Image Processing of Diffusion Tensor MRI Data - Advance Image Processing in Magnetic Resonance Imaging - page 425

which is the sum of the linear and planar measures. By normalizing with the
trace of the tensor instead of the norm, the measure will be more similar to the
RA measure (relative anisotropy, Equation 13.5).
The presented measures quantitatively describe the geometrical shape of the
diffusion tensor and, therefore, do not depend on the absolute level of the diffusion
present. However, in low-signal regions, where the noise level dominates these
shape measures, they make little sense. In practice, all shape measures should be
regularized by adding a constant in the denominator of size similar to that of the
noise level. For example, the λ normalized linear measure (Equation 13.11)

1

would be expressed as follows:

c l σ λλ

−
+

1 2

= (13.19)

λσ

1

in a low-signal
region. This expression has similarities to classical Wiener filtering, where the
noise level

where a suitable value for σ would be the expected value of λ

1

, but penalizes signals
that are smaller. When the normalization is done using the trace or norm,

σ has very little influence on signals larger than σ

should
have the expected value of the trace or norm, respectively, in low-signal regions.
When applied to white matter, the linear measure, c l , reflects the uniformity
of tract direction within a voxel. In other words, it will be high only if the diffusion
is restricted in two orthogonal directions. The anisotropy measure, c a , indicates
the relative restriction of the diffusion in the most restricted direction and empha-
sizes white matter tracts, which, within a voxel, exhibit at least one direction of
relatively restricted diffusion.

σ

13.3.2 M ACROSTRUCTURAL T ENSOR AND D IFFUSIVE M EASURES

In the previous section, we characterized the diffusion isotropy and anisotropy
within a voxel. Here, we will discuss methods for examining the pattern or
distribution of diffusion within a local image neighborhood. Basser and Pierpaoli
proposed a scalar measure for macrostructural diffusive similarity based on tensor
inner products between the center voxel tensor and its neighbors [3,16] (as in the
vector case, the inner product between two tensors measures their degree of
similarity). This intervoxel scalar measure is known as a lattice index and is
defined by*

〈, 〉 TT

3
8

3
4

〈, 〉
〈,〉〈 , 〉

TT
DD

∑

k

k

LI = a + (13.20)

k

〈, 〉 DD D D

k k k k

* Corrected formula; personal communication from Pierpaoli, 1997.