Biomedical Engineering Reference
In-Depth Information
by a zeroth-order Bessel function, where the blurring or the width of the Bessel
function was inversely proportional to the radius of the acquisition q-sphere.
QBI, therefore, made it possible to reconstruct the angular result of DSI, i.e. the
ODF, with fewer acquisitions and without assuming any models. QBI was further
boosted by Anderson [
2
], Hess [
32
], and Descoteaux et al. [
25
], where an analytical
solution was proposed, by using the
spherical harmonic
(SH) basis. It was shown
that the SHs are the eigenfunctions of the FRT [
25
]. Letting
Y
l
denote the SH of
order
l
and degree
m
, a modified real and symmetric SH basis
is defined. For even order
l
, a single index
j
in terms of
l
and
m
is used such that
j
(
m
=
−
l,
···
,l
)
l
2
+
(
l,m
)=(
l
+2)
/
2+
m
. The modified basis is given by:
⎧
⎨
√
2Re(
Y
|m|
l
)
,
if
m<
0
,
Y
j
=
Y
l
(6.25)
,
if
m
=0
,
⎩
√
2(
−
1)
m
+1
Im(
Y
l
)
,
if
m>
0
,
Y
l
)
Y
l
)
represent the real and imaginary parts of
Y
l
where
respec-
tively. This modified basis is designed to be real, symmetric and orthonormal, and it
is then possible to obtain an analytical estimate of the ODF in Eq. (
6.24
) with [
25
]:
Re(
and
Im(
L
Ψ
T
(
u
)=
2
πP
l
(
j
)
(0)
c
j
Y
j
(
u
)
,
(6.26)
j
=1
c
j
where
L
is the number of elements in the modified SH basis,
c
j
are the SH coefficients describing the input HARDI signal,
P
l
(
j
)
is the Legendre
polynomial of order
l
that is associated with
j
th element of the modified SH basis
and
c
j
=(
l
+1)(
l
+2)
/
2
are the SH coefficients describing the ODF
Ψ
T
.
Aganj et al. [
1
] recently proposed an analytical solution to QBI using SHs to
compute the ODF in Eq. (
6.23
), under a mono-exponential assumption of the signal.
The ODF in Eq. (
6.23
) takes into account the solid angle factor during the radial
integration, therefore, it is a true marginal density function of the EAP. This solution
was also proposed by Vega et al. in [
66
]. The ODF in Eq. (
6.24
) proposed by Tuch
on the other hand doesn't account for this solid angle, and therefore needs to be
numerically normalized after estimation [
68
](Fig.
6.8
).
6.5
Computational Framework for Processing Diffusion MR
Images
Diffusion MRI is a rich source of complex data in the form of images. Processing
dMRI data poses a challenging problem since diffusion images can range from
scalar images such as DWIs, where each voxel contains a scalar grey-level value, to
tensor images such as in DTI, where each voxel contains a second order tensor, to
Search WWH ::
Custom Search