Efrat Shimron^{1}, Andrew G. Webb^{2}, and Haim Azhari^{1}

Methods combining Compressed Sensing (CS) and Parallel MRI (PI) for accelerated MRI have shown great promise, yet they are commonly hindered by heavy iterative computations. This work introduces the novel CORE-Deblur method for accelerated MRI, which integrates CS and PI and offers fast computations with very few iterations. CORE-Deblur utilizes the recently introduced CORE-PI technique and introduces the novel concept of using CS for image deblurring. Experiments with in-vivo data show that for highly subsampled k-space (R=5) CORE-Deblur reduces the number of CS iterations by 10-fold (from 95 to about 5-7) and improves the reconstruction accuracy by 5%-8%.

Compressed Sensing (CS) and Parallel MRI (PI) have
emerged as two useful approaches for accelerating MRI scans by k-space
subsampling and subsequent reconstruction schemes. CS methods utilize image
sparsity in the domains of non-Fourier transforms (e.g. the wavelet transform),
and PI methods utilize multicoil acquisition and information about the coils
sensitivity maps for image reconstruction. While methods combining CS and
PI^{1–7} have shown great
promise, they are hindered by a heavy computational burden, which is mainly due
to massive iterative computations.

This work proposes a novel reconstruction method utilizing both CS and PI, with
simple computations and very few iterations. The proposed method utilizes: (1) the
recently introduced CORE-PI technique^{8}, which computes the convolution of
the unknown MR image with a known user-defined kernel, and (2) a unique
implementation of CS for reconstruction by deblurring the convolved image.

The proposed reconstruction method utilizes the
CORE-PI^{8} technique, which is a unique parallel MRI reconstruction method: it has as inputs highly
undersampled parallel k-space data, estimated sensitivity maps and a
user-defined kernel $$$g(x)$$$, and computes the convolution between the unknown
MR image $$$f(x,y)$$$ and the kernel:

$$f^{conv}(x,y)=f(x,y)*g(x) \quad\quad\quad(1)$$

CORE-PI is suitable for a 2D Cartesian k-space subsampled with a 1D subsampling scheme, i.e. using a subset of columns or rows. The method offers subsampling flexibility: both regular and random sampling schemes are possible. Furthermore, CORE-PI offers simple linear computations, and a flexible kernel choice: any kernel that can be represented by a curve containing $$$N_c$$$ points, where $$$N_c$$$ is the number of coils, is suitable.

The proposed reconstruction method, which is coined CORE-Deblur, consists of two steps: (i) computation of the convolved image $$$f^{conv}(x,y)$$$ using CORE-PI, and (ii) image reconstruction by implementation of a CS reconstruction that is initiated from the convolved image. The CS reconstruction solves the following convex optimization problem:

$$min\parallel{\boldsymbol{\Psi} f(x,y)}\parallel_1 \quad s.t.\quad \parallel{\boldsymbol{F_s} C_nf(x,y)-y_n}\parallel_2<\epsilon \quad n[1,N_c] \quad\quad\quad (2)$$

where $$$\boldsymbol{\Psi}$$$ is a
sparsifying transform, $$$y_n$$$ are the k-space
samples acquired by coil , $$$C_n$$$ is the sensitivity map
of coil $$$n$$$ and $$$\bf{F_s}$$$ is an operator
describing the Fourier transform and the subsampling. Eq. (2) describes a coil-by-coil CS image reconstruction process with
a multi-coil joint sparsity constraint. Once the individual coil images are
reconstructed, $$$f(x,y)$$$ is obtained by
merging them using Roemer’s optimal method^{9}.

Since the process described in eq. (2) is initiated from the convolved (i.e. blurred) image $$$f^{conv}(x,y)$$$ and produces the deblurred image $$$f(x,y)$$$, it may be conceived as an image deblurring process.

The proposed reconstruction method was implemented on in-vivo brain data obtained with a 32-coil array from two T1-weighted-7T
scans of healthy volunteers. Sensitivity maps were estimated from
low-resolution pre-scans. K-space data acquired in high-resolution scans were retrospectively
sub-sampled in one dimension using a regular scheme with a reduction factor of R=5. The CORE-Deblur method was implemented with a Gaussian kernel with
$$$\sigma=0.5$$$. The CS problem was solved using the Projection Onto Convex Sets
(POCS) approach^{10} with a Daubechies-2
wavelet. Computations were performed in Matlab on an HP-spectre-x360 PC.

The proposed method was compared to a coil-by-coil
CS-MRI^{11} reconstruction. The latter also solves the convex optimization problem of eq. (2) using POCS,
but it is initialized by the conventional method: k-space Zero Filling and an Inverse Fourier Transform.
Reconstruction errors were computed regarding gold standard images obtained from the fully
sampled k-space. Errors were measured using the Normalized Root Mean Square Error (NRMSE) measure.

Figures 1 and 2 demonstrate the reconstructions obtained by the proposed CORE-Deblur method from only 20% of k-space data (R=5), and compare them to the gold standard images and the CS reconstructions. The results show that the CORE-Deblur technique produces initial guesses which are quite close to the gold standards, without discernible artifacts.

Strikingly, CORE-Deblur converges rapidly, within only 5-7 iterations, whereas CS requires about 95 iterations (Figs. 1,2). In both experiments, CORE-Deblur also obtains highly accurate final reconstructions, with errors lower than CS by 7.6% (Fig 1) and 5.1% (Fig. 2).

The results therefore demonstrate that CORE-Deblur offers a significantly lower computational burden than CS, and a better reconstruction quality.

This work introduces the novel CORE-Deblur method for accelerated MRI, and the concept of using CS for deblurring.

Results from in-vivo data show that: (1) CORE-Deblur produces high quality reconstructions, better than those of CS by 5%-8%, and (2) requires significantly fewer iterations (90% less). The method is therefore suitable for real-time MRI applications.

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8. Shimron, E., Webb G., A. & Azhari, H. CORE-PI: Non-iterative Convolution-based Reconstruction for Parallel MRI in the Wavelet Domain. Med. Phys. (2018). doi:10.1002/MP.13260

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Figure 1.
Reconstructions from in-vivo 5-fold subsampled k-space data. (a) Comparison of
the gold standard image (left column) with reconstructions obtained by a
conventional CS process (middle column) and the proposed CORE-Deblur method
(right column). Upper row: the initial guess of each method. Second row: the
final reconstructions. Note that the initial CORE-Deblur guess is close to the
gold standard, and that CORE-Deblur obtained a lower final error than CS. (b)
Reconstruction error vs. iteration number. Note that CORE-Deblur begins with a much
smaller error than CS and converges after 5 iterations only, whereas the CS
method converges only after 95 iterations.

Figure 1.
Reconstructions from in-vivo 5-fold subsampled k-space data. (a) Comparison of
the gold standard image (left column) with reconstructions obtained by a
conventional CS process (middle column) and the proposed CORE-Deblur method
(right column). Upper row: the initial guess of each method. Second row: the
final reconstructions. Note that the initial CORE-Deblur guess is close to the
gold standard, and that CORE-Deblur obtained a lower final error than CS. (b)
Reconstruction error vs. iteration number. Note that CORE-Deblur begins with a much
smaller error than CS and converges after 7 iterations only, whereas the CS
method converges only after 95 iterations.