Chaoyi Zhang^{1}, Ukash Nakarmi^{1}, Hongyu Li^{1}, Yihang Zhou^{2}, Dong Liang^{3}, and Leslie Ying^{1,4}

Kernel Principal component analysis(KPCA) model has recently been proposed to accelerate dynamic cardiac imaging. In this abstract, we study the effectiveness of KPCA for MR T2 mapping from highly under-sampled data acquired at different echo time. Different from dynamic cardiac imaging where only morphological information is needed, the quantitative values are highly important in parameter mapping. Here we use a self-trained KPCA model to guarantee the accuracy of the reconstructed T2 maps. The experimental results show that the proposed method can recover the T2 map with high fidelity at high acceleration factors.

In MR parameter mapping, the mth parameter-weighted image $$$I_{m}$$$ and the k-space data $$$d_{m}$$$ at mth measurement time can be represented as $$$d_{m}=F_{m}I_{m}+n_{m}$$$, where $$$F_{m}$$$ denotes Fourier operator with undersampling, $$$n_{m}$$$ denotes the k-space measurement noise. In the meanwhile, the images $$$I_{m}$$$ also come from a point from a parametric model: $$$I_{m}=P_{m}(\theta)\rho$$$, where $$$P_{m}(\theta)$$$ is a parametric function of θ and ρ is a linearly related constant. For example, in T2 mapping, the image at the mth measurement time is: $$$I_{m}=\rho e^{-TE(m)\theta}$$$, where θ is 1/T_{2}. Our objective is to reconstruct the parameter map θ from the undersampled k-space data $$$d_{m}$$$ from all m. To find θ, we first reconstruct all $$$I_{m}s$$$ using KLR.

**Training Using kPCA:** We first construct a number of training datasets using the parametric model: $$$p_{t,m}=P_{m}(\theta t)\rho$$$ where the parameter θt take different values at different training sample t. We then perform kernel PCA on the training data. Specifically, we construct the kernel matrix $$$K_{p}$$$ whose (i,j)th element is given by $$$(<p_{ti},p_{tj}>+c)^{d}$$$, and then the principal components in the feature space is represented as $$$v_{l}=\sum_{t=1}^b\alpha_t^l\phi(p_{t})$$$, where $$$\alpha_t^l$$$ is the eigenvector of $$$K_{p}$$$.

**Low Rank Enforcement in Feature Space:** In this step, the zero-filled reconstruction from undersampled data is projected onto the feature space. Specifically, we construct a kernel vector between the training and testing data $$$k_{xp}=(<p_{ti},x_{q}>+c)^{b}$$$, q = 1,2,...,M, where M is the size of the image, and then find the projection coefficients $$$\beta_l^q=\alpha_l^Tk_{xp}$$$ will be calculated by projecting the $$$K_{xp}$$$ onto $$$v_{k}$$$. We assume that data can be sparsely represented by remaining only largest PCs in feature space and only K projection coefficients are calculated.

**Preimaging with Data Consistency and TV Constraint:** The images are then transferred back to the original data space using preimaging methods. The data consistency is enforced by replacing the values at the acquired k-space locations by the acquired values. The reconstructed image $$$I_{m}$$$ is finally smoothed using the total variation(TV) prior.

The above three steps are then repeated until convergence. Once we have all parameter-weighted images, we use Lavenberg-Marquardt algorithm to fit the parametric model and find the desired parameter map θ.

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