US 2013/0259343 A1 · ALTERNATING DIRECTION OF MULTIPLIERS METHOD FOR PARALLEL MRI RECONSTRUCTION

Abstract

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∥22+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, μ1 and μ3 are parameters of an augmented Lagrangian function, and bz and bv are dual variables of the augmented Lagrangian.

Claims26

Dependent claims

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∥22+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 FFH=I and H denotes a Hermitian adjoint matrix, W is a redundant Haar wavelet matrix satisfying WTW=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, μ1 and μ3 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 bz and bv are dual variables of the augmented Lagrangian.

251 more elements

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 FFH=I and H denotes a Hermitian adjoint matrix, W is a redundant Haar wavelet matrix satisfying WTW=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, μ1, μ2 and μ3 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 bz, bu and bv are dual variables of the augmented Lagrangian.

416 more elements

Dependencies are inferred from the claim text.

Alternating direction of multipliers method for parallel mri reconstruction

Life of this patent

Alternating direction of multipliers method for parallel mri reconstruction

Life of this patent

Abstract

Claims26