Sunrita Poddar^{1}, Yasir Mohsin^{1}, Bijoy Thattaliyath^{1}, Diedra Ansah^{1}, and Mathews Jacob^{1}

The main focus of this abstract is to introduce an efficient algorithm to recover a free breathing and ungated cardiac MR image series from highly undersampled measurements. The main contributions are (i) a kernel low-rank algorithm to estimate the manifold structure (Laplacian) from noisy navigator signals, (ii) a fast algorithm that uses the Laplacian basis functions to recover the data from highly undersampled measurements. The utility of the algorithm is demonstrated on radial acquisitions from patients with congenital heart disease; the results show that the framework is a promising alternative to self-gating methods.

We consider the recovery of $$$\mathbf X$$$, whose columns are images in the series. We estimate the manifold structure from the navigators $$$\mathbf Q= \boldsymbol \Phi \mathbf X$$$, where $$$\boldsymbol\Phi$$$ is the navigator acquisition operator. Our recent work shows that the non-linearly transformed original points $$$\boldsymbol\Psi(\mathbf q_i);i=1,..,N$$$ live in a subspace, if the original points $$$\mathbf q_i$$$ live on a band-limited surface. We consider the denoising of the navigator lines using the algorithm:\begin{equation}\mathbf Z^* = \arg\min_{\mathbf Z} \lbrace \|\mathbf Z - \mathbf Q\|_F^2 + \lambda~\|\mathbf \Psi(\mathbf Z)\|_* \rbrace,\end{equation}where $$$\mathbf \Psi(\mathbf Z)$$$ is the matrix whose columns are non-linear transforms of the original points and $$$\|\cdot\|_*$$$ denotes the nuclear norm. The kernel trick $$$\langle \mathbf \Psi(\mathbf x_1),\mathbf \Psi(\mathbf x_2)\rangle = \mathcal K(\mathbf x_1,\mathbf x_2)$$$, where $$$\mathcal K$$$ is the Gaussian kernel, allows us to solve the above problem using an iterative reweighted algorithm without performing the explicit mapping. In contrast, Nakarmi et al^{5} recently proposed a kernel low-rank technique, which requires computing the explicit images and pre-images. We solve the problem:\begin{equation}\mathbf Z^{(n)} = \arg\min_{\mathbf Z} \left\lbrace \|\mathbf Z - \mathbf Q\|_F^2 + \lambda~{\rm trace}\left(\mathbf Z~ \mathbf L^{(n)} ~\mathbf Z^{H} \right) \right\rbrace,\end{equation}where $$$\mathbf L^{(n)} = \mathbf D^{(n)} - \mathbf W^{(n)}$$$, $$$\mathbf W^{(n)} = \mathcal K(\mathbf Z^{(n-1)})\odot\mathbf P^{(n)}$$$, $$$\mathbf D^{(n)}_{ii} = \sum_j \mathbf W^{(n)}_{ij}$$$, and $$$\mathbf P^{(n)} = [\mathcal K(\mathbf Z^{(n-1)}) + \gamma^{(n)} \mathbf I]^{-\frac{1}{2}}$$$. This iterative algorithm is analogous to SToRM^{2}, where $$$\mathbf L^{(n)}$$$ is the Laplacian. Thus, the above algorithm provides a robust estimate for the Laplacian as a by-product; we propose to use this $$$\mathbf L$$$ in SToRM to recover $$$\mathbf X$$$:\begin{equation}\mathbf X^* = \arg\min_{\mathbf X} \left\lbrace \|\mathcal{A}(\mathbf X) - \mathbf B\|_F^2 + \lambda~{\rm trace}\left(\mathbf X \mathbf L \mathbf X^{H} \right) \right\rbrace,\end{equation}where $$$\mathcal A$$$ is the measurement operator and $$$\mathbf B$$$ are the measurements. To reduce the computational complexity, we perform an eigen decomposition of $$$\mathbf L=\mathbf V \boldsymbol \Sigma \mathbf V^T$$$. The eigen functions are analogous to Fourier exponentials and are ideally suited to represent smooth functions on the manifold; this allows us to rewrite the above expression as:\begin{equation}\mathbf U^{*} =\arg \min_{\mathbf U} \|\mathcal A(\mathbf U\mathbf V^H)-\mathbf b\|^{2}_{F} + \lambda~ \sum_{i=1}^k \sigma_i \|\mathbf u_i\|^2\end{equation}Our experiments show that only $$$30$$$ basis functions are sufficient, which greatly reduces the computational complexity. The proposed scheme is summarized in Fig 1. Denoising of data using the proposed iterative kernel low-rank technique is illustrated in Fig 2.

