Efficient redundant haar minimization for parallel MRI reconstruction
View Patent ↗A method for parallel magnetic resonance imaging (MRI) reconstruction of digital images includes providing a set of acquired k-space MR image data v, a redundant Haar wavelet matrix W satisfying W T W=I, wherein I is an identity matrix, a regularization parameter λ≧0, and a counter limit k, initializing a variable z 0 =Wv, and intermediate quantities p 0 =q 0 =0, calculating y i =arg min z ½∥z−(p i +z i )∥ 2 2 +λ∥z∥ 1 for 0≦i≦k, wherein z denotes values of an MR image sought to be reconstructed, updating p i+1 =(p i +z i )−y i , updating z i+1 =arg min z ½∥z−(q i +z i )∥ 2 2 +g(z), wherein g ( z ) = { 0 , z = WW T z , + ∞ , otherwise ; and updating q i+1 =(q i +y i )−z i−l , wherein x=W T z is a solution of min x 1 2 Wx - Wv 2 2 + λ Wx 1 that specifies a reconstruction of the MR image.
1. A method for parallel magnetic resonance imaging (MRI) reconstruction of digital images, comprising:
providing a set of acquired raw k-space MR image data v, a redundant Haar wavelet matrix W satisfying W T W=I, wherein I is an identity matrix, a regularization parameter λ≧0, and a counter limit k;
obtaining a proximal operator, the proximal operator obtained by the steps of:
initializing a variable z 0 =Wv, and intermediate quantities p 0 =q 0 =0;
calculating y i =arg min z ½∥z−(p i +z i )∥ 2 2 +λ∥z∥ 1 for 0≦i≦k, wherein z denotes values of an MR image sought to be reconstructed;
updating p i+1 =(p i +z i )−y i ;
updating z i+1 =arg min z ½∥z−(q i +z i )∥ 2 2 +g(z), wherein
g
(
z
)
=
{
0
,
z
=
WW
T
z
,
+
∞
,
otherwise
;
updating q i+1 =(q i +y i )−z i+1 ,
wherein x=W T z is a solution of
min
x
1
2
Wx
-
Wv
2
2
+
λ
Wx
1
that specifies a reconstruction of the MR image; and
reconstructing a digital MR image by applying the proximal operator to the acquired raw k-space MR image data, the digital MR image displayable on a computer system display.
2. The method of claim 1 , wherein
argmin
z
1
2
z
-
(
p
i
+
z
i
)
2
2
+
λ
z
1
=
{
(
p
i
+
z
i
)
-
λ
,
(
p
i
+
z
i
)
>
λ
,
(
p
i
+
z
i
)
+
λ
,
(
p
i
+
z
i
)
-
λ
,
0
,
otherwise
.
3. The method of claim 1 , wherein
arg min z ½∥ z −( q i +z i )∥ 2 2 +g ( z )= WW T ( q i +z i ).
4. A method for parallel MRI reconstruction of digital images, comprising:
providing a set of acquired raw k-space MR image data v, a redundant Haar wavelet matrix W satisfying W T W=I, wherein I is an identity matrix, a regularization parameter λ≧0, a counter limit k, an initial value of a dual variable p derived by forming a dual to a quadratic program
min
z
:
z
=
WW
T
z
1
2
z
-
Wv
2
2
+
λ
z
1
that represents parallel reconstruction of a magnetic resonance (MR) image, and an initial value of a Lagrange multiplier {tilde over (q)} of the constraint an initial value of z=WW T z;
obtaining a proximal operator, the proximal operator obtained by the steps of:
calculating
p
i
+
1
=
argmin
p
:
p
∞
≤
λ
1
2
p
-
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
2
2
for
0
≤
i
≤
k
;
calculating an intermediate value y i =(Wv−(I−WW T ){tilde over (q)} i )−p i+1 ;
updating
q
~
i
+
1
=
argmin
q
~
1
2
(
I
-
WW
T
)
q
~
i
-
(
Wv
-
p
i
)
2
2
;
calculating q i+1 =(I−WW T ){tilde over (q)} i+1 ;
calculating z i+1 =(Wv−p i+1 )−q i+1 ,
wherein x=W T z is a solution of
min
x
1
2
Wx
-
Wv
2
2
+
λ
Wx
1
that specifies a reconstruction of the MR image; and
reconstructing a digital MR image by applying the proximal operator to the acquired raw k-space MR image data, the digital MR image displayable on a computer system display.
5. The method of claim 4 , wherein
argmin
p
∞
≤
λ
1
2
p
-
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
2
2
=
{
λ
,
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
>
λ
,
-
λ
,
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
<
-
λ
,
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
,
otherwise
.
6. The method of claim 4 , wherein ½∥(I−Wv T ){tilde over (q)} i −(Wv−p i )∥ 2 2 is minimized by (I−WW T ){tilde over (q)} i+1 =(I−WW T )(Wv−p i ).
7. The method of claim 4 , wherein, if dual variable p and Lagrange multiplier {tilde over (q)} are both initialized to 0,
y
i
=
S
(
Wv
-
(
I
-
WW
T
)
q
~
i
,
λ
)
=
S
(
p
i
+
z
i
,
λ
)
=
argmin
z
1
2
z
-
(
p
i
+
z
i
)
2
2
+
f
(
z
)
,
wherein S is defined as
S
(
v
,
λ
)
=
{
v
-
λ
,
v
>
λ
,
v
+
λ
,
v
<
-
λ
,
0
,
otherwise
.
