Practical image reconstruction for magnetic resonance imaging
View Patent ↗A set of image-space data is reconstructed from a set of k-space data. The set of image-space data is generated by minimizing a cost functional by an iterative non-linear conjugate gradient process. The iterative process may be accelerated by introducing k-space weighting to the cost functional. With proper choice of k-space weighting, a block-Toeplitz matrix is generated which permits use of Fast Fourier Transform techniques. An image is rendered from the set of image-space data.
1. A computer-implemented-method for rendering an image from a set of k-space data, comprising the steps of:
constructing a cost functional based at least in part on said set of k-space data, said cost functional comprising
λ∥φ(ƒ)∥ 1 +∥W ( Aƒ−y )∥ 2 2
wherein ƒ is a vector representing said set of image-space data, λ is a Lagrange multiplier, y is a vector representing said set of k-space data, φ(ƒ) is an operator which transforms ƒ into a sparse representation, A is a Fourier operator which maps said set of image-space data into said set of k-space data, and W is said weighting matrix;
iteratively solving for a set of image-space data, wherein said step of iteratively solving comprises the step of minimizing said cost functional; and,
rendering said image from said set of image-space data.
2. The method of claim 1 wherein said step of iteratively solving further comprises the steps of:
generating a block-Toeplitz matrix based at least in part on said weighting matrix; and,
performing a plurality of first iterations based at least in part on said block-Toeplitz matrix elements, wherein each first iteration comprises the steps of:
(a) calculating the gradient of said cost functional;
(b) performing a plurality of second iterations, wherein each second iteration comprises the steps of:
(b1) performing a backtracking line search; and,
(b2) calculating a value of said cost functional; and,
(c) updating said set of image-space data.
3. The method of claim 2 , wherein said block-Toeplitz matrix comprises:
A † DA
wherein D=(AA † ) −1 .
4. The method of claim 3 wherein said gradient of said cost functional comprises:
∇ψ=λ∇∥φ(ƒ)∥ 1 +2 A † D ( Aƒ−y ).
5. The method of claim 3 wherein said step of iteratively solving further comprises the step of iteratively solving the equation:
f
k
≈
f
k
-
1
+
(
-
A
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DAf
k
-
1
+
A
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Dy
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=
A
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Dy
+
(
I
-
A
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DA
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f
k
-
1
wherein ƒ k is the set of f-values after the k-th iteration and I is the identity matrix.
6. The method of claim 1 wherein ∥φ(ƒ)∥ 1 is a total variation.
7. An image processing system for rendering an image from a set of k-space data, comprising:
means for constructing a cost functional based at least in part on said set of k-space data, said cost functional comprising:
λ∥φ(ƒ)∥ 1 +∥W ( Aƒ−y )∥ 2 2
wherein ƒ is a vector representing said set of image-space data, λ is a Lagrange multiplier, y is a vector representing said set of k-space data, φ(ƒ) is an operator which transforms f into a sparse representation, A is a Fourier operator which maps said set of image-space data into said set of k-space data, and W is said weighting matrix;
means for iteratively solving for a set of image-space data, wherein said step of iteratively solving comprises the step of minimizing said cost functional; and,
means for rendering said image from said set of image-space data.
8. The image processing system of claim 7 , further comprising:
means for generating a block-Toeplitz matrix based at least in part on said weighting matrix; and,
means for performing a plurality of first iterations based at least in part on said block-Toeplitz matrix elements, wherein each first iteration comprises the steps of:
(a) calculating the gradient of said cost functional;
(b) performing a plurality of second iterations, wherein each second iteration comprises the steps of:
(b1) performing a backtracking line search; and,
(b2) calculating a value of said cost functional; and,
(c) updating said set of image-space data.
9. The image processing system of claim 8 wherein said block-Toeplitz matrix comprises:
A † DA
wherein D=(AA†) −1 .
10. The image processing system of claim 9 wherein said gradient of said cost functional comprises:
∇ψ=λ∇∥φ(ƒ)∥ 1 +2 A † D ( Aƒ−y ).
11. The image processing system of claim 9 , further comprising means for solving the equation:
f
k
≈
f
k
-
1
+
(
-
A
†
DAf
k
-
1
+
A
†
Dy
)
=
A
†
Dy
+
(
I
-
A
†
DA
)
f
k
-
1
wherein ƒ k is the set of f-values after the k-th iteration and I is the identity matrix.
12. The image processing system of claim 7 wherein ∥φ(ƒ)∥ 1 is a total variation.
13. A computer readable medium storing computer program instructions for rendering an image from a set of k-space data, the computer program instructions defining the steps of:
constructing a cost functional based at least in part on said set of k-space data, said cost functional comprising
λ∥φ(ƒ)∥ 1 +∥W ( Aƒ−y )∥ 2 2
wherein ƒ is a vector representing said set of image-space data, λ is a Lagrange multiplier, y is a vector representing said set of k-space data, φ(ƒ) is an operator which transforms ƒ into a sparse representation, A is a Fourier operator which maps said set of image-space data into said set of k-space data, and W is said weighting matrix;
iteratively solving for a set of image-space data, wherein said step of iteratively solving comprises the step of minimizing said cost functional; and,
rendering said image from said set of image-space data.
14. The computer readable medium of claim 13 wherein said computer program instructions further comprise computer instructions defining the steps of:
generating a block-Toeplitz matrix based at least in part on said weighting matrix; and,
performing a plurality of first iterations based at least in part on said block-Toeplitz matrix elements, wherein each first iteration comprises the steps of:
(a) calculating the gradient of said cost functional;
(b) performing a plurality of second iterations, wherein each second iteration comprises the steps of:
(b1) performing a backtracking line search; and,
(b2) calculating a value of said cost functional;
(c) updating said set of image-space data.
15. The computer readable medium of claim 14 , wherein said block-Toeplitz matrix comprises:
A † DA
wherein D=(AA†) −1 .
16. The computer readable medium of claim 15 , wherein said gradient of said cost functional comprises:
∇ψ=λ∇∥φ(ƒ)∥ 1 +2 A † D ( Aƒ−y ).
17. The computer readable medium of claim 15 , wherein said computer instructions further comprise computer instructions for solving the equation
f
k
≈
f
k
-
1
+
(
-
A
†
DAf
k
-
1
+
A
†
Dy
)
=
A
†
Dy
+
(
I
-
A
†
DA
)
f
k
-
1
wherein ƒ k is the set of ƒ-values after the k-th iteration and I is the identity matrix.
18. The computer readable medium of claim 13 wherein ∥φ(ƒ)∥ 1 is a total variation.