Coil compression for three dimensional autocalibrating parallel imaging with cartesian sampling
A three dimensional image, in a phased array magnetic resonance imaging (MRI) system is provided. Three dimensional k-space data within an auto calibration signal (ACS) region and outside the ACS region are acquired. The k-space data within the ACS region are converted into hybrid space ACS data. Compression matrices and alignment matrices of the compression matrices for the hybrid space ACS data are found along a readout direction. Alignment matrices are multiplied to the compression matrices to achieve the properly-aligned compression matrices along the readout direction. All k-space data are converted into hybrid space. The properly-aligned compression matrices are applied to the hybrid space data to provide compressed data with fewer channels. The compressed data are used to form a three dimensional image.
1. A computer implemented method for providing a three dimensional image, in a phased array magnetic resonance imaging (MRI) system, comprising:
acquiring three dimensional k-space data within an autocalibration signal (ACS) region and outside the ACS region;
converting the k-space data within the ACS region into hybrid space ACS data;
finding compression matrices for the hybrid space ACS data along a readout direction;
finding alignment matrices of the compression matrices along the readout direction;
multiplying alignment matrices to the compression matrices to achieve the properly-aligned compression matrices along the readout direction;
converting the k-space data into hybrid space data;
applying the properly-aligned compression matrices to the hybrid space data to provide compressed data with fewer channels; and
using the compressed data to form a three dimensional image.
2. The computer implemented method, as recited in claim 1 , wherein within the ACS region the k-space data is fully sampled and wherein outside of the ACS region the k-space data is fully sampled in the readout direction but under sampled in both phase encoding directions.
3. The computer implemented method, as recited in claim 2 , wherein the using the compressed data to form a three dimensional image comprises converting the compressed data into k-space.
4. The computer implemented method, as recited in claim 3 , wherein the using the compressed data to form a three dimensional image comprises using autocalibrating parallel imaging or compressed sensing methods to reconstruct the compressed data.
5. The computer implemented method, as recited in claim 3 , wherein the finding compression matrices for the hybrid space ACS data along the readout direction, comprises:
minimi
z
ing
N
C
′
subject
to
v
′
(
r
)
=
v
(
r
)
P
,
P
P
H
=
P
H
P
=
I
,
|
v
i
′
(
r
)
|
2
<
ε
,
i
=
N
C
′
+
1
,
…
,
N
c
where N c is an original number of coils in data acquisition and at each location r in k-space at each readout position, and v(r)=[v 1 (r), v 2 (r), . . . , v Nc (r)] defines a vector that represents data at this location from all the original coils, and where P is a N c ×N c orthogonal matrix, and H is the conjugate transpose of a matrix, v′(r)=[v′ 1 (r), v′ 2 (r), . . . , v′ N C (r)], which is a vector at the same location from virtual coil 1 to N c , and N′ C is the number of effective virtual coils, and wherein ε is compression error.
6. The computer implemented method, as recited in claim 5 , wherein the finding the alignment matrices of the compression matrices, comprises:
minimizing
∑
i
=
2
N
x
||
A
i
-
A
i
-
1
||
F
2
subject
to
A
i
=
V
A
i
P
i
,
P
i
(
P
i
)
H
=
(
P
i
)
H
P
i
=
I
,
(
i
=
1
,
2
,
…
,
N
x
)
where the variables are A i and P i , and ∥A i −A i-1 ∥ F is the Frobenius norm of matrix (A i −A x-1 ) and V i A is a N c ×N c unitary input matrix for alignment matrix A i at the i-th slice.
7. The computer implemented method, as recited in claim 6 , wherein v(r) is prewhitened.
8. The method, as recited in claim 7 , wherein the using the compressed data to form a three dimensional image comprises using autocalibrating parallel imaging or compressed sensing to reconstruct the compressed data.
9. The computer implemented method, as recited in claim 3 , wherein the finding compression matrices A i for the hybrid space ACS data for each slice at each slice i (i=1, 2, . . . , N x ) along the readout direction, comprises:
a) at the ith slice, reformat the ACS in data matrix D h from each original coil into vectors X i 1 , X i 2 , . . . , X i Nc , which defines matrix X i =[X i 1 , X i 2 , . . . , X i Nc ];
b) perform a singular value decomposition (SVD) of X i :
X i =U i Σ i ( V i ) H (9)
where V i is the N c ×N c unitary input matrix, where the compression matrix A i is initialized as V A i ;
c) repeating steps a and b until compression matrices are calculated for all slides.
