Sofia Dimoudi^{1}, Matthew D Robson^{1}, Jared Tanner^{2}, and Aaron T Hess^{1}

We present our updated results in the evaluation of the Normalised Iterative Hard Thresholding Algorithm (NIHT) for parallel imaging and compressed sensing reconstructions of highly accelerated Cardiac cine MRI at 7 Tesla. We compare imaging performance with three other parallel imaging and compressed sensing methods, including regularisation in the temporal dimension.

**INTRODUCTION**

**METHODS**

Sparse signal recovery can be posed as the following optimisation problem:

$$x^*=\underset{x\,:\,\|x\|_0\leq K}{\operatorname{arg\,min}}{\|y-\Phi x\|}_2$$

where $$$y$$$ represents the k-space measurements, $$$x$$$ is the image or its sparse representation, and $$$\Phi$$$ is the linear mapping operator that maps $$$x$$$ to $$$y$$$. Assuming that the image vector, or its representation, is sparse, we are looking for a vector $$$x^*$$$ which minimises the error $$${\| y - \Phi x \|}_2$$$ with the constraint that $$$x$$$ has at most $$$K$$$ non-zero elements. The number of non-zeros in a vector is indicated by the operator $$${\| \cdot \|}_0$$$.

The NIHT algorithm performs reconstruction using gradient descent iterations each followed by the thresholding step:

$$x^{n+1}= H_K \left(x^n +\mu^n \Phi^T \left( y - \Phi x^n \right)\right)$$

where $$$H_K$$$ is the non-linear operator that keeps the $$$K$$$ largest magnitude elements of a vector and sets the rest to zero, and $$$\mu^n$$$ is an adaptive step size to maximally reduce the error at each iteration. We chose to use a representation of $$$x$$$ in the wavelet domain. We also consider thresholding to the $$$K$$$ largest elements consistent across the time dimension, which we refer to as joint thresholding.

We examined the
performance of NIHT, along with three other methods: $$$l_2$$$ regularisation with
conjugate gradients, Iterative Soft Thresholding (IST)^{4} using wavelets,
including joint thresholding in the time dimension, and Total Variation (TV)^{5} in the time dimension. All algorithms are applied with parallel imaging using
SENSE,^{6} where the sensitivities are estimated from the fully sampled k-space with
the ESPIRiT^{7} method.

We
performed tests on data acquired from a single healthy subject. A Fully
sampled, free breathing 2D GRE cine, was synthetically undersampled in 2D, so as to emulate undersampling in the two
phase-encoding directions of a 3D Cartesian acquisition. The relevant cine MRI
parameters^{8} are listed in the table of figure 1.

We applied 2D variable
density Poisson disk sampling masks^{9} on the data to produce acceleration
rates between 2.09 to 46.44. Images were then reconstructed with each method, and
compared to the fully sampled SENSE reconstruction over a region of interest,
containing mainly the heart. We calculated the Root Mean Squared Error (RMSE),
and the Structural Similarity Index (SSIM),^{10} the latter being generally
considered to provide a better measure of perceptual visual fidelity. The
algorithms were also run with a range of regularisation parameters for each
method, and the ones yielding the lowest RMS and highest SSIM were chosen.

**RESULTS**

The RMSE and SSIM are plotted in figure 2. Better RMSE performance is observed for the joint NIHT for accelerations above ~17.5, with a lower rate of RMSE increase across acceleration for both NIHT and joint NIHT, compared to IST. For accelerations lower than ~15, IST achieves the best RMSE. In terms of SSIM, the joint NIHT appears superior above an acceleration of ~20.

Figure 3 shows accelerations of 30-fold, along with a spatial-temporal profile for a line crossing the septum. We do not observe any significant advantage between the algorithms in their spatial-temporal profile, however TV appears to have increased temporal blurring in a region of boundary change.

**DISCUSSION**

**CONCLUSIONS**

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Figure 1: Scan parameters for cine.

Figure 2: Imaging performance across acceleration factors for conjugate gradients (CG), IST, NIHT, and temporal TV. A) RMSE, B) SSIM.

Figure 3: A) Full image of a fully sampled temporal slice at end systole, showing the region of interest used in the comparisons (red rectangle). B) Images of the region of interest, fully sampled and 30-fold accelerated resulting from each algorithm, as marked at their top. C) Spatial-temporal
profiles for each of the images in (A). White arrow marks an increased temporal blurring with TV reconstruction.