Water self-diffusion in tissue follows the Stejskal–Tanner relation between the diffusion-weighted MRI signal and a symmetric diffusion tensor D. In a single imaging plane D reduces to a real symmetric 2×2 matrix:
S(b,g) = S₀ · exp(−b · gᵀDg)
D = [ a b ]
[ b c ]
Inside myelinated axon bundles water diffuses freely along the fiber but is restricted across it, so D is anisotropic. Unlike the 3×3 tensor case — which needs an iterative Jacobi sweep — a symmetric 2×2 matrix diagonalizes in closed form: its eigenvalues and principal eigenvector direction follow directly from the quadratic formula, computed fresh at every point, every frame:
λ₁,₂ = (a+c)/2 ± √( ((a−c)/2)² + b² )
e₁ ∝ ( b , λ₁−a ) (unit-normalized)
The tensor ellipse glyphs you see are literally this eigenframe — semi-axes scaled by λ₁ and λ₂, rotated to e₁. A long thin ellipse means one dominant fiber direction; a circle means isotropic diffusion.
2D Fractional Anisotropy:
FA = |λ₁−λ₂| / (λ₁+λ₂)
This scene mixes two synthetic bundles — a left–right sheet and an anterior–posterior sheet — that overlap near the center, the 2D analogue of the classic "crossing-fiber" phantom used to stress-test DTI algorithms. Each point's tensor is the weighted sum of both bundles' single-fiber tensors, exactly how partial-volume averaging works in a real multi-fiber voxel; right at the crossing the two contributions cancel out most of the anisotropy and FA drops toward zero, even though every real fiber underneath is highly anisotropic — a genuine DTI pitfall this model reproduces on its own, not an artifact of only tracking two dimensions.
Streamline tractography (FACT algorithm) seeds a point, follows the principal eigenvector e₁ as a direction field, and takes discrete steps:
x_{n+1} = x_n + step · e₁(x_n)
- Anisotropy — sets how restricted diffusion is across a fiber (λ⊥) relative to along it (λ∥); low values wash the phantom back toward isotropic grey-matter-like diffusion.
- FA stop threshold — a streamline terminates once local anisotropy drops below this, the standard criterion for leaving white matter.
- Max turning angle — a streamline terminates if the direction would bend more than this per step, preventing tracks from jumping across unrelated bundles at the crossing.
- Step size — the Euler integration step; smaller steps track curvature more faithfully at the cost of more steps to reach the boundary.
Real-world relevance: this is the same principle behind clinical diffusion-MRI tractography, used to map white-matter connectivity and plan around eloquent tracts before neurosurgery — the 2D case here is exactly one imaging slice of that same pipeline.