ABOLFAZL MEHRANIAN^{1}, Claudia Prieto^{1}, Radhouene Neji^{1,2}, Colm J. McGinnity^{3}, Alexander J. Hammers^{3}, and Andrew J. Reader^{1}

We propose a simple and robust methodology for synergistic multi-contrast MR image reconstruction to improve image quality of undersampled MR data beyond what is achieved from conventional independent reconstruction methods. The advantages of the proposed methodology are threefold: i) it exploits quadratic priors that are mutually weighted using all available MR images, leading to preservation of unique features, ii) the weighting coefficients are independent of the relative signal intensity and contrast of different MR images and iii) the algorithm is based on a well-established parallel imaging iterative reconstruction, which makes the synergistic reconstruction of undersampled MR data clinically feasible

Synergistic SENSE reconstruction of $$$n$$$ undersampled MR contrasts,$$${\boldsymbol{v}}^{\left(k\right)}$$$, is given by the following minimization problem $${\left({\widehat{\boldsymbol{v}}}^{\left(1\right)},\dots ,{\widehat{\boldsymbol{v}}}^{\left(n\right)}\right)=\mathop{\mathrm{argmin}}_{{\boldsymbol{v}}^{\left(1\right)},\dots ,{\boldsymbol{v}}^{\left(n\right)}} \left\{\sum^n_{k=1}{{\left\|{\boldsymbol{E}}^{\left(k\right)}{\boldsymbol{v}}^{\left(k\right)}\mathrm{-}{\boldsymbol{s}}^{\left(k\right)}\right\|}^2_{{\boldsymbol{W}}^{\left(k\right)}}}\mathrm{+}\frac{{\beta }_k}{2}{\left\|{\boldsymbol{D}}^{(k)}{\boldsymbol{v}}^{\left(k\right)}\right\|}^2_{{\boldsymbol{\xi }}^{(k)}{\boldsymbol{\omega }}^{(k)}}\right\}\ }\ $$

where the first term includes $$$n$$$ data fidelity terms and the second term is a multi-modal weighted quadratic penalty function. $$${\boldsymbol{E}}^{\left(k\right)}$$$ includes coil sensitivity maps and k-space undersampling trajectories of $$$\textit{k}$$$th image contrast. $$${\boldsymbol{s}}^{\left(k\right)}$$$ is the acquired k-space data, $$${\boldsymbol{W}}^{\left(k\right)}$$$ is the noise-correlation matrix between coil channels, $$${\boldsymbol{D}}^{(k)}$$$ calculates intensity differences between voxels of the $$$\textit{k}$$$th image in a local neighbourhood, the elements of $$${\boldsymbol{\xi }}^{(k)}$$$ weight these voxel differences based on their distance.$$$\ {\beta }_k$$$ are regularization parameters. The elements of $$${\boldsymbol{\omega }}^{(k)}$$$ weight the difference between voxel $$$\textit{j}$$$ and $$$\textit{b}$$$ of the$$$ \textit{k}$$$th image in a local neighbourhood based on the product of Gaussian similarity coefficients derived from all $$$n$$$ MR images, defined as:

$${\omega }^{\ }_{jb}=\frac{\mathcal{G}\left({\tilde{v}}^{\left(1\right)}_j,{\tilde{v}}^{\left(1\right)}_b,{\sigma }_1\right)\dots \mathcal{G}\left({\tilde{v}}^{\left(n\right)}_j,{\tilde{v}}^{\left(n\right)}_b,{\sigma }_n\right)}{\sum^N_{j=1}{\mathcal{G}\left({\tilde{v}}^{\left(1\right)}_j,{\tilde{v}}^{\left(1\right)}_b,{\sigma }_1\right)}\dots \mathcal{G}\left({\tilde{v}}^{\left(n\right)}_j,{\tilde{v}}^{\left(n\right)}_b,{\sigma }_n\right)} \ \mathcal{G}\left(q,r,\sigma \right)=\frac{1}{\sqrt{2\pi }\sigma }{\mathrm{exp} \left(-\frac{{\left(q-r\right)}^2}{2{\sigma }^2}\right)\ }$$

As a result, Tikhonov regularization of the $$$\textit{k}$$$th image is guided by itself and the complementary information available in other image contrasts. For the calculation of the $$$\boldsymbol{\omega }$$$ coefficients, MR contrast images are mutually registered during their reconstruction. The minimization problem was solved using the conjugate gradient (CG) algorithm for each MR contrast at each iteration (implemented in MATLAB), using the $$$\boldsymbol{\omega }$$$ derived from all images from the previous iteration, i.e. $$$\widetilde{\boldsymbol{v}}$$$. The proposed algorithm was compared to iterative SENSE and TV regularized SENSE (TV-SENSE) reconstructions, optimized using the alternating direction method of multipliers and CG algorithms.

[1] Lustig M, Magn Reson Med. 2007, 58(6):1182-95.

[2] Ehrhardt MJ and Betcke MM SIAM J. Imaging Sciences, 2015, 9(3): 1084–1106.

[3] Bilgic B, et al. Magn. Reson. Med. 2011;66:1601–1615.