Fiber Tracing Algorithm

The algorithm starts with the user selecting a rectangular starting region. The fibers are traced starting from the points only where the anisotropy measure is bigger than the threshold, i.e., that are high enough on the mountainside. The initial direction will be determined by the ``largest'' eigenvector of locally filtered tensor field. At this point the filter is not oriented. The tracing will proceed in two opposite directions along the ``largest `` eigenvector. The tracing procedure integrates forward from the provided initial point and initial direction using forward or inverse Euler method. It then computes a filtered value of the tensor at the new point using the oriented filter (orientation and width of the filter is determined from the previous position: the filter is oriented along the ``largest'' eigenvector and is shaped according to the eigenvalues, with largest semi-axis along the ``largest'' eigenvector). If the anisotropy of the new point is greater than threshold value, the point is accepted and the tracing continues, otherwise the tracing is finished. The tracing routine also chooses the direction of tracing consistent with previous steps (no turn $>~90$ degrees is allowed).


 

We have also incorporated some simple mechanisms to ignore very short fibers and to stop tracing when the length of the fiber exceeds an allowed limit. The starting points are usually generated on a grid within user defined regions. We use numerical integration to evaluate the integrals (18)-(19) inside the filter. We use SVD and LU factorization routines from the ``Numerical Recipes'' [17] to solve the linear system (21). Evaluation of the tensor function ${\mathbf T}$ at the center of the filter (origin) requires only the first coefficient of the polynomial expansion (22), so we use only a single back-substitution procedure in LU factorization.