Self-constraint non-iterative GRAPPA reconstruction with closed-form solution

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 k0, 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=k0+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=k0−RNT(Q+NRNT)−1Nk0,
• 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.

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=k0+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=k0−RNT(Q+NRNT)−1Nk0,
• 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.