Image Processing Reference
In-Depth Information
To see the relation between the true signal distribution s(t,
r
) and the signal s
rec
(t,
r
),
the inverse DFT has to be computed. Substituting S(t,
k
l,m,n
) using Equation 11.8 and
inverting the order of the integral and the summation, the reconstructed signal s
rec
(t,
r
)
in Equation 11.13 can be expressed as a convolution of the true signal distribution
s(t,
r
) and the point-spread function (PSF)
∫
s
(,
r
t
)
=
s(t,
u
)PSF(
r
−
u
)d
u
(11.14)
rec
coil
where the PSF is defined as
1
N
1
N
1
N
∑
i
2π
k
r
PSF(r)
=
e
l,m,n
(11.15)
xyz
l,m,n
The PSF describes the signal of a hypothetical infinite small point object in
the reconstructed image and characterizes the efficiency of the employed recon-
struction method.
In the case of equally spaced Cartesian sampling, 3-D PSF(
r
) can be separated
into three 1-D PSF, each describing the corresponding dimension
PSF( ,y,z)
x
≡
PSF( )
r
=
psf (x)psf (y)psf (z)
x
(11.16)
y
z
Depending on the number of phase-encoding steps (odd or even number) and
the implementation of the gradient incrementing in the measuring sequence, the
gradients (and, hence, the
k
l,m,n
vector) can be sampled symmetrically or asym-
metrically with respect to the zero value. This symmetry influences the final shape
of psf
x
(x), psf
y
(y), and psf
z
(z). Direct computation of Equation 11.15 for psf
x
(x),
for example, yields the following results (44):
In the case of symmetric sampling about zero, when the index l ranges over
l
=
−
(N
x
−
1)/2,
…
, 0,
…
, (N
x
−
1)/2 for an odd N
x
or over l
=
−
(N
x
−
1)/2, …,
−
1/2,
1/2,
…
, (N
x
−
1)/2 for an even N
x
(
N
−
∑
12
)/
x
sin(
π
NxFOV
xFOV
/
)
1
1
N
psf
()
x
=
e
i
2
π
lxFOV
/
=
x
x
(11.17)
x
x
N
sin(
π
/
)
x
x
x
l
=−
(
N
−
12
)/
x
In the case of asymmetric sampling with an even N
x
, when the index l ranges
over l
=
-(N
x
/2),
…
, (N
x
/2) -1,
(
N
/
21
)
−
x
sin(
π
NxFOV
xFOV
/
)
1
∑
1
psf
()
x
=
e
i
2
π
lxFOV
/
=
e
− π
ixFOV
/
x
x
(11.18)
x
x
x
N
N
sin(
π
/
)
x
x
x
l
=−
(
N
/
2
)
x
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