Aurelien Bustin^{1}, Gastao Cruz^{1}, Giulia Ginami^{1}, Teresa Correia^{1}, Imran Rashid^{1}, Radhouene Neji^{1,2}, Rene Botnar^{1}, and Claudia Prieto^{1}

Free-breathing coronary MR angiography (CMRA) has shown great potential to visualize coronary stenosis. Three-dimensional (3D) CMRA, however, remains time consuming because a large amount of data is needed to accurately visualize all major coronary arteries. Scan acceleration using compressed sensing (CS) reconstruction has successfully been applied to coronary artery imaging. For high acceleration factors, however, CS-based techniques suffer from residual aliasing artifacts which compromise the diagnostic value of the reconstructed image. We propose a new 3D-patch-based reconstruction that exploits the complex 3D anatomy of the coronary arteries in an effective low-rank framework, which combined with 100% respiratory efficiency enables high-quality isotropic Cartesian CMRA images in ~3 mins.

**Acquisition:** A prototype undersampled Cartesian variable density 3D spiral-like
Cartesian sampling (VD-CASPR^{3}) was implemented on a 1.5T scanner
(Siemens, Aera). A 2D image-navigator precedes each VD-CASPR acquisition to enable beat-to-beat 2D translational respiratory motion estimation/correction without data rejection^{4}. Five healthy volunteers (30±4 years) underwent free-breathing CMRA. Data were acquired in mid-diastole with an ECG-gated 3D bSSFP sequence, 18-channel body and 32-channel spine coils, fat-saturation and T2-preparation
pulses (FOV=320x320x120mm^{3},TR/TE=3.35/1.47ms,FA=90°,bandwidth=890Hz/pixel,T2-preparation duration=40ms) with 1.2mm^{3}
isotropic resolution and undersampling factors of 5 and 9. An additional
free-breathing fully-sampled Cartesian acquisition was performed with identical
imaging parameters for comparison purposes.

**Reconstruction:** The proposed 3D-Patch-based low-rank ReconstructiOn with Self-similariTy
learning (3D-PROST) integrates anatomical structure information from 3D patch neighborhoods
through sparse representation, exploiting the redundancy of 3D patches in the acquired data itself. The
optimization problem iterates between a data consistency step, which reconstructs
a high-resolution isotropic MR volume , and a
low-complexity 3D patch-based denoising step, which provides a reconstructed volume
as prior for the next step (Fig.1). The two sub-problems are solved iteratively
in an effective Augmented Lagrangian (AL) scheme:
$$\mathcal{L}\left
(x,\alpha \right ):=\underset{x,\alpha}{\mathrm{argmin}}\,\frac{1}{2}\Vert
Ex-\rho\Vert_2^2+\lambda\Vert\alpha\Vert_0+\frac{\mu}{2}\Vert
x-D\alpha-b\Vert_2^2$$
where
$$$E$$$ is the encoding operator (including coils, Fourier operator and
sampling), $$$\rho$$$ denotes the undersampled data, $$$b$$$
denotes the AL multiplier, $$$D$$$ is the patch-grouping operator
and $$$\alpha$$$ are the associated sparse coefficients. The $$$l_0$$$
norm counts the number of non-zeros element in $$$\alpha$$$, and
$$$\lambda>0$$$ controls the strength of sparsity. The two problems are:

1)
MR reconstruction (Fig.1-Stage 1) is performed using a conjugate gradient (CG)
descent and uses the 3D denoised volume ($$$\omega=D\alpha$$$) obtained from
Stage 2 as prior knowledge ($$$\omega_0=b_0 = 0$$$):$$\mathcal{L}_1\left
(x\right):=\underset{x}{\mathrm{argmin}}\,\frac{1}{2} \Vert
Ex-\rho\Vert_2^2+\frac{\mu}{2}\Vert x-\omega-b\Vert_2^2$$
2)
A 3D block-matching^{2,5} algorithm is used to exploit redundancies in the
volume $$$x$$$. The L-most similar 3D blocks to a reference block are
extracted, vectorised and concatenated into a 2D matrix. Sparsity of this low-rank
matrix is enforced using complex SVD and by hard thresholding the singular
values below a specific threshold $$$\lambda$$$. The denoised 3D blocks are
then placed back to their original positions by averaging (Fig.1-Stage 2):
$$\mathcal{L}_2\left(\alpha\right):=\underset{\alpha}{\mathrm{argmin}}\,\frac{\mu}{2}\Vert
x-D\alpha-b\Vert_2^2+\lambda\Vert\alpha\Vert_0$$
**Implementation
and Analysis:** The following parameters were empirically selected to
provide the best reconstruction quality: patch_size=5x5x5voxels,
window_search=14x14x14voxels, AL_iterations=4,
CG_iterations=30,$$$\lambda=4$$$,$$$\mu=1$$$. The proposed approach was
compared to conventional zero-filled (ZF), iterative-SENSE, and Wavelet-based
CS^{6} reconstructions. Vessel sharpness of the right and left coronary arteries
was measured using SoapBubble^{7}, and image quality scoring was performed by two
experts.

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