Eigen-vector approach for coil sensitivity maps estimation
Abstract
A method for estimating a coil sensitivity map for a magnetic resonance (MR) image includes providing ( 61 ) a matrix A of sliding blocks of a 2D image of coil calibration data, calculating ( 62 ) a left singular matrix V ∥ from a singular value decomposition of A corresponding to τ leading singular values, calculating ( 63 ) P=V ∥ V ∥ H , calculating ( 64 ) a matrix S that is an inverse Fourier transform of a zero-padded matrix P, and solving ( 65 ) M H c r =(S r ) H c r for c r , where c r is a vector of coil sensitivity maps for all coils at spatial location r, and M ( ( 1 1 … 1 0 0 … 0 … … … 0 0 … 0 ) ( 0 0 … 0 1 1 … 1 … … … 0 0 … 0 ) … ( 0 0 … 0 0 0 … 0 … … … 1 1 … 1 ) ) .
Claims
exact text as granted — not AI-modifiedWhat is claimed is:
1 . A method for estimating a coil sensitivity map for a magnetic resonance (MR) image, comprising the steps of:
providing a matrix A of sliding blocks of a 2D n x ×n y image of coil calibration data, wherein
A
=
(
a
1
,
1
a
1
,
2
…
a
1
,
n
b
a
2
,
1
a
2
,
2
…
a
2
,
n
b
…
…
…
a
n
c
,
1
a
n
c
,
2
…
a
n
c
,
n
b
)
,
n c is a number of coils, n b is a number of sliding blocks extracted from the coil calibration data, and a i,j is a k x k y ×1 column vector that represents a jth sliding block of an ith coil;
determining a unit eigenvector α that maximizes (S r ) H α,M H α ,
wherein S r is an inverse Fourier transform of a zero-padded matrix P=V ∥ V ∥ H at spatial location r in the 2D n x ×n y image, wherein
V
=
[
v
1
,
v
2
,
…
,
v
τ
]
=
(
v
1
,
1
v
1
,
2
…
v
1
,
τ
v
2
,
1
v
2
,
2
…
v
2
,
τ
…
…
…
v
n
c
,
1
v
n
c
,
2
…
v
n
c
,
τ
)
is a matrix composed of left singular vectors of A corresponding to τ leading singular values wherein v i,k is a k x k y ×1 column vector that is an i-th block in vector v k ,
M
=
(
(
1
1
…
1
0
0
…
0
…
…
…
0
0
…
0
)
(
0
0
…
0
1
1
…
1
…
…
…
0
0
…
0
)
…
(
0
0
…
0
0
0
…
0
…
…
…
1
1
…
1
)
)
is a n c ×k x k y n c sparse matrix of the same size as S r ; and
finding an optimizer c r that maximizes a correlation between (S r ) H α and M H α wherein said optimizer c r is a column vector of coil sensitivities of a coil at spatial position r in the 2D n x ×n y image.
2 . A method for estimating a coil sensitivity map for a magnetic resonance (MR) image, comprising the steps of:
providing a matrix A of sliding blocks of a 2D n x ×n y image of coil calibration data, wherein
A
=
(
a
1
,
1
a
1
,
2
…
a
1
,
n
b
a
2
,
1
a
2
,
2
…
a
2
,
n
b
…
…
…
a
n
c
,
1
a
n
c
,
2
…
a
n
c
,
n
b
)
,
n c is a number of coils, n b is a number of sliding blocks extracted from the coil calibration data, and a i,j is a k x k y ×1 column vector that represents a jth sliding block of an ith coil;
calculating a left singular matrix V ∥ from a singular value decomposition of A, wherein A=VΣU H and
V
=
[
v
1
,
v
2
,
…
,
v
τ
]
=
(
v
1
,
1
v
1
,
2
…
v
1
,
τ
v
2
,
1
v
2
,
2
…
v
2
,
τ
…
…
…
v
n
c
,
1
v
n
c
,
2
…
v
n
c
,
τ
)
is a matrix of left singular vectors of A corresponding to τ leading singular values wherein v i,k is a k x k y ×1 column vector that is an i-th block in vector v k ;
calculating
P
=
V
V
H
=
(
(
p
1
,
1
,
1
p
1
,
1
,
2
…
p
1
,
1
,
k
x
k
y
p
1
,
1
,
1
p
1
,
2
,
2
…
p
1
,
2
,
k
x
k
y
…
…
…
…
p
1
,
n
c
,
1
p
1
,
n
c
,
2
…
p
1
,
n
c
,
k
x
k
y
)
…
(
p
n
c
,
1
,
1
…
p
n
c
,
1
,
k
x
k
y
p
n
c
,
2
,
1
…
p
n
c
,
2
,
k
x
,
k
y
…
…
…
p
n
c
,
n
c
,
1
…
p
n
c
,
n
c
,
k
x
,
k
y
)
)
wherein p i,j,t is a k x k y ×1 column vector;
calculating S i,j,t =F H (P t (p i,j,t )) wherein F H represents an inverse Fourier transform and P t represents a zero-padding operator; and
solving M H c r (S r ) H c r for c r , where c r is a vector of coil sensitivity maps for all coils at spatial location r,
S
r
=
(
(
s
1
,
1
,
1
r
s
1
,
1
,
2
r
…
s
1
,
1
,
k
x
k
y
r
s
1
,
2
,
1
r
s
1
,
2
,
2
r
s
1
,
2
,
k
x
k
y
r
…
…
…
…
s
1
,
n
c
,
1
r
s
1
,
n
c
,
2
r
…
s
1
,
n
c
,
k
x
k
y
r
)
…
(
s
n
c
,
1
,
1
r
…
s
n
c
,
1
,
k
x
k
y
r
s
n
c
,
2
,
1
r
…
s
n
c
,
2
,
k
x
,
k
y
r
…
…
…
s
n
c
,
n
c
,
1
r
…
s
n
c
,
n
c
,
k
x
,
k
y
r
)
)
and
M
=
(
(
1
1
…
1
0
0
…
0
…
…
…
0
0
…
0
)
(
0
0
…
0
1
1
…
1
…
…
…
0
0
…
0
)
…
(
0
0
…
0
0
0
…
0
…
…
…
1
1
…
1
)
)
.
