High-dimensional fast convolutional framework (HICU) for calibrationless MRI
The present disclosure is directed to a computational procedure for accelerated, calibrationless magnetic resonance image (CI-MRI) reconstruction that is fast, memory efficient, and scales to high dimensional imaging. The computational procedure, High-dimensional fast ConvolUtional framework (HICU), provides fast, memory-efficient recovery of unsampled k-space points.
1. A method for Magnetic Resonance Imaging (MRI) reconstruction comprising:
acquiring undersampled k-space MRI data;
interpolating and extrapolating the undersampled k-space MRI data to form a completed k-space data array;
computing, by convergent iteration, the completed data array to minimize a cost function defined as the weighted distance of a multi-level block Hankel matrix to a manifold of matrices with a rank, r, that is an integer value associated with a cost function;
converting the completed data array into reconstructed images; and
displaying the reconstructed images.
2. The method in claim 1 , wherein, at each iteration, an element in the linear variety of multi-level block Hankel matrices is parsimoniously parametrized and updated using the k-space data array and the principal r right singular vectors from the previous iterate, without recourse to the full tangent space to the low-rank manifold.
3. The method in claim 2 where the cost function is augmented by one or more functionals differentiable with respect to the entries in the data array; examples include barrier function for exact match of observed k-space samples and a weighted norm of mismatch between estimated and observed k-space samples.
4. The method in claim 2 where an image or k-space denoising subroutine is called within each iteration.
5. The method in claim 2 , wherein computation of a descent direction on the variety is accelerated by reducing the dimensionality of the multi-level Hankel matrix nullspace from r to p, constructed either explicitly or implicitly for the purpose of computing a descent direction.
6. The method in claim 2 , wherein the time to convergence is accelerated by interpolating a small subregion of the data array, then progressively expanding the region to encompass the entirety of the data array.
7. The method in claim 2 , wherein k-space data array has dimensions beyond spatial frequency, including time, coil, diffusion encoding, and velocity encoding.
8. The method in claim 2 , wherein update of multi-level Hankel matrix nullspace is stopped, while continuing descent iterations to update the data array.
9. The method in claim 2 , wherein additional nullspace vectors are approximated at initialization by collapsing one dimension of the completed data array via averaging.
10. The method in claim 2 , wherein the kernel size used to construct the multi-level Hankel matrix has, in any dimension, any integer size no less than 1 and no greater than the size of the data array in the corresponding dimension.
11. The method of claim 2 , wherein the kernel size used to define the multi-level Hankel matrix is modified from iteration to iteration.
12. The method of claim 5 , wherein the dimension, p, of the reduced-dimension nullspace is modified from iteration to iteration.
13. The method of claim 2 , wherein the coil dimension of the data array is compressed to a smaller dimension or expanded using virtual coils via conjugate symmetry.
14. The method in claim 2 , wherein the procedure is applied independently and in parallel to planes of dimension N-1 extracted from the N-dimensional data array.
15. The method of claim 2 , wherein initial iterations for a data array of dimension N are accelerated by operating on (N-q)-dimensional planes of the data array for 1<q<N.
16. A non-transitory computer-readable storage medium storing instructions that when executed by a computer cause the computer to perform a method for Magnetic Resonance Imaging (MRI) reconstruction, comprising:
acquiring undersampled k-space MRI data;
interpolating and extrapolating the undersampled k-space MRI data to form a completed k-space data array;
computing, by convergent iteration, the completed data array to minimize a cost function defined as the weighted distance of a multi-level block Hankel matrix to a manifold of matrices with a rank, r, that is an integer value associated with a cost function:
converting the completed data array into reconstructed images; and
displaying the reconstructed images.
17. The non-transitory computer-readable storage medium as recited in claim 16 , further comprising instructions wherein, at each iteration, an element in the linear variety of multi-level block Hankel matrices is parsimoniously parametrized and updated using the k-space data array and the principal r right singular vectors from the previous iterate, without recourse to the full tangent space to the low-rank manifold.
18. The non-transitory computer-readable storage medium as recited in claim 17 , further comprising instructions wherein the cost function is augmented by one or more functionals differentiable with respect to the entries in the data array; examples include barrier function for exact match of observed k-space samples and a weighted norm of mismatch between estimated and observed k-space samples.
19. The non-transitory computer-readable storage medium as recited in claim 17 , further comprising instructions wherein computation of a descent direction on the variety is accelerated by reducing the dimensionality of the multi-level Hankel matrix nullspace from r to p, constructed either explicitly or implicitly for the purpose of computing a descent direction.
20. The non-transitory computer-readable storage medium as recited in claim 17 , further comprising instructions wherein the time to convergence is accelerated by interpolating a small subregion of the data array, then progressively expanding the region to encompass the entirety of the data array.