10. The computer implemented method, as recited in claim 9 , wherein the finding the alignment matrices, comprises finding an orthogonal matrix P i at each slice, so that the resulting compression matrices A i are smooth in x direction, comprising:
a) assuming P i-1 (and A i-1 ) is known at slice i−1, and P i is to be solved for slice I;
b) defining matrix C i =(A i-1 ) H V A i ;
c) performing SVD of matrix C i (size: N′ c ×N′ c );
C
i
=
U
C
i
∑
C
i
(
V
C
i
)
H
;
d) set P i =V C i (U i C ) H and A i =V A i P i ;
e) repeat steps b-d for all slices.
11. The computer implemented method, as recited in claim 10 , wherein coil compression matrix A has the property of A H A=I.
12. The computer implemented method, as recited in claim 11 , wherein the using the compressed data to form a three dimensional image comprises using autocalibrating parallel imaging or compressed sensing to reconstruct the compressed data.
13. The computer implemented method, as recited in claim 10 , wherein the using the compressed data to form a three dimensional image comprises using l 1 -SPIRiT or ARC.
14. The computer implemented method, as recited in claim 3 , wherein the using the compressed data to form a three dimensional image comprises using l 1 -SPIRiT or ARC.
15. A computer implemented method for providing a three dimensional image, in a phased array magnetic resonance imaging (MRI) system, comprising:
acquiring three dimensional k-space data within an autocalibration signal (ACS) region and outside the ACS region, wherein within the ACS region the k-space data are fully sampled and wherein outside of the ACS region the k-space data are fully sampled in a readout direction but under sampled in both phase encoding directions;
converting the k-space data within the ACS region into hybrid space ACS data;
finding compression matrices for the hybrid space ACS data along the readout direction;
finding alignment matrices of the compression matrices along the readout direction;
multiplying alignment matrices to the compression matrices to achieve the properly-aligned compression matrices along the readout direction;
converting the entire k-space data into hybrid space data;
applying the properly-aligned compression matrices to the hybrid space data to provide compressed data with fewer channels; and
using the compressed data to form a three dimensional image, using autocalibrating parallel imaging or compressed sensing methods to reconstruct the compressed data.
16. The computer implemented method, as recited in claim 15 , wherein the using the compressed data to form a three dimensional image comprises using l 1 -SPIRiT or ARC.
17. The computer implemented method, as recited in claim 15 , wherein the finding compression matrices for the hybrid space ACS data along the readout direction, comprises:
minimizing
N
C
′
subject
to
v
′
(
r
)
=
v
(
r
)
P
,
P
P
H
=
P
H
P
=
I
,
|
v
i
′
(
r
)
|
2
<
ε
,
i
=
N
C
′
+
1
,
…
,
N
c
where N is an original number of coils in data acquisition and at each location r in k-space at each readout position, and v(r)=[v 1 (r), v 2 (r), . . . , v Nc (r)] defines a vector that represents data at this location from all the original coils, and where P is a N c ×N c orthogonal matrix, and H is the conjugate transpose of a matrix, v′(r)=[v′ 1 (r), v′ 2 (r), . . . , v′ N c (r)], which is a vector at the same location from virtual coil 1 to N c , and N′ C is the number of effective virtual coils, and wherein cis compression error.
18. The computer implemented method, as recited in claim 17 , wherein the finding the alignment matrices of the compression matrices, comprises:
minimizing
∑
i
=
2
N
x
||
A
i
-
A
i
-
1
||
F
2
subject
to
A
i
=
V
A
i
P
i
,
P
i
(
P
i
)
H
=
(
P
i
)
H
P
i
=
I
,
(
i
=
1
,
2
,
…
,
N
x
)
where the variables are A i and P i , and ∥A i −A i-1 ∥ F is the Frobenius norm of matrix (A i −A i-1 ) and V i A is a N c ×N c unitary input matrix for alignment matrix A i at the i-th slice.
19. The computer implemented method, as recited in claim 15 , wherein v(r) is prewhitened.