Guanxiong Luo^{1}, Xiaoqing Wang^{1}, Volkert Roeloffs^{1}, Zhengguo Tan^{1}, and Martin Uecker^{1,2}

^{1}Institute for Diagnostic and Interventional Radiology, University Medical Center Göttingen, Germany, Göttingen, Germany, ^{2}Campus Institute Data Science (CIDAS), University of Göttingen, Germany, Göttingen, Germany

Parallel imaging for reduction of scanning time is now routinely used in clinical practice. The spatial information from the coils’ profiles are exploited for encoding. The nonlinear inversion reconstruction is a calibrationless parallel imaging technique, which jointly estimate coil sensitivities and image content. In this work, we demonstrate how to combine such a calibrationless parallel imaging technique with an advanced neural network based image prior for efficient MR imaging.

$$F(\rho, c):=(\mathcal{F}_S(\rho\cdot c_1), \cdots, \mathcal{F}_S(\rho\cdot c_N)) = {y},$$

where $$$\mathcal{F}_S$$$ is an undersampled Fourier transform operator and the corresponding k-space data is $$${y} = (y_1, \cdots, y_N)^T$$$, $$$\rho$$$ denotes the spin density and $$${c}=(c_1, \cdots, c_N)^T$$$ denotes coil sensitivities. Proposed in the nonlinear inverse reconstruction (nlinv) [1], this problem can be solved with the Iteratively Regularized Gauss Newton Method (IRGNM) by estimating $$$\delta m:=(\delta \rho, \delta c)$$$ in each step $$$k$$$ for given $$$m^k:=(\rho^k, c^k)$$$ with the following minimization problem

$$\underset{\delta x}{\min} \frac{1}{2}\|F'(m^k)\delta m+F(m^k) - \boldsymbol{y}\|_2^2 + \frac{\alpha_k}{2}\mathcal{W}(\boldsymbol{c}+\delta c) + \beta_k{R}(\rho^k+\delta \rho),\quad (1)$$

where $$$\mathcal{W}({c})=\|W{c}\|^2=\|w\cdot\mathcal{F}{c}\|$$$ is a penalty on the high Fourier coefficients of the coil sensitivities and $$$R(\rho)$$$ is a regularization term on $$$\rho$$$. The $$$\alpha_k$$$ and $$$\beta_k$$$ decay based on reduction factor over iteration steps. In this work, the neural network based log-likelihood prior was investigated [2], formulated with following joint distribution

$$\log P(\hat{\Theta}, \boldsymbol{x}) = \log p(\boldsymbol{x};\mathrm{NET}(\hat{\Theta}, \boldsymbol{x}))=\log p(x^{(1)})\prod_{i=2}^{n^2} p(x^{(i)}\mid x^{(1)},..,x^{(i-1)}),$$

where the neural network $$$\mathrm{NET}(\hat{\Theta}, \boldsymbol{x})$$$ outputs the distribution parameters of the mixture of logistic distribution which was used to model images. For each step, the fast iterative gradient descent method (FISTA) [3] is used to minimize Eq (1). The proximal operation on $$$\log P(\hat{\Theta}, \boldsymbol{x})$$$ was approximated using gradient updates. The gradient of $$$\log P(\hat{\Theta}, \boldsymbol{x})$$$ is computed via backpropagation.

For validation, T2$$$^*$$$-weighted data (TE=16ms, TR=770ms, 3T) from a human brain was acquired with a GRE sequence. The image matrix was 256$$$\times$$$256 and the resolution was 1mm$$$\times$$$1mm. We acquired 160 radial k-space spokes using golden angle radial trajectory (2.5-fold acceleration). The gradient delay of radial trajectories was estimated with RING [4]. The number of channels was compressed to eight. At last, we reconstructed images from a different number of spokes (50, 70, 160) and made comparisons of different regularization terms that includes $$$\ell_2$$$, $$$\ell_1$$$-wavelet and learned log-likelihood.

[1] Uecker M et al., "Image reconstruction by regularized nonlinear inversion—joint estimation of coil sensitivities and image content." Magnetic Resonance in Medicine 60.3 (2008): 674-682.

[2] Luo G et al., "MRI reconstruction using deep Bayesian estimation." Magnetic Resonance in Medicine 84.4 (2020): 2246-2261.

[3] Beck A et al., "A fast iterative shrinkage-thresholding algorithm for linear inverse problems." SIAM journal on imaging sciences 2.1 (2009): 183-202.

[4] Rosenzweig S, Holme HCM, Uecker M. "Simple auto‐calibrated gradient delay estimation from few spokes using Radial Intersections (RING)." Magnetic resonance in medicine 81.3 (2019): 1898-1906.

[5] Lustig M et al. "Compressed sensing MRI." IEEE signal processing magazine 25.2 (2008): 72-82.

[6] Uecker M et al., BART Toolbox for Computational Magnetic Resonance Imaging, DOI:10.5281/zenodo.592960