MRI is an important tool for clinical diagnosis. Although recognized for being non-invasive and producing images of high quality and excellent soft tissue contrast, its long acquisition times and high cost are problematic. Recently, deep learning techniques have been developed to help solve these issues by improving acquisition speed and image quality.
The multi-coil measurement process is modeled by a linear operator, the SENSE encoding model
$$
\begin{aligned}
A:\mathbb{C}^{N_x\times N_y}&\to \mathbb{C}^{N_S \times N_C}\\
x &\mapsto y=\mathcal{PFC}x.
\end{aligned}
$$
The discretized image $x$ corresponds to the complex-valued transversal magnetization in the tissue. In the encoding process, it is first weighted with the coil-sensitivity maps of the $N_C$ receive $\mathcal{C}$oils, then $\mathcal{F}$ourier transformed and finally projected to the $N_S$ sample points of the acquired sampling $\mathcal{P}$attern. Unrolled model-based deep learning approaches are motivated by classical optimization algorithm of the linear inverse problem and integrate learned prior knowledge by learned regularization terms. Typical examples of end-2-end trained networks from the field of MRI are the Variational Network [1] or MoDL [2].
Despite MRI reconstruction often being treated linearly, there are many applications that require non-linear approaches. For instance, the estimation of coil-sensitivity maps can be challenging. An alternative to the use of calibration measurements or pre-estimation of the sensitivity maps from fully-sampled auto-calibration regions is to integrate the estimation into the reconstruction problem. This results in a non-linear - in fact, bilinear - forward model of the form
$$
\begin{aligned}
F:\mathbb{C}^{N_x\times N_y}\times \mathbb{C}^{N_x\times N_y\times N_c}&\to \mathbb{C}^{N_S \times N_C}\\
x = \begin{pmatrix}
x_{\mathrm{img}}\\x_{\mathrm{col}}
\end{pmatrix} &\mapsto y=\mathcal{PF}\left(x_{\mathrm{img}}\odot x_{\mathrm{col}}\right)\,.
\end{aligned}
$$
A possible approach to solve the corresponding inverse problem is the iteratively regularized Gauss-Newton method (IRGNM) [3], which can in turn be combined with deep-learning based regularization [4] similarly to MoDL.
Another source of non-linearity in the reconstruction is the temporal evolution of transverse magnetization. The magnetization follows the Bloch equations, which are parametrized by tissue-specific relaxation parameters $T_1$ and $T_2$. In quantitative (q)MRI, parameter maps $x_{\mathrm{par}}$ are estimated instead of qualitative images $x_{\mathrm{img}}$ of transverse magnetization. In model-based qMRI, physical models that map the parameter maps $x_{\mathrm{par}}$ to the transverse magnetization are combined with encoding models to create non-linear forward models of the form [5]:
$$
\begin{aligned}
F:\mathbb{C}^{N_x\times N_y\times N_p}\times \mathbb{C}^{N_x\times N_y\times N_c}&\to \mathbb{C}^{N_S \times N_C}\\
x = \begin{pmatrix}
x_{\mathrm{par}}\\x_{\mathrm{col}}
\end{pmatrix} &\mapsto y=\mathcal{PF}\left(\mathcal{M}(x_{\mathrm{par}})\odot x_{\mathrm{col}}\right)
\end{aligned}
$$
An efficient way to solve a particular class of such non-linear inverse problems is the approximation of the non-linear signal model in linear subspaces, which in turn can be well combined with deep-learning based regularization [6].
This talk will cover deep-learning based approaches to solve the non-linear inverse problems defined above.
[1] K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson, T. Pock, F. Knoll. Learning a variational network for reconstruction of accelerated MRI data, Magn. Reson. Med. 79: 3055-3071, 2018.
[2] H. K. Aggarwal, M. P. Mani, M. Jacob. MoDL: Model-Based Deep Learning Architecture for Inverse Problems, IEEE Trans. Med. Imaging. 38: 394-405, 2019.
[3] M. Uecker, T. Hohage, K. T. Block, J. Frahm. Image reconstruction by regularized nonlinear inversion—Joint estimation of coil sensitivities and image content, Magn. Reson. Med. 60: 674-682, 2018.
[4] M. Blumenthal, G. Luo, M. Schilling, M. Haltmeier, M. Uecker, NLINV-Net: Self-Supervised End-2-End Learning for Reconstructing Undersampled Radial Cardiac Real-Time Data, Proc. Intl. Soc. Mag. Reson. Med. 28: 0499, 2022
[5] X. Wang et al. Physics-based reconstruction methods for magnetic resonance imaging. Phil. Trans. R. Soc. A. 379: 20200196, 2021
[6] M. Blumenthal et al. Deep Subspace Learning for Improved T1 Mapping using Single-shot Inversion-Recovery Radial FLASH. Proc. Intl. Soc. Mag. Reson. Med. 28: 0241, 2022
