Yudu Li^{1,2} and Zhi-Pei Liang^{1,2}

Constrained image reconstruction incorporating prior information has been widely used to overcome the ill-posedness of reconstruction problems. In this work, we propose a novel "kernel+sparse" model for constrained image reconstruction. This model represents the desired image as a function of features "learned" from prior images plus a sparse component that captures localized novel features. The proposed method has been validated using multiple MR applications as example. It may prove useful for solving a range of image reconstruction problems in various MR applications where both prior information and localized novel features exist.

Image reconstruction is known to be an ill-posed mathematical problem because most imaging operators are ill-conditioned and its feasible solutions are not unique due to finite sampling. To address this issue, constrained reconstruction incorporating prior information has been widely used. A popular approach to constrained reconstruction is to use regularization in which priori information is incorporated implicitly in a regularization functional. In this work, we propose a novel “kernel+sparse” model for constrained reconstruction. This model represents the desired image as a function of features “learned” from prior images plus a sparse component that captures localized novel features. The proposed representation has been validated using multiple MR applications as a testbed.

**Kernel+Sparse Model**

We decompose the spatial variations of a desired image function into two terms, one absorbing prior information (using a kernel model) and the other capturing localized sparse features:

$$\hspace{14em}\rho(\boldsymbol{x}_n)=\sum_{i=1}^N\alpha_{i}k(i,n)+\tilde{\rho}(\boldsymbol{x}_n).\hspace{14em}(1)$$

The kernel component was motivated by the success of kernel models in machine learning. More specifically, this component models the desired image value at spatial location $$$\boldsymbol{x}_n$$$ as a function of a set of low-dimensional features $$$\boldsymbol{f}_n\in\mathbb{R}^m$$$:

$$\hspace{16.5em}\rho(\boldsymbol{x}_n)=\Omega(\boldsymbol{f}_n).\hspace{16.5em}(2)$$

The features $$$\{\boldsymbol{f}_n\}_{n=1}^N$$$
are learned/extracted from prior images, which
leads to implicit incorporation of priori information. However, the
function $$$\Omega(\cdot)$$$
is often highly complex in practice and cannot be
accurately described as a linear operator in the original feature space^{1-2}.
Inspired by the "kernel trick" in machine learning, we linearize $$$\Omega(\cdot)$$$
in a high-dimensional transformed space spanned by $$$\{\phi(\boldsymbol{f}_n):\boldsymbol{f}_n\in\mathbb{R}^m\}$$$:

$$\hspace{16.0em}\Omega(\boldsymbol{f}_n)=\omega^T\phi(\boldsymbol{f}_n).\hspace{16.0em}(3)$$

In the sense of empirical risk minimization (ERM), the optimal $$$\omega$$$ should minimize the empirical risk:

$$\hspace{12.8em}r_{amp}(\omega)=\frac{1}{N}\sum_{n=1}^{N}l(\omega^T\phi(\boldsymbol{f}_n),\rho(\boldsymbol{x}_n)),\hspace{12.8em}(4)$$

where $$$l(\cdot)$$$ is some loss function (e.g., square-error loss).
The well-known representer theorem ensures that this optimal $$$\omega$$$ takes the following form^{3}:

$$\hspace{16.2em}\omega=\sum_{n=1}^N\alpha_i\phi(\boldsymbol{f}_i).\hspace{16.2em}(5)$$

Hence we obtain the kernel-based representation for $$$\rho(\boldsymbol{x}_n)$$$ as:

$$\hspace{11.7em}\rho(\boldsymbol{x}_n)=\sum_{i=1}^N\alpha_i\phi^T(\boldsymbol{f}_i)\phi(\boldsymbol{f}_n)=\sum_{i=1}^N\alpha_ik(i,n),\hspace{11.7em}(6)$$

where $$$k(i,n)=\phi^T(\boldsymbol{f}_i)\phi(\boldsymbol{f}_n)$$$ is a kernel function. However, Eq. (6) alone may bias the model towards prior information. To avoid this potential problem, we introduce a sparsity term into Eq. (6) to capture localized novel features as described in Eq. (1) with the requirement that $$$||M\{\tilde{\rho}(\boldsymbol{x}_n)\}||_0\leq{\epsilon}$$$ where $$$M(\cdot)$$$ is some sparsifying transform.