3 . The method of claim 2 , further comprising finding an optimizer c r that maximizes a correlation between (S r ) H α and M H α wherein said optimizer c r is vector of coil sensitivity maps for all coils at spatial location r in the 2D n x ×n y image.
4 . The method of claim 2 , wherein solving M H c r =(S r ) H c r comprises solving eigenvalue equations
M
(
S
r
)
H
k
x
k
y
β
=
λ
_
β
,
and
(
S
r
(
S
r
)
H
)
γ
=
λ
~
γ
,
wherein λ and {tilde over (λ)} denote eigenvalues, and β and γ denote eigenvectors.
5 . The method of claim 4 , further comprising calculating S r M H using
m
i
,
j
=
∑
t
=
1
k
x
k
y
s
i
,
j
,
t
=
∑
t
=
1
k
x
k
y
F
H
(
P
t
(
p
i
,
j
,
t
)
)
=
F
H
(
∑
t
=
1
k
x
k
y
P
t
(
p
i
,
j
,
t
)
)
,
wherein m i,j is an n x n y ×1 column vector containing the entries of m i,j r at all spatial locations, and m i,j r is the i,j elements of the 2D MR image at spatial location r.
6 . The method of claim 4 , further comprising calculating S r (S r ) H as S r (S r ) H =k x k y Z r (Z r ) H , wherein
Z
r
=
(
z
1
,
1
r
z
1
,
2
r
z
1
,
τ
r
z
2
,
1
r
z
2
,
2
r
z
2
,
τ
r
z
n
c
,
1
r
z
n
c
,
2
r
z
n
c
,
τ
r
)
is an n c ×τ matrix with the j-row and k-th column being z j,k r , z j,k r denotes the r-th entry of z j,k =F H (P c (v j,k )), and P c denotes the zero-padding operator that maps a center of v j,k to a center of z j,k .
7 . The method of claim 6 , further comprising computing the i-th row and j-th column of
Z
r
(
Z
r
)
H
as
∑
k
=
1
τ
z
i
,
k
r
z
i
,
k
r
_
wherein z i,k r denotes the i-th row and k-th column in Z r .
8 . The method of claim 2 , wherein the coil calibration data is acquired from a center square of frequency space data.
9 . The method of claim 2 , wherein the matrix A is constructed from calibration data of size c x ×c y ×c z by gathering sliding blocks of kernel size k x ×k y ×k z , wherein A has [(c x −k x +1)×(c y −k y +1)×(c z −k z +1)] columns and k x k y k z n c rows.