8. The method of claim 4 , wherein, if dual variable p and Lagrange multiplier {tilde over (q)} are both initialized to 0,
z
i
+
1
=
WW
T
(
Wv
-
p
i
+
1
)
=
WW
T
(
q
i
+
y
i
)
=
argmin
z
1
2
z
-
(
q
i
+
y
i
)
2
2
+
g
(
z
)
,
wherein
g
(
z
)
=
{
0
,
z
=
WW
T
z
,
+
∞
,
otherwise
.
9. 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 parallel magnetic resonance imaging (MRI) reconstruction of digital images, the method comprising the steps of:
providing a set of acquired raw k-space MR image data v, a redundant Haar wavelet matrix W satisfying W T W=I, wherein I is an identity matrix, a regularization parameter λ≧0, and a counter limit k;
obtaining a proximal operator, the proximal operator obtained by the steps of:
initializing a variable z 0 =Wv, and intermediate quantities p 0 =q 0 =0;
calculating y i =arg min z ½∥z−(p i +z i )∥ 2 2 +λ∥z∥ 1 for 0≦i≦k, wherein z denotes values of an MR image sought to be reconstructed;
updating p i+1 =(p i +z i )−y i ;
updating z i+1 =arg min z ½∥z−(q i +z i )∥ 2 2 +g(z), wherein
g
(
z
)
=
{
0
,
z
=
WW
T
z
,
+
∞
,
otherwise
;
updating q i+1 =(q i +y i )−z i+l ,
wherein x=W T z is a solution of
min
x
1
2
Wx
-
Wv
2
2
+
λ
Wx
1
that specifies a reconstruction of the MR image; and
reconstructing a digital MR image by applying the proximal operator to the acquired raw k-space MR image data, the digital MR image displayable on a computer system display.
10. The computer readable program storage device of claim 9 , wherein
argmin
z
1
2
z
-
(
p
i
+
z
i
)
2
2
+
λ
z
1
=
{
(
p
i
+
z
i
)
-
λ
,
(
p
i
+
z
i
)
>
λ
,
(
p
i
+
z
i
)
+
λ
,
(
p
i
+
z
i
)
<
-
λ
,
0
,
otherwise
.
11. The computer readable program storage device of claim 9 wherein
arg min z ½∥ z −( q i +z i )∥ 2 2 +g ( z )= WW T ( q i +z i ).
12. 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 parallel magnetic resonance imaging (MRI) reconstruction of digital images, the method comprising the steps of:
providing a set of acquired raw k-space MR image data v, a redundant Haar wavelet matrix W satisfying W T W=I, wherein I is an identity matrix, a regularization parameter λ≧0, a counter limit k, an initial value of a dual variable p derived by forming a dual to a quadratic program
min
z
:
zWW
T
z
1
2
z
-
Wv
2
2
+
λ
z
1
that represents parallel reconstruction of a magnetic resonance (MR) image, and an initial value of a Lagrange multiplier {tilde over (q)} of the constraint an initial value of z=WW T z;
obtaining a proximal operator, the proximal operator obtained by the steps of:
calculating
p
i
+
1
=
argmin
p
:
p
∞
≤
λ
1
2
p
-
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
2
2
for
0
≤
i
≤
k
;
calculating an intermediate value y i =(Wv−(I−WW T ){tilde over (q)} i )−p i+1 ;
updating
q
~
i
+
1
=
arg
min
q
~
1
2
(
I
-
WW
T
)
q
~
i
-
(
Wv
-
p
i
)
2
2
;
calculating q i+1 =(I−WW T ){tilde over (q)} i+1 ;
calculating z i+1 =(Wv−p i+1 )−q i+1 ,
wherein x=W T z is a solution of
min
x
1
2
Wx
-
Wv
2
2
+
λ
Wx
1
that specifies a reconstruction of the MR image; and
reconstructing a digital MR image by applying the proximal operator to the acquired raw k-space MR image data, the digital MR image displayable on a computer system display.
13. The computer readable program storage device of claim 12 , wherein
argmin
p
∞
≤
λ
1
2
p
-
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
2
2
=
{
λ
,
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
>
λ
,
-
λ
,
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
<
-
λ
,
(
Wv
-
(
I
-
WW
T
)
q
~
i
)
,
otherwise
.
14. The computer readable program storage device of claim 12 , wherein ½∥(I−WW T ){tilde over (q)} i −(Wv−p i )∥ 2 2 is minimized by (I−WW T ){tilde over (q)} i+1 =(I−WW T )(Wv−p i ).
15. The computer readable program storage device of claim 12 , wherein, if dual variable p and Lagrange multiplier {tilde over (q)} are both initialized to 0,
y
i
=
S
(
Wv
-
(
I
-
WW
T
)
q
~
i
,
λ
)
=
S
(
p
i
+
z
i
,
λ
)
=
argmin
z
1
2
z
-
(
p
i
+
z
i
)
2
2
+
f
(
z
)
,
wherein S is defined as
S
(
v
,
λ
)
=
{
v
-
λ
,
v
>
λ
,
v
+
λ
,
v
<
-
λ
,
0
,
otherwise
.
16. The computer readable program storage device of claim 12 , wherein, if dual variable p and Lagrange multiplier {tilde over (q)} are both initialized to 0,
z
i
+
1
=
WW
T
(
Wv
-
p
i
+
1
)
=
WW
T
(
q
i
+
y
i
)
=
argmin
z
1
2
z
-
(
q
i
+
y
i
)
2
2
+
g
(
z
)
,
wherein
g
(
z
)
=
{
0
,
z
=
WW
T
z
,
+
∞
,
otherwise
.