Alternating direction of multipliers method for parallel MRI reconstruction
View Patent ↗A method for reconstructing parallel magnetic resonance images includes providing a set of acquired k-space MR image data y, and finding a target MR image x that minimizes ½∥Fv−y∥ 2 2 +λ∥z∥ 1 where v=Sx and z=Wx where S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix, W is a redundant Haar wavelet matrix, and λ≧0 is a regularization parameter, by updating x k + 1 = ( μ 1 I + μ 3 S H S ) - 1 [ μ 1 W H ( z k - b z k ) + μ 3 S H ( v k - b v k ) ] , z k + 1 = soft ( Wx k + 1 + b z k , 1 μ 1 ) where soft ( x , T ) = { x + T if x ≤ - T , 0 if x ≤ T , x - T if x ≥ T , and v k + 1 = ( F H F + μ 3 I ) - 1 [ F H y + μ 3 ( Sx k + 1 + b v k ) ] , where k is an iteration counter, μ 1 and μ 3 are parameters of an augmented Lagrangian function, and b z and b v are dual variables of the augmented Lagrangian.
1. A method for reconstructing parallel magnetic resonance images (MRI), comprising:
providing a set of acquired k-space MR image data y; and
finding a target MR image x that minimizes ½∥Fv−y∥ 2 2 +λ∥z∥ 1 wherein v=Sx and z=Wx wherein S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix wherein FF H =I and H denotes a Hermitian adjoint matrix, W is a redundant Haar wavelet matrix satisfying W T W=I, wherein I is an identity matrix, and λ≧0 is a regularization parameter, by updating
x
k
+
1
=
(
μ
1
I
+
μ
3
S
H
S
)
-
1
[
μ
1
W
H
(
z
k
-
b
z
k
)
+
μ
3
S
H
(
v
k
-
b
v
k
)
]
,
z
k
+
1
=
soft
(
W
x
k
+
1
+
b
z
k
,
1
μ
1
)
wherein
soft
(
x
,
T
)
=
{
x
+
T
if
x
≤
-
T
,
0
if
x
≤
T
,
x
-
T
if
x
≥
T
,
and
v
k
+
1
=
(
F
H
F
+
μ
3
I
)
-
1
[
F
H
y
+
μ
3
(
Sx
k
+
1
+
b
v
k
)
]
,
wherein k is an iteration counter, μ 1 and μ 3 are parameters of an augmented Lagrangian function
L
(
x
,
z
,
v
,
b
z
,
b
v
)
=
1
2
Fv
-
y
2
2
+
λ
z
1
+
μ
1
2
Wx
-
z
2
2
+
μ
3
2
Sx
-
v
2
2
+
μ
1
〈
b
z
,
Wx
-
z
+
y
〉
+
μ
3
〈
b
v
,
Sx
-
v
〉
,
and b z and b v are dual variables of the augmented Lagrangian.
2. The method of claim 1 , wherein v k+1 can be updated as
v
k
+
1
=
(
Sx
k
+
1
+
b
v
k
)
+
τ
τ
+
1
F
H
(
y
-
F
(
Sx
k
+
1
+
b
v
k
)
)
wherein
τ
=
1
/
μ
3
.
3. The method of claim 1 , wherein λ≦∥WA H y∥ ∞ .
4. The method of claim 1 , wherein μ 1 =μ 3 <1.
5. The method of claim 1 , further comprising updating the dual variables as b z k+1 =b z k +(Wx k+1 −z k+1 ) and b v k+1 =b v k +(Sx k+1 −v k+1 ).
6. The method of claim 5 , wherein x k , z k , and v k , and dual variables b z k and b v k are initialized to zero.
7. The method of claim 1 , wherein x k , z k , and v k are updated until a relative difference between a current iteration estimate and a previous iteration estimate fall below a predefined threshold.
8. A method for reconstructing parallel magnetic resonance images (MRI), comprising:
providing a set of acquired k-space MR image data y; and
finding a target MR image x that optimizes
min
x
z
1
+
S
ɛ
(
u
)
s
.
t
.
