Image Processing Reference
In-Depth Information
because due to the integral number of averages A
lmn
, the acquisition weighting
will not exactly match the required value of w(l, m, n) given by Equation 11.23.
However, it can be shown (49) that pure acquisition weighting, according to
Equation 11.23, represents an optimal method of producing the desired PSF shape
in terms of the highest SNR
in a given measurement time. Alternative methods
of acquisition weighting have been proposed, such as weighting achieved by the
variable repetition time (50) or, for spiral-based k-space sampling, by variable
density of the sampled spiral in k-space (51).
The extreme case of weighted sampling is reduced k-space sampling, when
some parts of the k-space are not sampled at all. A typical example is circular
(spherical) sampling when only points of the k-space inside the circular (spher-
ical) region are sampled and the remaining points are zero-filled (52).This leads
to the reduction of the measurement time and also to the improvement of the
PSF profile. The side lobes of PSF are reduced in circular sampling compared
to rectangular sampling. This is, however, at the expense of a slight broadening
of the central PSF lobe. Also, circular sampling leads to an isotropic PSF in
comparison to rectangular sampling, in which PSF side lobes are propagated only
along the principal axis. This can be important when potential signal contamina-
tion from problematic areas could be reduced by the proper orientation of the
CSI grid. Variations of circular sampling to achieve further improvements have
been suggested (53,54).
Reduced sampling can be combined with both acquisition k-space weighting
and postacquisition filtering, resulting in various PSFs with different data collec-
tion efficiencies (48).
11.3.5
CSI P
RE
P
ROCESSING
After spatial reconstruction of CSI data has been performed, spectra in all voxels of
the spectroscopic grid are available. Performing N
x
, N
y
, and N
z
phase-encoding steps
along orthogonal directions results in N
x
* N
y
* N
z
spatially resolved spectra along
corresponding axes. However, it is possible to increase the number of voxels artifi-
cially after the measurement. This operation, called
zero filling
, consists in appending
zeros to S(t,
k
l,m,n
either symmetrically or asymmetrically, the latter leading to a phase shift in spectra
among voxels. Zero filling represents an interpolation method and does not affect
the PSF. Therefore, even if the voxel size is decreased, the real resolution of the CSI
experiment is not improved.
Another unique feature of the FT reconstruction is the possibility of adjusting
the exact position of the grid after the measurement. This operation is based on
the shift theorem of FT
) values prior to the FT (
Figure 11.19
). Zeros can be appended
−
i
2
π
kxN
1j
/
s(t, x
−=
x )
DFT
−
1
{ (
S t, k
) exp
}
(11.24)
′
l
j
l
′
′
where x
j
represents generally the subvoxel size shift.
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