Image Processing Reference
In-Depth Information
which is the phase-modulated projection of the sensitivity-weighted image onto
the x axis. Equation 3.15 can be discretized along
y
and written as follows:
N
∑
1
i
)
(
jGn
y
γ
τ
).
SGx
(,
)
=
IxnWxne
(,
)(,
(3.16)
k
y
k
n
=
This expression can be converted to matrix form for each position x along
the horizontal direction of the image, as follows:
1
1
!
Wx e
jG
y
1
( , )
1
γ
(
1
τ
)
.........
WxNe
(
)
jGN
γ
(
τ
)
y
SG x
(,
1
)
!
1
1
y
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
Ix
(,1
!
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
y
F
y
F
Wx e
( , )
1
jG
γ
(
1
τ
)
.......
...
WxNe
( ,
)
jGN
γ
(
τ
)
SG x
(,
y
F
)
I
((, )
x
2
1
1
1
1
y
F
Wx e
( , )
1
jG
γ
(
1
τ
)
.....
.....
WxNe
( ,
)
jGN
γ
(
τ
)
SGx
(,
1
)
y
Ix
(, )
3
2
2
2
y
y
F
y
F
SG x
(,
F
)
=
Wx e
( , )
1
jG
γ
(
1
τ
)
...
.......
WxNe
( ,
)
jG
γ
(
1
τ
)
•
(3.17)
2
y
2
2
SG
(,
x
)
1
y
F
1
Wx e
(,)
1
jG
γ
(
1
τ
)
.
.........
WxNe
( ,
)
jG
γ
(
1
τ
)
y
Ky
K
K
IxN
(, )
"
SGx
(,
)
Ky
F
jG
γ
(
1
1
τ
)
jG
y
F
γ
(
1
τ
)
"
Wx e
(,)
1
..........
WxNe
K
( ,
)
y
"
1
where F is the number of phase encodes used in the experiment, [
W
1
,…,
W
k
] represent
the sensitivity profiles of the coils, I(x, y) represents the image, and [S
1
, …,S
k
]
represent the discrete Fourier transform of the signals received in all the coils.
Equation 3.17 is a matrix equation where the term on the left side is a K
F element
vector containing the F phase-encoded values for all K coils. The term on the far
right is an
n
-element vector representing the “image” for one column. The middle
term in Equation 3.17 is a matrix with K
×
F rows and N columns; it is constructed
based on the sensitivity profiles and phase encodes used. Solving Equation 3.17
for each position along the x axis yields a column-by-column reconstruction of the
image. Increasing F results in an increase of the rank of the matrices, yielding
system matrices that are better conditioned. SPACE RIP allows for the arbitrary
choice of phase encodes, making it well suited to getting real-time sensitivity profile
estimations by fully sampling the center of k-space. These estimates are more
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