Image reconstruction

1. A method of reconstructing an image through solution of a data fitting problem, wherein the data fitting problem is not susceptible to efficient solution as a whole, comprising:

gathering, via a computing device, k-space image data;

selecting a data fitting problem solution algorithm for the k-space data;

decomposing the data fitting problem solution into a plurality of sub-problem solutions each susceptible to efficient solution separately for the k-space data, wherein decomposing the data fitting problem comprises splitting a first variable into at least two variables;

obtaining, via the computing device, the plurality of sub-problem solutions for the k-space data, wherein obtaining the plurality of sub-problem solutions comprises performing operator splitting over each of the at least two variables independently; and

reconstructing, via the computing device, the image based upon a weighted average of the plurality of sub-problem solutions for the k-space, wherein reconstructing the image comprises obtaining a solution for the first variable by a linear combination of the plurality of sub-problem solutions.

2. The method of claim 1 , wherein, the image is a magnetic resonance image.

3. The method of claim 1 wherein the data tilting problem comprises a minimization of a plurality of linear terms of a least square data fitting solution.

4. The method of claim 1 wherein the image comprises under-sampled k-space image data.

5. The method of claim 1 wherein the fitting problem comprises a composite regularization problem.

6. The method of claim 4 , wherein the image comprises a magnetic resonance image comprising more samples within a first frequency band than within a second frequency band.

7. The method of claim 6 further comprising the first frequency band comprising lower frequencies than the second frequency band.

8. The method of claim 1 further comprising:

minimizing, via the computing device, a linear combination of three terms corresponding to a least square data fitting, a total variation (TV) and an L1 normalization regularization.

9. The method of claim 1 further comprising:

decomposing a problem solution into the TV and L1 normalization regularization sub-problems respectively; and

solving, via the computing device, the sub-problems.

10. The method of claim 1 , further comprising:

solving x k =prox ρ (α∥x∥ TV +β∥Φx∥ 1 )(x g ), via the computing device, where x is an unknown target, k is an integer from 1 to K, ρ is a scalor quantity, α and β are regularization parameters and [Φx] is a wavelet transform; and

applying, via the computing device, FISTA to the results to obtain a ε-optimal solution.

11. A system for reconstructing an image through solution of a data fitting problem, wherein the data fitting problem is not susceptible to efficient solution as a whole, comprising:

a computing device configured to gather k-space image data;

the computing device configured to run a data fitting problem solution algorithm for the k-space data, wherein the data fitting problem solution is decomposed into a plurality of sub-problem solutions each susceptible to efficient solution separately for the k-space data, wherein decomposing the data fitting problem comprises splitting a first variable into at least two variables;

the computing device configured to obtain the plurality of sub-problem solutions for the k-space data, wherein obtaining the plurality of sub-problem solutions comprises performing operator splitting over each of the at least two variables independently; and

the computing device configured to reconstruct the image based upon a weighted average of the plurality of sub-problem solutions for the k-space data, wherein reconstructing the image comprises obtaining a solution for the first variable by a linear combination of the plurality of sub-problem solutions.

12. The system of claim 10 , wherein, the image is a magnetic resonance image.

13. The system of claim 12 wherein the data fitting problem comprises a minimization of a plurality of linear terms of a least square data fitting solution.

14. The system of claim 12 wherein the image comprises under-sampled k-space image data.

15. The system of claim 12 wherein the fitting problem comprises a composite regularization problem.

16. The system of claim 14 , wherein the image comprises a magnetic resonance image comprising more samples within a first frequency band than within a second frequency band.

17. The system claim 11 further comprising the first frequency band comprising lower frequencies than the second frequency band.

18. A method of reconstructing an image through solution of a data fitting problem, wherein the data fitting problem is not susceptible to efficient solution as a whole, comprising:

gathering, via a computing device, k-space image data;

selecting a data fitting problem solution algorithm for the k-space data;

decomposing the data fitting problem solution into a plurality of sub-problem solutions each susceptible to efficient solution separately for the k-space data;

obtaining, via the computing device, the plurality of sub-problem solutions for the k-space data;

reconstructing, via the computing device, the image based upon a weighted average of the plurality of sub-problem solutions for the k-space;

solving x k =prox ρ (α∥x∥ TV +β∥Φx∥ 1 )(x g ), via the computing device, where x is an unknown target, k is an integer from 1 to K, ρ is a scalor quantity, α and β are regularization parameters and [Φx] is a wavelet transform; and

applying, via the computing device, FISTA to the results to obtain a ε-optimal solution for reconstructing the image.