Yudu Li^{1,2}, Fan Lam^{2}, Bryan Clifford^{1,2}, Rong Guo^{1,2}, Xi Peng^{2,3}, and Zhi-Pei Liang^{1,2}

Reconstruction for MR spectroscopic imaging (MRSI) is a challenging problem where incorporation of spatiospectral prior information is often necessary. While spectral constraints have been effectively utilized in the form of temporal basis functions, spatial constraints are often imposed using spatial regularization. In this work, we present a new kernel-based method to incorporate a priori spatial information, which was motivated by the success of kernel-based methods in machine learning. It provides a new mechanism for constrained image reconstruction, effectively incorporating a priori spatial information. The proposed method has been evaluated using both simulation and *in vivo* data, producing very impressive results. This new reconstruction scheme can be used to process any MRSI data, especially those from high-resolution MRSI experiments.

**Kernel-based signal model**

Exploiting the partial separability (PS) property of MRSI signals, the desired spatiotemporal function can be expressed as^{5}

$$\hspace{8em}s(\boldsymbol{x}_n,t_m)=\sum_{l=1}^{L}c_{l}(\boldsymbol{x}_n)\psi_{l}(t_m),\hspace{0.3em}n=1,2,...,N\hspace{0.3em}\mathrm{and}\hspace{0.3em}m=1,2,...,M\hspace{7.5em}(1)$$

where $$$L$$$ is usually a small number in practice, $$$\{\psi_l(t_m)\}_{l=1}^L$$$ are temporal basis functions and $$$\{c_{l}(\boldsymbol{x}_n)\}_{l=1}^{L}$$$ the corresponding spatial coefficients.

In the previous works^{2-3}, the $$$c_l(\boldsymbol{x}_n)$$$ is represented using the conventional Fourier model; in this work, we use a kernel-based model to represent $$$c_l(\boldsymbol{x}_n)$$$. More specifically, we assume that $$$c_{l}(\boldsymbol{x}_n)$$$ at each location $$$\boldsymbol{x}_n$$$ can be viewed as a function of a set of low-dimensional features $$$\hspace{0.2em}\boldsymbol{f}_n\in\mathbb{R}^p$$$:

$$\hspace{16.6em}c_{l}(\boldsymbol{x}_n)=\Omega_l(\boldsymbol{f}_n).\hspace{16.6em}(2)$$

The features $$$\{\boldsymbol{f}_n\}_{n=1}^N$$$ can be extracted or "learned" from water images (such as those obtained from a reference scan or the companion water images in MRSI experiments without water suppression). However, $$$\Omega_l(\cdot)$$$ is often highly complex in practice and cannot be accurately described as a linear operator in the original feature space^{4,6-7}. Inspired by the "kernel trick" in machine learning, we express $$$\Omega_l(\cdot)$$$ in a transformed space spanned by $$$\{\phi(\boldsymbol{f}_n):\boldsymbol{f}_n\in{\mathbb{R}^p}\}$$$ as:

$$\hspace{16.2em}\Omega_l(\boldsymbol{f}_n)=\omega_l^T\phi(\boldsymbol{f}_n),\hspace{16.2em}(3)$$

where $$$\phi(\cdot)$$$ is some mapping function. The well-known representer theorem ensures that the optimal $$$\omega_l$$$ takes the following form^{8}:

$$\hspace{16.25em}\omega_l=\sum_{i=1}^N\alpha_{l,i}\phi(\boldsymbol{f}_i).\hspace{16.25em}(4)$$

Hence we obtain the kernel representation for $$$c_l(\boldsymbol{x}_n)$$$ as

$$\hspace{11.6em}c_l(\boldsymbol{x}_n)=\sum_{i=1}^N\alpha_{l,i}\phi^T(\boldsymbol{f}_i)\phi(\boldsymbol{f_n})=\sum_{i=1}^N\alpha_{l,i}k(i,n),\hspace{11.6em}(5)$$

where $$$k(i,n)=\phi^T(\boldsymbol{f}_i)\phi(\boldsymbol{f_n})$$$ is a kernel function. Substituting Eq. (5) into Eq. (1) leads to the kernel-based representation for the MRSI signals:

$$\hspace{12.8em}s(\boldsymbol{x}_n,t_m)=\sum_{l=1}^{L}\{\sum_{i=1}^N\alpha_{l,i}k(i,n)\}\psi_{l}(t_m).\hspace{12.8em}(6)$$

The equivalent matrix-vector form of Eq. (6) is

$$\hspace{17.5em}\boldsymbol{s}=KA\Psi,\hspace{17.5em}(7)$$

where $$$A\in\mathbb{C}^{N×L}$$$ is formed by $$$\{\alpha_{l,i}\}$$$ appropriately. Note that each column of $$$K$$$ can also be viewed as a spatial basis function.

**Kernel-based Image Reconstruction**

As in the existing works, the temporal basis functions in Eq. (6) are determined from training data^{2-3}. To determine the kernel matrix, we choose the radial Gaussian kernel function:

$$k(\boldsymbol{f}_i,\boldsymbol{f}_j)=\exp(-\frac{||\boldsymbol{f}_i-\boldsymbol{f}_j||_2^2}{2\sigma^2}),$$

which corresponds to an infinite-dimensional mapping function. The image feature vectors $$$\{\boldsymbol{f}_i\}$$$ contain image intensities and edge information.

After $$$K$$$ is determined, kernel-based reconstruction for MRSI is performed using a penalized maximum likelihood (ML) formulation. Specifically, we solve the following regularized least-squares problem:

$$\hspace{10.25em}\{A^*\}=\min_{\{A\}}||d-ΠF_B{KA\Psi}||_2^2+{\lambda}R(A),\hspace{10.25em}(8)$$

where $$$d$$$ is the measured MRSI data, $$$Π$$$ the k-space sampling operator, $$$F_B$$$ the Fourier encoding operator including B_{0} inhomogeneity, $$$R(A)$$$ a regularization term and $$$\lambda$$$ a tunable parameter. Many choices can be used for $$$R(A)$$$, and in this work we choose $$$R(A)=||KA\Psi-\hat{S}||$$$ where $$$\hat{S}$$$ is the initial reconstruction obtained by the use of edge-preserving regularization. But note that $$$R(A)$$$ mainly serves to improve the conditioning rather than imposing spatial smoothing, as in the standard regularization techniques. The final reconstruction of the desired spatiotemporal function can be obtained by substituting $$$A^*$$$ into Eq. (7).

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Figure 1: Simulation results: (a) ground truth, (b) subspace projection, (c) weighted-L_{2} regularized reconstruction, and (d) the proposed method. The left column shows the spatial distributions of the frequency component at 2.0 ppm. The right two columns show the spectra corresponding to the voxels marked by the red and blue dots, respectively. The reconstruction errors are also shown in red. Note the significant improvement in the reconstruction accuracy achieved by the proposed method.

Figure 2: Illustration of spatial basis functions, showing how the water side information is incorporated into the signal model: (a) a sample of spatial bases for one slice obtained from 12 columns of the kernel matrix respectively, and (b) the companion anatomical image used for the extraction of features.

Figure 3: *In vivo *results: (a) subspace projection and (b) reconstruction by the proposed method. The left column illustrates the spatial distributions of the frequency component at 2.0 ppm. The right columns are representative spectra from the voxels marked by the red and blue dots, respectively. Note that the reconstruction by subspace projection has much more spatial and spectral variations, which have been significantly reduced by the proposed method.