverse of my verse, a null domain.

Ellipse of bliss, converge, O lips divine!
The product of our scalars is defined!
Cyberiad draws nigh, and the skew mind
Cuts capers like a happy haversine.

I see the eigenvalue in thine eye,
I hear the tender tensor in thy sigh.
Bernoulli would have been content to die,
Had he but known such a2 cos 2 phi!