Magnetic resonance image reconstruction for undersampled data acquisitions

1. A magnetic resonance imaging (MRI) system comprising:

an MRI gantry including static and gradient magnetic field generators, at least one radio frequency (RF) coil and RF transmitter and receiver circuits configured to transmit nuclear magnetic resonance (NMR) RF excitation pulses into an imaging volume and to receive NMR RF response signals from patient tissue located in the imaging volume; and

at least one control circuit connected to said gradient magnetic field generators and said RF transmitter and receiver circuits, said control circuits being configured to:

(a) perform a sparse acquisition of k-space data corresponding to NMR signals received by each RF coil;

(b) estimate, using the acquired sparse k-space data, a sensitivity map for each RF coil;

(c) reconstruct an MR image using compressed sensing based upon the at least one sensitivity map by minimizing a predetermined function which is based upon the MR image and coefficients of the at least one sensitivity map;

(d) update the at least one sensitivity map by minimizing the predetermined function;

(e) iteratively perform the reconstruct and the update processes (c) and (d) until at least one of the following conditions occur: (1) an operator selects to end iterating, or (2) the predetermined function converges; and

(f) output the reconstructed MR image when at least one of the conditions occur.

2. The MRI system according to claim 1 , wherein the update is based upon a polynomial function of sensitivities of the at least one RF coils.

3. The MRI system according to claim 2 , wherein estimate is based upon the polynomial function.

4. The MRI system according to claim 3 , wherein the polynomial function, when applied to the sensitivity map for each of the at least one RF coils, performs smoothing by reducing sharp differences between pixels located at outer edges of a reconstructed image.

5. The MRI system according to claim 1 , wherein the predetermined function minimizes, at each iteration, a sum of at least (1) a difference between the acquired k-space data and the reconstructed MR image, and (2) a sparsification of the reconstructed MR image.

6. The MRI system according to claim 5 , wherein the sum further includes (3) a penalty for a sum of absolute variations in the reconstructed MR image.

7. The MRI system according to claim 1 , wherein the compressed sensing includes finding a minimum sparsification of the reconstructed image such that a difference between the acquired k-space data and the reconstructed MR image is less than a configured threshold.

8. The MRI system according to claim 1 , wherein the compressed sensing includes finding a minimum sparsification of the reconstructed MR image and a minimum sum of absolute variations in the reconstructed MR image such that a difference between the acquired k-space data and the reconstructed MR image is less than a configured threshold.

9. The MRI system according to claim 1 , wherein the predetermined function includes

arg min a l ,x Σ l=1 c ∥d−E ( a l ) x∥ 2 +λ 1 ∥φx∥ 1

wherein φ represents a sparsifying transform, wherein d represents the acquired k-space data, wherein E represents an operator which integrates both the Fourier transform with a specified k-space data acquisition trajectory and a function representing sensitivities of the at least one RF coil, wherein x represents the reconstructed MR image, wherein C is the number of RF coils including the at least one RF coil, wherein a l represents the sensitivities of RF coil l, and wherein λ 1 and λ 2 are predetermined constants.

10. The MRI system according to claim 1 , wherein the predetermined function is represented by

arg min a l ,x Σ l=1 c ∥d−E ( a l ) x∥ 2 +λ 1 ∥φx∥ 1 +λ 2 TV ( x )

wherein φ represents a sparsifying transform, wherein TV represents taking the gradient of the image along at least two of the directions phase encode (PE), frequency encode (FE), and spatial encode (SE), wherein d represents the acquired k-space data, wherein E represents an operator which integrates both the Fourier transform with a specified k-space data acquisition trajectory and a function representing sensitivities of the at least one RF coil, wherein x represents the reconstructed MR image, wherein C is the number of RF coils including the at least one RF coil, wherein a l represents the sensitivities of RF coil l, and wherein λ 1 and λ 2 are predetermined constants.

11. The MRI system according to claim 1 , wherein the at least one control circuits are further configure to, after each iteration, display the reconstructed image on a display device and receive an operator input before proceeding to the next iteration.

12. The MRI system according to claim 1 , wherein the MRI gantry comprises a plurality of RF coils, and wherein the sparse acquisition is performed in parallel for each of the plurality of RF coils.

13. The MRI system according to claim 1 , wherein the sparse acquisition is a radial acquisition.

14. A method comprising:

(a) accessing a sparse acquisition of k-space data corresponding to nuclear magnetic resonance (NMR) signals received by each Radio Frequency (RF) coil of a MRI system which includes static and gradient magnetic field generators, at least one RF coil and RF transmitter and receiver circuits configured to transmit NMR RF excitation pulses into an imaging volume and to receive NMR RF response signals from patient tissue located in the imaging volume, and at least one control circuit connected to said gradient magnetic field generators; and

(b) estimating, using the acquired sparse k-space data, a sensitivity map for each RF coil;

(c) reconstructing an MR image using compressed sensing based upon the at least one sensitivity map by minimizing a predetermined function which is based upon the MR image and coefficients of the at least one sensitivity map;

(d) updating the at least one sensitivity map by minimizing the predetermined function;

(e) iteratively performing the reconstructing and the updating steps (c) and (d) until at least one of the following conditions occur: (1) an operator selects to end iterating, or (2) the predetermined function, converges; and

(f) outputting the reconstructed MR image when at least one of the conditions occur.

15. The method according to claim 14 , wherein the updating is based upon a polynomial function of sensitivities of the at least one RF coils.

16. The method according to claim 14 , wherein the predetermined function is directed to minimizing, at each iteration, a sum of at least (1) a difference between the acquired k-space data and the reconstructed MR image, and (2) a sparsification of the reconstructed MR image.

17. The method according to claim 13 , wherein the compressed sensing includes finding a minimum sparsification of the reconstructed image such that a difference between the acquired k-space data and the reconstructed MR image is less than a configured threshold.

18. The method according to claim 13 , wherein the predetermined function is represented by

arg min a l ,x Σ l=1 c ∥d−E ( a l ) x∥ 2 +λ 1 ∥φx∥ 1

wherein φ represents a sparsifying transform, wherein d represents the acquired k-space data, wherein E represents an operator which integrates both the Fourier transform with a specified k-space data acquisition trajectory and a function representing sensitivities of the at least one RF coil, wherein x represents the reconstructed MR image, wherein C is the number of RF coils including the at least one RF coil, wherein a l represents the sensitivities of l and wherein λ 1 and λ 2 are predetermined constants.

19. A non-transitory computer readable storage medium having stored thereon instructions which, when executed by one or more control circuits controls a magnetic resonance imaging (MRI) system to perform operations comprising:

(a) accessing a sparse acquisition of k-space data corresponding to nuclear magnetic resonance (NMR) signals received by each radio frequency (RF) coil in an MRI system which includes static and gradient magnetic field generators, at least one RF coil configured to transmit NMR RF excitation pulses into an imaging volume and to receive NMR RF response signals from patient tissue located in the imaging volume and at least one control circuit;

(b) estimating, using the acquired sparse k-space data, a sensitivity map for each RF coil;

(c) reconstructing an MR image using compressed sensing based upon the at least one sensitivity map by minimizing a predetermined function which is based upon the MR image and coefficients of the at least one sensitivity map;

(d) updating the at least one sensitivity map by minimizing the predetermined function;

(e) iteratively performing the reconstructing and the updating steps (c) and (d) until at least one of the following conditions occur: (1) an operator selects to end iterating, or (2) the predetermined function converges; and

(f) outputting the reconstructed MR image when at least one of the conditions occur.