Streamline Integration

The fiber tract trajectory $\mathbf{s}(\tau)$ can be computed as a parametric 3D curve through linear integration of the filtered principal eigenvector:

$$\mathbf{s}(\tau) = \int_0^\tau \mathbf{ \bar e}_1(t) dt$$ (26)

where $t$ is a parameter of the curve and has corresponding $t = t(x,y,z)$ values and $\mathbf{\bar e}_1$ is the MLS filtered principal direction (unit) eigenvector as a function of position.

$$\mathbf{ \bar T} \mathbf{\bar e}_1 = \bar \lambda_1 \mathbf{\bar e}_1$$ (27)

The discrete integration can be done numerically using explicit or implicit methods depending on the converging/diverging nature of the tensor field. The simplest approaches are a forward (for diverging fiber fields)

$$\mathbf{r}_{new} = \mathbf{r}_{old} + \mathbf{ \bar e}_1[\mathbf{\bar T}(\mathbf{ r}_{old})] \Delta t$$ (28)

or inverse Euler schemes (for converging fiber fields):

$$\mathbf{ r}_{new} = \mathbf{r}_{old} + \mathbf{\bar e}_1[\mathbf{\bar T}(\mathbf{ r}_{new})] \Delta t$$ (29)

One can easily employ higher order integration schemes, but they still should be chosen according to the local properties of the tensor field (converging or diverging) that are associated with the ``stiffness'' of the differential equation, bifurcations and the desired geometry.