Methods and systems for fast auto-calibrated reconstruction with random projection in parallel MRI
View Patent ↗Methods and systems for fast auto-calibrated reconstruction with random projection in parallel MRI are disclosed. In one embodiment, calibration and reconstruction are accelerated by reducing the number of k-space data channels before calibration. During calibration, a random projection is performed in order to reduce the dimensionality of the calibration equation. The reduced-dimensionality equation is used to obtain the reconstruction coefficients, which are then used to synthesize reconstructed k-space data. The received and reconstructed k-space data together construct a parallel MRI image. Systems for fast GRAPPA reconstruction are disclosed comprising a processor configured, for example, through software, to collect k-space data, reduce the number of data channels, perform the random projection, solve a reduced calibration equation, and synthesize the k-space data to construct a final parallel MRI image.
1. A method for constructing a parallel magnetic resonance image comprising:
receiving, at a processor, a plurality of k-space signals, each signal corresponding to a channel of a receiver coil;
selecting, using the processor, a subset of the plurality of k-space signals, wherein the subset does not include all of the plurality of k-space signals;
receiving, at the processor, a plurality of auto-calibration signals;
selecting, using the processor, a subset of the plurality of auto-calibration signals;
determining, using the processor, a first calibration equation associated with the selected auto-calibration signals and the selected k-space signals;
reducing, using the processor, the first calibration equation to a reduced calibration equation having a lesser dimensionality than the first calibration equation;
determining, using the processor, a plurality of reconstruction coefficients based on the reduced calibration equation, each coefficient corresponding to a channel of the receiver coil;
synthesizing, using the processor, reconstructed k-space data based on the plurality of reconstruction coefficients and received k-space signals; and
constructing, using the processor, a parallel magnetic resonance image based on the reconstructed k-space data.
2. The method of claim 1 , wherein each k-space signal comprises a combination of sample data corresponding to the parallel magnetic resonance image and sensitivity information of a channel.
3. The method of claim 1 , wherein the step of selecting a subset of the plurality of k-space signals is performed using principal component analysis.
4. The method of claim 1 , wherein the first calibration equation is reduced by randomly selecting a subset of the first calibration equation or multiplying the first calibration equation with a random projection matrix.
5. The method of claim 4 , wherein the random projection matrix comprises elements from the following distribution:
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6. The method of claim 1 , wherein the plurality of reconstruction coefficients are determined using a least-squares method.
7. The method of claim 1 , wherein the plurality of reconstruction coefficients are determined using a cost function minimization algorithm.
8. The method of claim 6 , wherein the cost function minimization algorithm calculates the plurality of reconstruction coefficients under a specified constraint.
9. The method of claim 7 , wherein the constraint is 1 1 -norm or 1 2 -norm.
10. The method of claim 1 , wherein the plurality of reconstruction coefficients are also based on a sensitivity of the receiver coil.
11. The method of claim 1 , wherein the step of selecting a subset of the plurality of k-space signals comprises:
defining, using the processor, a set of N vectors, where N equals the total number of channels associated with the plurality of k-space signals;
calculating, using the processor, a mean corresponding to each vector in the set of N vectors;
subtracting from each vector in the set of N vectors, using the processor, the calculated mean from the corresponding vector;
calculating, using the processor, a plurality of eigenvectors of a covariance matrix;
creating, using the processor, a compression matrix based on the plurality of eigenvectors; and
selecting, using the processor, a subset of the plurality of k-space signals by multiplying the plurality of k-space signals by the compression matrix.