Sunrita Poddar^{1}, Xiaoming Bi^{2}, Dingxin Wang^{2}, and Mathews Jacob^{1}

We introduce an image manifold smoothness regularization, coupled with spatial regularization, for high-resolution free-breathing and ungated cardiac cine imaging. Prior work in this area relied on additional navigators within each image frame to estimate the manifold structure. In this abstract, we focus on eliminating the need for navigators, which will provide improved sampling efficiency.

We extend the image manifold smoothness regularization (STORM)^{1} scheme, where the similarity of images at the same cardiac/respiratory phases was exploited. STORM was enabled by a modified golden angle acquisition, where 3-4 spokes are acquired at the same k-space locations every frame. The measurement for the $$$i^{th}$$$ image frame is specified by $$\underbrace{\begin{bmatrix}\mathbf z_i\\ \mathbf y_i\end{bmatrix} }_{\mathbf b_i}= \underbrace{\begin{bmatrix}\mathbf \Phi\\ \mathbf B_i\end{bmatrix} }_{\mathbf A_i} \mathbf x_i + \mathbf n_i$$ where $$$\mathbf z_i$$$ denotes the navigator measurements. $$$\mathbf y_i$$$ are the measurements that are different every frame. The Laplacian of the image manifold $$$\mathbf L$$$ is estimated from navigator data ($$$\mathbf Z = \mathbf \Phi\mathbf X$$$) as $$$\mathbf L = \mathbf D-\mathbf W$$$, where$$\mathbf W_{i,j} = \exp\left(-{\| \mathbf z_i - \mathbf z_j\|^2}/{\sigma^{2}}\right);~~~ \mathbf D = {\rm diag}(\mathbf 1^T \mathbf W)$$While STORM exploits the non-local similarities between images in the time series, it fails to capture the redundancies within each image. We improve STORM by adding a spatial total variation prior:$$\mathbf X^* = \arg\min_{\mathbf X} \left\lbrace \|\mathcal{A}(\mathbf X) - \mathbf b\|_F^2 + \lambda_1~{\rm trace}\left(\mathbf X \mathbf L \mathbf X^{H} \right) + \lambda_2 ~\|\mathbf T \mathbf X \|_{\ell_1}\right\rbrace$$Here, $$$\mathbf T$$$ is the spatial gradient matrix. We use FISTA^{2} to solve the above optimization scheme.

We now introduce a technique (illustrated in Fig. 1) to estimate the weight matrix $$$\mathbf W$$$, when navigators are not available. We split the radial spokes in the entire golden angle acquisition to $$$K$$$ groups, denoted by $$$G_k$$$, each of size $$$180/K$$$ degrees. We evaluate approximate weight matrices $$$\mathbf W^{(k)};k=1,..,K$$$ by comparing spokes in the images that are in the $$$k^{\rm th}$$$ group (Fig. 2):$$\mathbf W_{i,j}^{(k)} = \frac{1}{N_{ij}^k}\sum_{\{\mathbf l_m^{(i)},\mathbf l_n^{(j)} \in G_k\}}\exp\left(-{\|\mathbf l_m^{(i)} - \mathbf l_n^{(j)}\|^2}/{\sigma^{2}}\right)$$Here, $$$\mathbf l_m^{(i)}$$$ and $$$\mathbf l_n^{(j)}$$$ are spokes in the $$$i^{\rm th}$$$ and $$$j^{\rm th}$$$ image frames, which are in group $$$G_k$$$. $$$N_{i,j}^k$$$ denotes total number of spoke pairs in $$$k^{\rm th}$$$ group between frames $$$i$$$ and $$$j$$$. If there is no pair of spokes between two image frames, the corresponding weight is set to zero. The complete weight matrix is calculated as $$\mathbf W = \frac{1}{N} \sum_{k=1}^{N} \mathbf W^{(k)}$$To prevent over-smoothing, $$$\mathbf W$$$ is post-processed to keep only 3-5 neighbours per frame.

1. S. Poddar, M. Jacob, "Dynamic MRI using Smoothness Regularization on Manifolds (SToRM)", IEEE Transactions on Medical Imaging, Vol 35, No 4, April 2016.

2. A. Beck, M. Teboulle, "Fast Gradient-based algorithms for Constrained Total Variation Image Denoising and Deblurring Problems", IEEE Transactions on Image Processing, Vol 18, No 11, Nov 2009.

3. S. Bhadra, S. Kaski, J. Rousu, "Multi-view Kernel Completion", arXiv:1602.02518.