1. Li Feng, Leon Axel, Hersh Chandarana, Kai Tobias Block, Daniel K. Sodickson, Ricardo Otazo. XD-GRASP: Golden-Angle Radial MRI with Reconstruction of Extra Motion-State Dimensions Using Compressed Sensing. Magnetic Resonance in Medicine. 2016; 75(2): 775–788.

2. Sunrita Poddar, Mathews Jacob. Dynamic MRI using smoothness regularization on manifolds (SToRM). IEEE Transactions on Medical Imaging. 2016; 35(4):1106 - 1115.

3. M. Usman, D. Atkinson, C. Kolbitsch, T. Schaeffter, C. Prieto. Manifold learning based ECG-free free-breathing cardiac CINE MRI. Journal of Magnetic Resonance Imaging. 2015; 41(6):1521–1527.

4. Kanwal K. Bhatia, Jose Caballero, Anthony N. Price, Ying Sun, Jo V. Hajnal, Daniel Rueckert. Fast Reconstruction of Accelerated Dynamic MRI Using Manifold Kernel Regression. International Conference on Medical Image Computing and Computer-Assisted Intervention 2015.

5. Ukash Nakarmi, Yanhua Wang, Jingyuan Lyu, Dong Liang, Leslie Ying. A Kernel-based Low-rank (KLR) Model for Low-dimensional Manifold Recovery in Highly Accelerated Dynamic MRI. IEEE Transactions on Medical Imaging. 2017; 36(11): 2297 - 2307.

Outline of the proposed technique. We acquire four radial navigators for each temporal frame; the corresponding multichannel sampling operator is denoted by $$$\boldsymbol \Phi$$$. The navigator signals (in blue) are used to estimate the manifold Laplacian using the iterative kernel low-rank technique. The matrix of temporal basis functions $$$\mathbf V$$$ are obtained as the important eigen-vectors of the estimated Laplacian matrix. The basis images $$$\mathbf U$$$ corresponding to the eigen vectors are then obtained by solving the SToRM optimization problem. The reconstructed images are recovered as $$$\mathbf X = \mathbf U \mathbf V^{H}$$$.

Illustration of denoising and Laplacian estimation using kernel low-rank algorithm. Noisy points on a TigerHawk logo in (a) are denoised using the algorithm to obtain (b). (c) shows the eigen-vectors of the Laplacian matrix estimated using $$$50^{th}$$$ iteration (left), $$$1^{st}$$$ iteration (center) and exponential approach (right), respectively. The higher order basis functions estimated by the proposed scheme are smooth and reliable, while the ones estimated using Exponential approach and from the first iteration are rough/not reliable.

We show the denoising of the navigator signals from three coils in (3). The Laplacian estimation from the denoised data provides improved results as shown in the next figure.

We compare the images reconstructed using the new approach (right column) and our previous STORM reconstruction (left column). The improved estimation of the Laplacian, along with the reconstruction using the temporal basis functions, translated to higher quality reconstructions. Specifically, the images reconstructed using the proposed technique look sharper and preserve the details much better. This can also be observed from the temporal profiles shown along the blue dotted line.

The use of the basis function based approach provided a significant speedup; the original SToRM reconstruction took around 30 mins, while the proposed reconstruction took only 2 mins.

We demonstrate the proposed algorithm on a dataset with extensive breathing induced out-of-plane motion. This can be observed from the temporal profile shown at the top. We show examples of reconstructed image series from different respiratory phases. The results show the ability of the algorithm to yield good reconstructions even in the presence of large respiratory motion. This approach may enable us to simultaneously study cardiac and respiratory function in subjects with pulmonary hypertension and/or CoPD.

Manifold embedding of the images using the eigen vectors of the Laplacian matrix. The second and third eigen vectors in (a) provide information about the respiratory and cardiac phases, which can be used to segment the data into different phases. (b) shows the embedding of the images in the 2d plane. The images corresponding to the dots in the center of the colored squares (representative image with a specific cardiac and respiratory phase) are shown on the left. Note from (c) that these images actually come from different time points in the time series with 900 images.