Robust reconstruction method for parallel magnetic resonance images
View Patent ↗Methods and systems for reconstruction of an image from parallel Magnetic Resonance Image (pMRI) data are disclosed. A reconstructed pMRI image may suffer from noise and aliasing. A method for reducing aliasing by applying a bounded error function is disclosed. A method for reducing noise in a reconstruction by applying an error term is also disclosed. Error terms are included in an expression that can be solved as a minimization problem. Creating a solution in an iterative way is also disclosed. Examples of specific solutions are provided. A system applying the methods is also provided.
1. A method for parallel Magnetic Resonance Image (pMRI) reconstruction from image data, comprising:
a processor processing acquired lines in k-space to reconstruct an image from image data that is obtained in parallel from a plurality of Magnetic Resonance receivers, each Magnetic Resonance receiver acquiring a subset of lines associated with the image; and
the processor optimizing a quality measure in the reconstructed image by applying a bounded penalty.
2. The method as claimed in claim 1 , wherein the bounded penalty is determined by the processor by applying an error function
σ
α
(
arg
)
=
∑
i
(
1
-
ⅇ
-
α
arg
i
2
)
wherein α is an application dependent parameter and arg is an argument vector which contains elements arg i .
3. The method as claimed in claim 1 , further comprising:
the processor applying a discontinuity preserving spatial coherency function to image data for reducing noise.
4. The method as claimed in claim 3 , wherein the function is expressed as:
ƒ( x )=λσ α 2 ( Ax ); wherein
α 2 is a gradient level defining a discontinuity;
λ denotes a strength of smoothing: x denotes the reconstructed image; and
A denotes a node-edge incidence matrix.
5. The method as claimed in claim 4 , further comprising the processor solving
min
x
f
(
x
)
=
min
x
∑
i
σ
α
1
(
K
y
i
-
K
F
S
i
x
)
+
λσ
α
2
(
A
x
)
;
wherein
K is a diagonal matrix reflecting known frequencies;
i relates to a coil i out of N coils;
F is a discrete Fourier transformation matrix;
S i represents a sensitivity map for coil i;
α 1 expresses a point at which an acquired line is defined an outlier;
α 2 expresses a gradient level defining a discontinuity;
λ controls a smoothing;
y i is a known component of a coil i in a MRI machine,
x represents the reconstructed image.
6. The method as claimed in claim 1 , wherein an error data term due to aliasing is expressed as
f
(
x
)
=
∑
i
=
1
N
σ
α
1
(
K
y
i
-
K
F
S
i
x
)
with
σ
α
(
arg
)
=
∑
j
(
1
-
ⅇ
-
α
*
arg
j
2
)
;
wherein
K is a diagonal matrix reflecting known frequencies;
i relates to a coil i out of N coils;
F is a discrete Fourier transformation matrix;
S i represents a sensitivity map for coil i;
α 1 express a point at which an acquired line is defined an outlier;
α 2 expresses a gradient level defining a discontinuity;
y i is a known component of a coil i in a MRI machine,
x represents the reconstructed image; and
arg is an argument vector which contains elements arg j .
7. The method as claimed in claim 1 , wherein the bounded penalty is applied iteratively by the processor.
8. The method as claimed in claim 1 , further comprising:
estimating C 1,i by C α 1 (FS i x t −y i );
estimating C 2 by C α 2 (Ax t );
solving in an iterative way an equation
(
∑
i
S
i
T
F
T
K
C
1
,
i
F
S
i
+
λ
A
T
C
2
A
)
x
t
=
(
∑
i
S
i
T
F
T
K
C
l
,
i
y
i
)
for X t ,
and
stopping solving when ∥x t −x t−1 ∥≧ε or when a maximum number of iterations has been reached, and wherein
C α (argument) is a diagonal matrix representation with exponential terms representing elements of a vector argument;
F is a discrete Fourier transformation matrix;
S i represents a sensitivity map for coil i;
A denotes a node-edge incidence matrix;
T denotes a matrix transposition;
α 1 expresses a point at which an acquired line is defined an outlier;
α 2 expresses a gradient level defining a discontinuity;
K a diagonal matrix reflecting known frequencies;
λ controls a smoothing;
y i is a known component of a coil i in a MRI machine, and
x t represents the reconstructed image at an iteration t.
9. The method as claimed in claim 8 , whereinα 1 =20,000, α 2 =800, λ=0.0025 and
ε=1e−7.
10. A method for parallel Magnetic Resonance Image (pMRI) reconstruction from image data, comprising:
a processor processing acquired lines in k-space to reconstruct an image from image data that is obtained in parallel from multiple Magnetic Resonance receivers, each Magnetic Resonance receiver acquiring a subset of lines associated with the image; and
the processor adjusting acquired lines in k-space to meet a quality measure of the reconstructed image by applying a bounded penalty.