z
=
Wx
,
u
=
Fv
-
y
,
v
=
Sx
,
wherein v=Sx and z=Wx wherein S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix wherein FF H =I and H denotes a Hermitian adjoint matrix, W is a redundant Haar wavelet matrix satisfying W T W=I, wherein I is an identity matrix, λ≧0 is a regularization parameter,
S
ɛ
(
u
)
=
{
0
,
u
2
≤
ɛ
,
∞
u
2
>
ɛ
,
and ε represents an allowed discrepancy between the target image x and the observed k-space data y, by updating
x
k
+
1
=
(
μ
1
I
+
μ
3
S
H
S
)
-
1
[
μ
1
W
H
(
z
k
-
b
z
k
)
+
μ
3
S
H
(
v
k
-
b
v
k
)
]
,
z
k
+
1
=
soft
(
Wx
k
+
1
+
b
z
k
,
1
μ
1
)
wherein
soft
(
x
,
T
)
=
{
x
+
T
if
x
≤
-
T
,
0
if
x
≤
T
,
x
-
T
if
x
≥
T
,
,
v
k
+
1
=
(
μ
2
F
H
F
+
μ
3
I
)
-
1
[
μ
2
(
F
H
(
y
+
u
k
+
b
u
k
)
)
+
μ
3
(
Sx
k
+
1
+
b
v
k
)
]
,
and
u
k
+
1
=
shrink
(
Fv
k
+
1
-
y
-
b
u
k
,
ɛ
)
wherein
shrink
(
x
,
T
)
=
{
0
if
x
≤
T
,
x
if
x
T
,
,
wherein k is an iteration counter, μ 1 , μ 2 and μ 3 are parameters of an augmented Lagrangian function
L
(
x
,
z
,
u
,
v
,
b
z
,
b
u
,
b
v
)
=
z
1
+
S
ɛ
(
u
)
+
μ
1
2
Wx
-
z
2
2
+
μ
2
2
Fv
-
y
-
u
2
2
+
μ
3
2
Sx
-
v
2
2
+
μ
1
〈
b
z
,
Wx
-
z
〉
+
μ
2
〈
b
u
,
u
-
Fv
+
y
〉
+
μ
3
〈
b
v
,
Sx
-
v
〉
,
and b z , b u and b v are dual variables of the augmented Lagrangian.
9. The method of claim 8 , further comprising updating the dual variables as
b z k+1 =b z k +( Wx k+1 −z k+1 ),
b u k+1 =b u k +( u k+1 −Fv k+1 +y ), and
b v k+1 =b v k +( Sx k+1 −v k+1 ).
10. The method of claim 8 , wherein ε≦∥y∥ 2 .
11. The method of claim 8 , wherein μ 1 , μ 2 , and μ 3 are set as μ 1 =μ 3 =0.1ρ and μ 2 =ρ wherein
ρ
=
1
0.01
WA
H
y
∞
.
12. The method of claim 9 , wherein x k , z k , v k , and u k , and dual variables b z k , b u k , and b v k are initialized to zero.
13. The method of claim 8 , wherein x k , z k , v k and u k are updated until a relative difference between a current iteration estimate and a previous iteration estimate fall below a predefined threshold.
14. 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 reconstructing parallel magnetic resonance images (MRI), the method comprising the steps of:
providing a set of acquired k-space MR image data y; and
finding a target MR image x that minimizes ½∥Fv−y∥ 2 2 +λ∥z∥ 1 wherein v=Sx and z=Wx wherein S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix wherein FF H =I and H denotes a Hermitian adjoint matrix, W is a redundant Haar wavelet matrix satisfying W T W=I, wherein I is an identity matrix, and λ≧0 is a regularization parameter, by updating
x
k
+
1
=
(
μ
1
I
+
μ
3
S
H
S
)
-
1
[
μ
1
W
H
(
z
k
-
b
z
k
)
+
μ
3
S
H
(
v
k
-
b
v
k
)
]
,
z
k
+
1
=
soft
(
Wx
k
+
1
+
b
z
k
,
1
μ
1
)
wherein
soft
(
x
,
T
)
=
{
x
+
T
if
x
≤
-
T
,
0
if
x
≤
T
,
x
-
T
if
x
≥
T
,
and
v
k
+
1
=
(
F
H
F
+
μ
3
I
)
-
1
[
F
H
y
+
μ
3
(
Sx
k
+
1
+
b
v
k
)
]
,
wherein k is an iteration counter, μ 1 and μ 3 are parameters of an augmented Lagrangian function
L
(
x
,
z
,
v
,
b
z
,
b
v
)
=
1
2
Fv
-
y
2
2
+
λ
z
1
+
μ
1
2
Wx
-
z
2
2
+
μ
3
2
Sx
-
v
2
2
+
μ
1
〈
b
z
,
Wx
-
z
+
y
〉
+
μ
3
〈
b
v
,
Sx
-
v
〉
,
and b z and b v are dual variables of the augmented Lagrangian.
