Self-constraint non-iterative GRAPPA reconstruction with closed-form solution
Parallel magnetic resonance imaging (pMRI) reconstruction techniques are commonly used to reduce scan time by undersampling the k-space data. In GRAPPA, a k-space based pMRI technique, the missing k-space data are estimated by solving a set of linear equations; however, this set of equations does not take advantage of the correlations within the missing k-space data. All k-space data in a neighborhood acquired from a phased-array coil are correlated. The correlation can be estimated easily as a self-constraint condition, and formulated as an extra set of linear equations to improve the performance of GRAPPA. We propose a modified k-space based pMRI technique call self-constraint GRAPPA (SC-GRAPPA) which combines the linear equations of GRAPPA with these extra equations to solve for the missing k-space data. Since SC-GRAPPA utilizes a least-squares solution of the linear equations, it has a closed-form solution that does not require an iterative solver.
1. A method of determining a Self-Constraint (SC) GeneRalized Autocalibrating Partially Parallel Acquisition (GRAPPA) reconstruction, comprising:
sampling k-space data as a set of ACS lines acquired from a magnetic resonance imaging (MRI) apparatus;
estimating a GRAPPA kernel from the ACS lines;
estimating a null projection matrix N from the ACS lines;
performing a GRAPPA reconstruction to calculate k 0 , which represents the GRAPPA reconstruction;
calculating a k-space of the SC-GRAPPA in accordance with the GRAPPA reconstruction and a self-constraint condition defined by local k-space correlations among acquired and unacquired k-space samples, the self-constraint condition being a set of linear equations determined in accordance with correlations within missing k-space data, as follows:
k=k 0 +w , and
0= Nk+v,
wherein w and v are random noise terms, wherein k is vectorized local k-space data, and
wherein calculating the k-space of the SC-GRAPPA is performed in accordance with:
k=k 0 −RN T ( Q+NRN T ) −1 Nk 0 ,
wherein Q and R are covariance matrices of w and v, respectively; and
performing image reconstruction by transforming the k-space of the SC-GRAPPA into MRI image data that is displayed in a human comprehensible format.
2. The method of claim 1 , further comprising estimating the GRAPPA kernel using a linear regression.
3. The method of claim 1 , further comprising estimating the null projection matrix N using a parallel reconstruction using null operations (PRUNO) method.
4. The method of claim 3 , wherein the null projection matrix is defined by:
[
0
0
]
=
[
N
11
N
12
N
21
N
22
]
[
k
acq
k
unacq
]
,
wherein k unacq and k acq are vectorized unacquired and acquired k-space data, respectively.
5. The method of claim 1 , wherein the k-space of the SC-GRAPPA is reconstructed using a least squares problem with prior estimation.
6. The method of claim 5 , wherein the prior estimation is k 0 .
7. The method of claim 1 , further comprising applying a channel combination using either the sum-of-squares or B1-weighted summation.
8. A nontransitory computer-readable medium having computer executable instructions stored thereon that when executed by a processor of a computing device determines a Self-Constraint (SC) GeneRalized Autocalibrating Partially Parallel Acquisition (GRAPPA) reconstruction, comprising:
sampling k-space data, including a set of ACS lines acquired from a magnetic resonance imaging (MRI) apparatus;
estimating a GRAPPA kernel from the ACS lines;
estimating a null projection matrix N from the ACS lines;
performing a GRAPPA reconstruction to calculate k 0 , which represents the GRAPPA reconstruction;
calculating a k-space of the SC-GRAPPA in accordance with the GRAPPA reconstruction and a self-constraint condition defined by local k-space correlations among acquired and unacquired k-space samples, the self-constraint condition being a set of linear equations determined in accordance with correlations within missing k-space data, as follows:
k=k 0 +w , and
0= Nk+v,
wherein w and v are random noise terms, wherein k is vectorized local k-space data, and
wherein calculating the k-space of the SC-GRAPPA is performed in accordance with:
k=k 0 −RN T ( Q+NRN T ) −1 Nk 0 ,
wherein Q and R are covariance matrices of w and v, respectively; and
performing image reconstruction by transforming the k-space of the SC-GRAPPA into MRI image data that is displayed in a human comprehensible format.
9. The nontransitory computer-readable medium of claim 8 , further comprising estimating the null projection matrix N using a parallel reconstruction using null operations (PRUNO) method.
10. The nontransitory computer-readable medium of claim 8 , wherein the k-space of the SC-GRAPPA is reconstructed using a least squares problem with prior estimation.
11. The nontransitory computer-readable medium of claim 10 , wherein the prior estimation is k 0 .
12. The nontransitory computer-readable medium of claim 8 , further comprising applying a channel combination using either the sum-of-squares or B1-weighted summation.
13. An apparatus for determining a Self-Constraint (SC) GeneRalized Autocalibrating Partially Parallel Acquisition (GRAPPA) reconstruction, comprising:
a memory;
a processor executing computer executable instructions in the memory, the computer executable instructions,
wherein the apparatus samples k-space data on a set of ACS lines acquired from a magnetic resonance imaging (MRI) apparatus, estimates a GRAPPA kernel from the ACS lines, estimates a null projection matrix N from the ACS lines, performs a GRAPPA reconstruction to calculate k0, which represents the GRAPPA reconstruction, and calculates a k-space of the SC-GRAPPA in accordance with the GRAPPA reconstruction and a self-constraint condition defined by local k-space correlations among acquired and unacquired k-space samples,
the self-constraint condition being a set of linear equations determined in accordance with correlations within missing k-space data, as follows:
k=k 0 +w , and
0= Nk+v,
wherein w and v are random noise terms, wherein k is vectorized local k-space data, and
wherein calculating the k-space of the SC-GRAPPA is performed in accordance with:
k=k 0 −RN T ( Q+NRN T ) −1 Nk 0 ,
wherein Q and R are covariance matrices of w and v, respectively; and
performing image reconstruction by transforming the k-space of the SC-GRAPPA into MRI image data that is displayed in a human comprehensible format.