Method for estimating a GRAPPA reconstruction kernel
A method for improving the signal-to-noise ratio (SNR) of TGRAPPA. The SNR of the ACS lines is proportional to the condition number of the GRAPPA kernel encoding equations. Therefore, the GRAPPA kernel estimated from higher SNR ACS lines amplifies the random noise in GRAPPA reconstruction. In TGRAPPA reconstruction of dynamic image series, a widely used method to acquire ACS lines is to average-all-frame (AAF). The present disclosure utilizes a tile-all-frame (TAF) as ACS lines to improve the SNR of the reconstructed images.
1. A method of determining a k-space convolution kernel in Generalized Auto-calibrating Partially Parallel Acquisitions (GRAPPA) reconstruction of magnetic resonance imaging, comprising:
acquiring plural frames of 2-D images or 3-D images at a same slice location (2-D) or slab location (3-D);
acquiring k-space data of the plural frames of 2-D images or 3-D images;
providing at least two sets of auto-calibration signal (ACS) lines from the acquired k-space data;
utilizing a number of linear equations to estimate the k-space convolution kernel that is greater than a number of linear equations that could be derived from one set of ACS lines; and
wherein the linear equations are a linear regression to at least two sets of ACS lines to estimate the k-space convolution kernel,
wherein the linear regression is defined by the relationship:
AG=b,
wherein an m×n matrix A is the input of the GRAPPA reconstruction or other k-space based reconstructions, wherein each row of the matrix A represents a sliding window in k-space, m is the number of reconstructed k-space points/sliding window, wherein G is a vectorized GRAPPA kernel with size n×1, and wherein b is a vectorized output of the GRAPPA reconstruction with size m×1.
2. The method of claim 1 , wherein a well-posedness of the linear equations used to estimate the k-space convolution kernel are determined in accordance with a condition number.
3. The method of claim 2 , wherein the condition number is affected by the signal-to-noise ratio in the ACS lines.
4. The method of claim 2 , further comprising combining the linear equations with any type of regularization to estimate the k-space convolution kernel.
5. The method of claim 1 , further comprising determining multiple k-space convolution kernels.
6. A method of determining a k-space convolution kernel in a k-space based reconstruction method that has at least one implicit k-space convolution kernel estimation step, comprising:
acquiring k-space data of plural frames of images at a same location;
providing at least two sets of auto-calibration signal (ACS) lines from the acquired k-space data;
utilizing a number of linear equations to estimate the k-space convolution kernel that is greater than a number of linear equations that could be derived from one set of ACS lines; and
wherein the linear equations are a linear regression to at least two sets of ACS lines to estimate the k-space convolution kernel,
wherein the linear regression is defined by the relationship:
AG=b,
wherein an m×n matrix A is the input of a Generalized Auto-calibrating Partially Parallel Acquisitions (GRAPPA) reconstruction, wherein each row of the matrix A represents a GRAPPA sliding window in k-space, m is the number of reconstructed k-space points/sliding window, wherein G is a vectorized GRAPPA kernel with size n×1, and wherein b is a vectorized output of the GRAPPA reconstruction with size m×1.
7. The method of claim 6 , wherein a well-posedness of the linear equations used to estimate the k-space convolution kernel are determined in accordance with a condition number.
8. The method of claim 7 , wherein the condition number is affected by the signal-to-noise ratio in the ACS lines.
9. The method of claim 7 , further comprising combining the linear equations with any type of regularization to estimate the k-space convolution kernel.
10. The method of claim 6 , wherein a number of equations m is larger than a number of equations constructed from the one set of ACS lines.
11. A non-transitory computer readable medium containing computer-executable instruction that when executed by a processor of a computing device performs a method of determining a k-space convolution kernel in Generalized Auto-calibrating Partially Parallel Acquisitions (GRAPPA) reconstruction of magnetic resonance imaging, comprising:
acquiring plural frames of 2-D images or 3-D images at a same slice location (2-D) or slab location (3-D);
acquiring k-space data of the plural frames of 2-D images or 3-D images;
providing at least two sets of auto-calibration signal (ACS) lines from the acquired k-space data;
utilizing a number of linear equations to estimate the k-space convolution kernel that is greater than a number of linear equations that could be derived from one set of ACS lines; and
wherein the linear equations are a linear regression to at least two sets of ACS lines to estimate the k-space convolution kernel,
wherein the linear regression is defined by the relationship:
AG=b,
wherein an m×n matrix A is the input of the GRAPPA reconstruction, wherein each row of the matrix A represents a GRAPPA sliding window in k-space, m is the number of reconstructed k-space points/sliding window, wherein G is a vectorized GRAPPA kernel with size n×1, and wherein b is a vectorized output of the GRAPPA reconstruction with size m×1.
12. The non-transitory computer readable medium of claim 11 , wherein the linear equations used to estimate the k-space convolution kernel are determined in accordance with a condition number.
13. The non-transitory computer readable medium of claim 12 , wherein the condition number is affected by a signal-to-noise ratio in the ACS lines.
14. The non-transitory computer readable medium of claim 12 , further comprising instructions for combining the linear equations with any type of regularization to estimate the k-space convolution kernel.
15. The non-transitory computer readable medium of claim 11 , wherein a number of equations m is larger than a number of equations constructed from the one set of ACS lines.