**Image Reconstruction**

Image reconstruction using the proposed model requires specification of the kernel function and features. In this work, we choose the radial Gaussian kernel function:

$$\hspace{13.9em}k(\boldsymbol{f}_i,\boldsymbol{f}_n)=\exp(-\frac{||\boldsymbol{f}_i-\boldsymbol{f}_n||_2^2}{2\sigma^2})\hspace{13.9em}(7)$$

which
corresponds to an infinite-dimensional mapping function^{2}. Choices of
features are rather flexible, such as image intensities and edge information,
making the proposed model even more powerful in absorbing a large range of
priors.

The proposed kernel-based signal model results in maximum likelihood reconstruction by solving:

$$\hspace{5.7em}\{\alpha_i^*,\tilde{\rho}^*\}=\arg\max_{\{\alpha,\tilde{\rho}\}}L(d,I(\{\sum_{i=1}^{N}\alpha_ik(i,n)+\tilde{\rho}(\boldsymbol{x}_n)\})),\mathrm{s.t.}||M\{\tilde{\rho}(\boldsymbol{x}_n)\}||_0\leq{\epsilon}.\hspace{5.7em}(8)$$

where $$$d$$$ denotes the measured data, $$$I(\cdot)$$$ the imaging operator, and $$$L(\cdot,\cdot)$$$ the likelihood function.

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2. Wang, G., & Qi, J. (2015). PET image reconstruction using kernel method. IEEE transactions on medical imaging, 34(1), 61-71.

3. Schölkopf, B., Herbrich, R., & Smola, A. (2001). A generalized representer theorem. In Computational learning theory (pp. 416-426). Springer Berlin/Heidelberg.

4. Peng, X., Lam, F., Li, Y., Clifford, B., & Liang, Z. P. (2017). Simultaneous QSM and metabolic imaging of the brain using SPICE. Magnetic Resonance in Medicine.

5. Lam, F., Ma, C., Clifford, B., Johnson, C. L., & Liang, Z. P. (2016). High‐resolution 1H‐MRSI of the brain using SPICE: Data acquisition and image reconstruction. Magnetic resonance in medicine, 76(4), 1059-1070.

6. Liu,
T., Liu, J., De Rochefort, L., Spincemaille, P., Khalidov, I., Ledoux, J. R.,
& Wang, Y. (2011). Morphology enabled dipole inversion (MEDI) from a single‐angle acquisition: comparison with COSMOS in
human brain imaging. Magnetic resonance in medicine, 66(3), 777-783.

Figure 1: Simulation results demonstrating the role of the sparse term in the proposed model: (a) ground truth image, (b) the prior image, (c) kernel-based reconstruction without the sparse term, and (d) reconstruction by the proposed method. Note the loss of some details (e.g., edges) in (c) has been recovered in (d), indicating that the sparse term indeed well captures the novel features.

Figure 2: Simulation study for CS-MRI comparing the proposed method to conventional algorithms including results from (a) traditional CS reconstruction penalizing the total variation, (b) reconstruction with edge-preserving regularization, and (c) the proposed method. The peak signal-to-noise-ratio (PSNR) is also shown in red. Note the significant improvement in the reconstruction accuracy by the proposed method.

Figure 3: *In vivo* results for dynamic imaging of breast: (a) Fourier reconstruction, and (b) reconstruction by the proposed method. Note that the proposed method produces less noisy reconstruction, compared to the Fourier Method.

Figure 4: In vivo results for QSM including reconstructions by (a) conventional method with edge-preserving regularization, and (b) the proposed method. Note the proposed method significantly improves the estimated susceptibility maps with clearer tissue contrast and better-defined edges.