11. The method as claimed in claim 10 , wherein the quality measure is expressed as an error function
f
(
x
)
=
∑
i
=
1
N
σ
α
1
(
K
y
i
-
K
F
S
i
x
)
with
σ
α
(
arg
)
=
∑
j
(
1
-
ⅇ
-
α
*
arg
j
2
)
wherein:
K a diagonal matrix reflecting known frequencies;
i relates to a coil i out of N coils;
F is a discrete Fourier transformation matrix;
S i represents a sensitivity map for coil i;
α 1 expresses a point at which an acquired line is defined an outlier;
y i is a known component of a coil i in a MRI machine,
x represents the reconstructed image; and
arg is an argument vector which contains elements arg j .
12. The method as claimed in claim 10 , further comprising:
using a discontinuity preserving spatial coherency function for reducing noise.
13. The method as claimed in claim 12 , wherein the function is expressed as:
f
(
x
)
=
λ
σ
α
2
(
A
x
)
with
σ
α
(
arg
)
=
∑
j
(
1
-
ⅇ
-
α
*
arg
j
2
)
wherein
λ denotes a strength of smoothing;
α 2 is a gradient level defining a discontinuity;
x denotes the reconstructed image;
A denotes a node-edge incidence matrix; and
arg is an argument vector which contains elements arg j .
14. The method as claimed in claim 10 , further comprising solving
min
x
f
(
x
)
=
min
x
∑
i
σ
α
1
(
K
y
i
-
K
F
S
i
x
)
+
λσ
α
2
(
A
x
)
with
σ
α
(
arg
)
=
∑
j
(
1
-
ⅇ
-
α
*
arg
j
2
)
wherein
K is a diagonal matrix reflecting known frequencies;
i relates to a coil i out of N coils;
F is a discrete Fourier transformation matrix;
S i presents a sensitivity map for coil i;
α l expresses a point at which an acquired line is defined an outlier;
α 2 expresses a gradient level defining a discontinuity;
λ controls a smoothing;
y i is a known component of a coil i in a MRI machine,
x represents the reconstructed image; and
arg is an argument vector which contains elements arg j .
15. The method as claimed in claim 10 , wherein the bounded penalty is applied iteratively.
16. A system for parallel Magnetic Resonance Image (pMRI) reconstruction from image data, comprising:
a processor;
software operable on the processor to:
processing acquired lines in k-space to reconstruct an image from image data that is obtained in parallel from multiple Magnetic Resonance receivers, each Magnetic Resonance receiver acquiring a subset of lines associated with the image; and
optimizing a quality measure in the reconstructed image by applying a bounded penalty.
17. The system as claimed in claim 16 , wherein the bounded penalty is applied iteratively.
18. The system as claimed in claim 16 , wherein the bounded penalty is determined by an error function
σ
α
(
x
)
=
∑
i
(
1
-
ⅇ
-
α
x
i
2
)
wherein α is an application dependent parameter and x is an argument vector which contains elements x i .
19. The system as claimed in claim 16 , further comprising:
using a discontinuity preserving spatial coherency function for reducing noise.
20. The system as claimed in claim 19 , wherein the function is expressed as:
ƒ( x )=λσ α 2 ( Ax ); wherein
α 2 gradient level defining a discontinuity;
λ denotes a strength of smoothing; x denotes the reconstructed image; and
A denotes a node-edge incidence matrix.
21. The system as claimed in claim 16 , wherein an error data term due to aliasing is expressed as
f
(
x
)
=
∑
i
=
1
N
σ
α
1
(
K
y
i
-
K
F
S
i
x
)
with
σ
α
(
arg
)
=
∑
j
(
1
-
ⅇ
-
α
*
arg
j
2
)
;
wherein
K is a diagonal matrix reflecting known frequencies;
i relates to a coil i out of N coils;
F is a discrete Fourier transformation matrix;
S i represents a sensitivity map for coil i;
α 1 expresses a point at which an acquired line is defined an outlier;
α 2 expresses a gradient level defining a discontinuity;
y i is a known component of a coil i in a MRI machine,
x represents the reconstructed image; and
arg is an argument vector which contains elements arg j .
22. The system as claimed in claim 16 , further comprising solving
min
x
f
(
x
)
=
min
x
∑
i
σ
α
1
(
K
y
i
-
K
F
S
i
x
)
+
λσ
α
2
(
A
x
)
min
x
f
(
x
)
=
min
x
∑
i
σ
α
1
(
K
y
i
-
K
F
S
i
x
)
+
λσ
α
2
(
A
x
)
;
wherein
K is a diagonal matrix reflecting known frequencies;
i relates to a coil i out of N coils;
F is a discrete Fourier transformation matrix;
S i represents a sensitivity map for coil i;
α 1 expresses a point at which an acquired line is defined an outlier;
α 2 expresses a gradient level defining a discontinuity;
λ controls a smoothing;
y i is a known component of a coil i in a MRI machine,
x represents the reconstructed image.