Image Processing Reference
In-Depth Information
From Equation 11.1 to Equation 11.3, Equation 11.4 can be generalized for
3-D phase encoding
∫
−
i
2
π
k
St,
(
k
)
=
st
( ,
r
)
e
l,m,n
d
r
(11.8)
r
l,m,n
coil
where the integration is performed over the sensitive area of the coil (denoted as
coil
) and
r
represents the position vector relative to the gradient isocenter.
ri
=++
xyz
x
i
i
(11.9)
y
z
S(t,
k
l,m,n
) represents the analogy of k-space in MR imaging, with the differ-
ence that not only is one k-space acquired, as in the case of MR imaging, but
one k-space for each time point t is sampled. Because gradient amplitudes are
incremented discretely, S(t,
k
l,m,n
) is a discrete function of
k
l,m,n
. The reconstructed
signal s
rec
(t,
r
) is usually calculated by the discrete Fourier transform (DFT)
1
N
1
N
1
N
∑
i
2π
k
r
st
(, )
r
=
St
(,
k
)
e
l,m,n
(11.10)
rec
l,m,n
xyz
l,
m,n
The truncation of the k-space (due to the finite number N
x
, N
y
, N
z
of phase-
encoding steps) leads to
r
-space blurring, resulting in finite spatial resolutions
∆
x,
∆
y,
∆
z along
i
x
,
i
y
, and
i
z
(43).
FOV
N
1
∆
u
=
u
=
u
=
{x, y, z}
(11.11)
∆
k N
u
u
u
The number of voxels equals the number of corresponding phase-encoding
steps. The second equality in Equation 11.11 gives the relation between the size
of the FOV and the step
k between k values as a consequence of discrete
k-space sampling (Nyquist criterion).
Following Equation 11.10 and Equation 11.11, signals from voxels at posi-
tions
r
l′,m′,n′
∆
r
l',m',n'
=
l'
∆
x
i
x
+
m'
∆
y
i
y
+
n'
∆
z
i
z
,
l'
=
−
N
x
/2 …. (N
x
/2)
−
1; m', n' correspondingly
(11.12)
can be reconstructed by
1
N
1
N
1
N
∑
i
2
π
l,m,n l',m',n'
k
r
st
(,
r
)
=
S(t,
k
)e
(11.13)
rec
l',m',n'
l,m,n
xyz
l,m,n
Equation 11.13 represents the basic formula for the reconstruction of the
voxel signals in the CSI experiment.
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