Signal Processing - May 2016 - 43

linearly associated with their location. However, we also
have a considerable amount of fat in the scalp whose protons precess with a much higher gyromagnetic ratio, in
long chains of carbons with one to three hydrogen atoms.
Because spatial location in MR imaging is encoded with
phase or equivalently frequencies, fat signals in the reconstructed image will not appear additively at the location of
scalp but, instead, map to locations where water protons
precess with the same Larmor frequency as they do. In a
magnetic field of 3 T, fat protons precess with a frequency
400 Hz higher than water protons. The phase-encoding
dimension of a typical EPI has around 30 Hz/pixel, so the
400-Hz difference will show as a shift of a dozen pixels in
the phase-encoding direction.
There is another reason that fat shift is more harming to
dMR imaging than other modalities. The diffusion coefficient
of fat is much lower than that of water, so its signal attenuation,
according to the equation S = S 0 e -bD, is stronger than that of
water protons. Therefore, fat shift appears as bright curved
lines in dMR images or as dark lines in derived diffusion coefficient images, as shown in Figure 8.
This fat-shift effect can be suppressed by various methods, and there is no simple answer to which is best. The most
widely used is to first excite fat protons at their frequency and
remove their phase coherence with a dephasing gradient pulse
before imaging. Although they still spin and precess, as the fat
protons are dephased, their MR signals become very weak in
comparison with those of water protons.

reconstruction of foDs
Diffusion signals captured by dMR imaging distinguish
from fibers' spatial distribution in the following senses.
First, they reflect the Brownian motion restricted by neuronal fibers rather than neuronal fibers themselves. Second, they are the average of diffusion signals within voxels,
not a detailed microscopic image. The voxel resolution of
dMR images is usually 2 mm, and that which was acquired
in the Human Connectome Project is 1.25 mm [25]. On the

other hand, the diameter of the axon is at the micrometer
level [26]. Therefore, it is impossible to reconstruct the exact
fiber structure with dMR images. However, it is possible to
estimate statistical properties of neuronal fibers from dMR
images by modeling diffusion properties of brain tissues.
As an inverse problem, such estimation topically involves a
representation of FODs and a forward model to relate FODs
to diffusion signals. After briefly introducing two popular
representations of FODs, diffusion tensors and spherical
harmonics, we discuss the essential part of reconstruction:
signal generation models.

Diffusion tensors
In the early 1990s, it was feasible to scan the brain only
in a few directions. The limited angular resolution did not
support complicated models, so FODs were depicted with
the most concise anisotropic model: Gaussian distributions
determined by symmetric, positive definite matrices, which
are called diffusion tensors [8]. The principal eigenvector of
a diffusion tensor reflects the dominant fiber direction, and
its eigenvalues characterize rotation-invariant properties.
The tensor model is unable to effectively account for crossing fibers, as shown in Figure 9. In the presence of crossing fibers, it usually reduces to a nearly isotropic diffusion
"ball" or a thin and round "plate." Such a side effect will lead
fiber track simulation to propagate in the wrong directions
when fibers actually cross each other, which is not rare in
the brain. To solve this problem, high-order diffusion tensors
have been proposed [27].

Spherical harmonics
Breakthroughs in dMR imaging have made it practical to
scan the brain in a hundred or more directions in a reasonable time, resolving the problem of crossing fibers [10]. To
fully utilize such high angular resolution in data acquisition,
a probabilistic distribution F defined on a unit sphere S 2 has
been employed, replacing diffusion tensors. Similarly, every
smooth function in a linear space can be represented with

(a)

(b)

FIGURe 8. An example of fat shift. (a) An image with fat shift (left) and the derived apparent diffusion coefficient (ADC) image (right). (b) An image of the
same subject with fat shift suppressed (left) and the derived ADC image. (Figure reprinted from [24] with permission.)
IEEE Signal Processing Magazine

May 2016

Table of Contents for the Digital Edition of Signal Processing - May 2016