Image Processing Reference
In-Depth Information
its edges. The operation is performed for all time points t. After the application
of the filter w, Equation 11.10 can be rewritten as:
1
N
1
N
1
N
i
k
r
st
(, )
r
=
w(l,m,n)S(t,
k
)e
l,m,n
(11.19)
rec
l,m,n
xyz
l,m,n
where w(l, m, n) describes the value of the filter function for the l-th, m-th, and
n-th encoding steps.
From Equation 11.14 and Equation 11.15 the resulting PSF is given by
1
N
1
N
1
N
i
k
r
PSF( )
r
=
w(l,m,n)e
l,m,n
(11.20)
xyz
l,m,n
Therefore, by using a proper filter function w(l, m, n), the PSF profile can
be improved. Various filter functions can be used (47), such as the cosine filter
2
l
wl
()
=
cos
(11.21)
l
max
or the Hamming filter
2 π
l
wl
()
=
054
.
+
046
.
cos
(11.22)
l
max
where l ranges over N x sampled values, and l max stands for the maximal sampled
value of l. 3-D extensions are, in the case of Cartesian sampling, given by the
product of the corresponding 1-D expressions.
The effect of filter functions on PSF shape is shown in Figure 11.18 . Gener-
ally, apodization is always a compromise between PSF side lobe reduction and
the increase of the width of the main lobe and, hence, worsens resolution. In this
respect, the Hamming function is the optimal filter (47). Because the real reso-
lution of the CSI experiment is related to the width of the main lobe of the PSF,
apodization influence the final resolution. Therefore, the details of the applied
filter should be provided whenever k-space filtering is used.
Postacquisition filtering is not an efficient method of k-space apodization in
terms of signal-to-noise ratio (SNR) or time, because the sampled signal from
the edges of the k-space is eventually reduced. If more averages are needed,
k-space apodization can be performed directly during the measurement by varying
the number of averages A lmn for each phase-encoding step (excitation) in propor-
tion to the desired filter function w(l, m, n):
,,
wlmn
(, , )
=
A
/
N
;
N
=
A
(11.23)
lmn
exc
exc
lmn
lmn
 
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