Image Processing of Diffusion Tensor MRI Data - Advance Image Processing in Magnetic Resonance Imaging - page 435

FIGURE 13.13 Relation between, the diffusion tensor, and the metric tensor.

A natural interpretation of the degree of connectivity between two points
is the distance between the points in some metric space. For our application,
the distance between two anatomical locations should depend on the alignment
of their connecting paths with the diffusion tensor field. The diffusion operator
(Equation 13.3) can naturally be associated with a Riemannian metric tensor
G via the relation G D −1 , allowing us to compute geometric quantities such

as geodesic paths and distances between points in the brain. Unlike tracto-
graphic methods based on following the flow of principal eigenvectors of D ,
these geodesic paths are well defined even in regions where the tensor diffusion
is isotropic.
The inverse relation between the diffusion and metric tensors is intuitive: large
eigenvalues in the original tensor create short metric distances along the direction
of the corresponding eigenvector (see Figure 13.13). Or in general, the path length
is locally inversely proportional to the diffusion coefficient in the direction tangent
to the path. So, short paths are those that are aligned with the tensor field (not just
the major eigenvector field).

=

We will limit ourselves here to a brief discussion of the theory; see Reference 11

for a more rigorous and thorough treatment of the connection between diffusion
and Riemannian geometry. The Laplace-Beltrami operator is the generalization
of the Laplacian to manifolds. In matrix notation, the Laplace-Beltrami operator
can be written as

1
2

1
2

−

2 − 1

∇ |

G uG

| ∇⋅ | GG u | ∇ (13.29)

Here, we seek the relation between isotropic diffusion on a manifold
(Equation 13.29) and anisotropic diffusion in Euclidean space (Equation 13.3).
The following relation exists between the diffusion operator in Equation 13.3 and
a diffusion operator in the Riemannian space characterized by G :

1
2

2

∇⋅ ∇ =∇ − 〈∇ | ( Du ) u log G |,∇ 〉 u (13.30)

G G