15. The computer readable program storage device of claim 14 , wherein v k+1 can be updated as
v
k
+
1
=
(
Sx
k
+
1
+
b
v
k
)
+
τ
τ
+
1
F
H
(
y
-
F
(
Sx
k
+
1
+
b
v
k
)
)
wherein
τ
=
1
/
μ
3
.
16. The computer readable program storage device of claim 14 , wherein λ≦∥WA H y∥ ∞ .
17. The computer readable program storage device of claim 14 , wherein μ 1 =μ 3 <1.
18. The computer readable program storage device of claim 14 , the method further comprising updating the dual variables as b z k+1 =b z k +(Wx k+1 −z k+1 ) and b v k+1 =b v k +(Sx k+1 −v k+1 ).
19. The computer readable program storage device of claim 18 , wherein x k , z k , and v k , and dual variables b z k and b v k are initialized to zero.
20. The computer readable program storage device of claim 14 , wherein x k , z k , and v k are updated until a relative difference between a current iteration estimate and a previous iteration estimate fall below a predefined threshold.
21. 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 reconstructing parallel magnetic resonance images (MRI), the method comprising the steps of:
providing a set of acquired k-space MR image data y; and
finding a target MR image x that optimizes
min
x
z
1
+
S
ɛ
(
u
)
s
.
t
.
z
=
Wx
,
u
=
Fv
-
y
,
v
=
Sx
.
wherein v=Sx and z=Wx wherein S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix wherein FF H =I and H denotes a Hermitian adjoint matrix, W is a redundant Haar wavelet matrix satisfying W T W=I, wherein I is an identity matrix, λ≧0 is a regularization parameter,
S
ɛ
(
u
)
=
{
0
,
u
2
≤
ɛ
,
∞
u
2
>
ɛ
,
and ε represents an allowed discrepancy between the target image x and the observed k-space data y, by updating
x
k
+
1
=
(
μ
1
I
+
μ
3
S
H
S
)
-
1
[
μ
1
W
H
(
z
k
-
b
z
k
)
+
μ
3
S
H
(
v
k
-
b
v
k
)
]
,
z
k
+
1
=
soft
(
Wx
k
+
1
+
b
z
k
,
1
μ
1
)
wherein
soft
(
x
,
T
)
=
{
x
+
T
if
x
≤
-
T
,
0
if
x
≤
T
,
x
-
T
if
x
≥
T
,
,
v
k
+
1
=
(
μ
2
F
H
F
+
μ
3
I
)
-
1
[
μ
2
(
F
H
(
y
+
u
k
+
b
u
k
)
)
+
μ
3
(
Sx
k
+
1
+
b
v
k
)
]
,
and
u
k
+
1
=
shrink
(
Fv
k
+
1
-
y
-
b
u
k
,
ɛ
)
wherein
shrink
(
x
,
T
)
=
{
0
if
x
≤
T
,
x
if
x
>
T
,
,
wherein k is an iteration counter, μ 1 , μ 2 and μ 3 are parameters of an augmented Lagrangian function
L
(
x
,
z
,
u
,
v
,
b
z
,
b
u
,
b
v
)
=
z
1
+
S
ɛ
(
u
)
+
μ
1
2
Wx
-
z
2
2
+
μ
2
2
Fv
-
y
-
u
2
2
+
μ
3
2
Sx
-
v
2
2
+
μ
1
〈
b
z
,
Wx
-
z
〉
+
μ
2
〈
b
u
,
u
-
Fv
+
y
〉
+
μ
3
〈
b
v
,
Sx
-
v
〉
,
and b z , b u and b v are dual variables of the augmented Lagrangian.
22. The computer readable program storage device of claim 21 , the method further comprising updating the dual variables as
b z k+1 =b z k +( Wx k+1 −z k+1 ),
b u k+1 =b u k +( u k+1 −Fv k+1 +y ), and
b v k+1 =b v k +( Sx k+1 −v k+1 ).
23. The computer readable program storage device of claim 21 , wherein ε≦∥y∥ 2 .
24. The computer readable program storage device of claim 21 , wherein μ 1 , μ 2 , and μ 3 are set as μ 1 =μ 3 =0.1ρ and μ 2 =ρ wherein
ρ
=
1
0.01
WA
H
y
∞
.
25. The computer readable program storage device of claim 22 , wherein x k , z k , v k , and u k , and dual variables b z k , b u k , and b v k are initialized to zero.
26. The computer readable program storage device of claim 21 , wherein x k , z k , v k and u k are updated until a relative difference between a current iteration estimate and a previous iteration estimate fall below a predefined threshold.