A singular K-space model for fast reconstruction of magnetic resonance images from undersampled data

Abstract

Reconstructing magnetic resonance images from undersampled k-space data is a challenging problem. This paper introduces a novel method of image reconstruction from undersampled k-space data based on the concept of singularizing operators and a novel singular k-space model. Exploring the sparsity of an image in the k-space, the singular k-space model (SKM) is proposed in terms of the k-space functions of a singularizing operator. The singularizing operator is constructed by combining basic difference operators. An algorithm is developed to reliably estimate the model parameters from undersampled k-space data. The estimated parameters are then used to recover the missing k-space data through the model, subsequently achieving high-quality reconstruction of the image using inverse Fourier transform. Experiments on physical phantom and real brain MR images have shown that the proposed SKM method constantly outperforms the popular total variation (TV) and the classical zero-filling (ZF) methods regardless of the undersampling rates, the noise levels, and the image structures. For the same objective quality of the reconstructed images, the proposed method requires much less k-space data than the TV method. The SKM method is an effective method for fast MRI reconstruction from the undersampled k-space data.

Appendix: The implemented TV Algorithm

The TV method is detailed as follows:

Input:	Undersampled k-space data G z (k i , k j ).
TV	Find $g=\arg \underset{g}{\min }{\sum}_{i=1}^N{\sum}_{j=1}^N\left|\nabla g\left(i,j\right)\right|$, s.t. ℱ[g(i, j)]H(k i , k j ) = G z (k i , k j ), where ∇ designates the gradient operator, H(k i , k j ) is the scanning matrix.
Step 1	Initialize the image to be reconstructed as g 0(i, j) = ℱ−1[G z (k i , k j )] and ϵ = ‖g 0(i, j)‖2
Step 2	Reconstruction loop: for r = 1 to N r ${g}_r^0\left(i,j\right)= real\left({g}^0\left(i,j\right)\right),\kern0.5em {g}_i^0\left(i,j\right)= imaginary\left({g}^0\left(i,j\right)\right),$ where g r (i, j) and g i (i, j) are the real and imaginary images of g(i, j) a) Perform TV gradient descent on the real and imaginary images independently: for t = 1 to N t ${g}_r^t\left(i,j\right)={g}_r^{t-1}\left(i,j\right)-\alpha \in \nabla \left[\left|\nabla {g}_r^{t-1}\left(i,j\right)\right|\right]$ ${g}_i^t\left(i,j\right)={g}_i^{t-1}\left(i,j\right)-\alpha \epsilon \nabla \left[\left|\nabla {g}_i^{t-1}\left(i,j\right)\right|\ \right]$ b) Apply consistency to the sampled k-space data and evaluate the change ϵ of the reconstructed image ${G}^0\left({k}_i,{k}_j\right)=\left\{\begin{array}{ll}F\left[{g}^{N_t}\left(i,j\right)\right],& H\left({k}_i,{k}_i\right)=0\\ {}{G}_z\left({k}_i\cdot {k}_j\right),& H\left({k}_i,{k}_j\right)=1\end{array}\right.$ ${g}^0\left(i,j\right)={\mathcal{F}}^{-1}\left[{G}^r\left({k}_i,{k}_j\right)\right]$ $\epsilon =\left\Vert {g}^0\left(i,j\right)-{g}^{N_t}\left(i,j\right)\right\Vert$ c) if ϵ < 10−8, continue the r-loop.
Step 3	Output the reconstructed image g(i, j) = g 0(i, j)

In the above TV method, N_r is the maximum number of iterations for the reconstruction of the image. It is empirically determined and set to 100 in the present study, beyond which there is no appreciable change in the image. N_t is the total number of gradient descent steps and is chosen as 20. α is a controlling parameter and is set to 0.08.