10 . A non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executed by the computer to perform the method steps for estimating a coil sensitivity map for a magnetic resonance (MR) image, the method comprising the steps of:
providing a matrix A of sliding blocks of a 2D n x ×n y image of coil calibration data, wherein
A
=
(
a
1
,
1
a
1
,
2
…
a
1
,
n
b
a
2
,
1
a
2
,
2
…
a
2
,
n
b
…
…
…
a
n
c
,
1
a
n
c
,
2
…
a
n
c
,
n
b
)
,
n c is a number of coils, n b is a number of sliding blocks extracted from the coil calibration data, and a i,j is a k x k y ×1 column vector that represents a jth sliding block of an ith coil;
calculating a left singular matrix V ∥ from a singular value decomposition of A, wherein A=VΣU H and
V
=
[
v
1
,
v
2
,
…
,
v
τ
]
=
(
v
1
,
1
v
1
,
2
…
v
1
,
τ
v
2
,
1
v
2
,
2
…
v
2
,
τ
…
…
…
v
n
c
,
1
v
n
c
,
2
…
v
n
c
,
τ
)
is a matrix of left singular vectors of A corresponding to τ leading singular values wherein v i,k is a k x k y ×1 column vector that is an i-th block in vector v k ;
calculating
P
=
V
V
H
=
(
(
p
1
,
1
,
1
p
1
,
1
,
2
…
p
1
,
1
,
k
x
k
y
p
1
,
1
,
1
p
1
,
2
,
2
…
p
1
,
2
,
k
x
k
y
…
…
…
…
p
1
,
n
c
,
1
p
1
,
n
c
,
2
…
p
1
,
n
c
,
k
x
k
y
)
…
(
p
n
c
,
1
,
1
…
p
n
c
,
1
,
k
x
k
y
p
n
c
,
2
,
1
…
p
n
c
,
2
,
k
x
,
k
y
…
…
…
p
n
c
,
n
c
,
1
…
p
n
c
,
n
c
,
k
x
,
k
y
)
)
wherein p i,j,t is a k x k y ×1 column vector;
calculating S i,j,t =F H (P t (p i,j,t )) wherein F H represents an inverse Fourier transform and P t represents a zero-padding operator; and
solving M H c r =(S r ) H c r for c r , where c r is a vector of coil sensitivity maps for all coils at spatial location r,
S
r
=
(
(
s
1
,
1
,
1
r
s
1
,
1
,
2
r
…
s
1
,
1
,
k
x
k
y
r
s
1
,
2
,
1
r
s
1
,
2
,
2
r
s
1
,
2
,
k
x
k
y
r
…
…
…
…
s
1
,
n
c
,
1
r
s
1
,
n
c
,
2
r
…
s
1
,
n
c
,
k
x
k
y
r
)
…
(
s
n
c
,
1
,
1
r
…
s
n
c
,
1
,
k
x
k
y
r
s
n
c
,
2
,
1
r
…
s
n
c
,
2
,
k
x
,
k
y
r
…
…
…
s
n
c
,
n
c
,
1
r
…
s
n
c
,
n
c
,
k
x
,
k
y
r
)
)
and
M
=
(
(
1
1
…
1
0
0
…
0
…
…
…
0
0
…
0
)
(
0
0
…
0
1
1
…
1
…
…
…
0
0
…
0
)
…
(
0
0
…
0
0
0
…
0
…
…
…
1
1
…
1
)
)
.
11 . The computer readable program storage device of claim 10 , the method further comprising finding an optimizer c r that maximizes a correlation between (S r ) H α and M H α wherein said optimizer c r is vector of coil sensitivity maps for all coils at spatial location r in the 2D n x ×n y image.
12 . The computer readable program storage device of claim 10 , wherein solving M H c r =(S r ) H c r comprises solving eigenvalue equations
M
(
S
r
)
H
k
x
k
y
β
=
λ
_
β
,
and
( S r ( S r ) H )γ={tilde over (λ)}γ,
wheren λ and {tilde over (λ)} denote eigenvalues, and β and γ denote eigenvectors.
13 . The computer readable program storage device of claim 12 , the method further comprising calculating S r M H using
m
i
,
j
=
∑
t
=
1
k
x
k
y
s
i
,
j
,
t
=
∑
t
=
1
k
x
k
y
F
H
(
P
t
(
p
i
,
j
,
t
)
)
=
F
H
(
∑
t
=
1
k
x
k
y
P
t
(
p
i
,
j
,
t
)
)
,
wherein m i,j is an n x n y ×1 column vector containing the entries of m i,j r at all spatial locations, and m i,j r is the i,j elements of the 2D MR image at spatial location r.
14 . The computer readable program storage device of claim 12 , the method further comprising calculating S r (S r ) H as S r (S r ) H =k x k y Z r (Z r ) H , wherein
Z
r
=
(
z
1
,
1
r
z
1
,
2
r
…
z
1
,
τ
r
z
2
,
1
r
z
2
,
2
r
…
z
2
,
τ
r
…
…
…
z
n
c
,
1
r
z
n
c
,
2
r
…
z
n
c
,
τ
r
)
is an n c ×τ matrix with the j-row and k-th column being z j,k r , z j,k r denotes the r-th entry of z j,k =F H (P c (v j,k )), and P c denotes the zero-padding operator that maps a center of v j,k to a center of z j,k .
15 . The computer readable program storage device of claim 14 , the method further comprising computing the i-th row and j-th column of Z r (Z r ) H as
∑
k
=
1
τ
z
i
,
k
r
z
i
,
k
r
_
wherein z i,k r denotes the i-th row and k-th column in Z r .
16 . The computer readable program storage device of claim 10 , wherein the coil calibration data is acquired from a center square of frequency space data.
17 . The computer readable program storage device of claim 10 , wherein the matrix A is constructed from calibration data of size c x ×c y ×c z by gathering sliding blocks of kernel size k x ×k y ×k z , wherein A has [(c x −k x +1)×(c y −k y +1)×(c z −k z +1)] columns and k x k y k z n c rows.Join the waitlist — get patent alerts
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