Emre Kopanoglu^{1}, Alper Gungor^{1}, Toygan Kilic^{2,3}, Emine Ulku Saritas^{2,3}, Tolga Cukur^{2,3}, and H. Emre Guven^{1}

In many clinical settings, multi-contrast images of a patient are acquired to maximize complementary information. With the underlying anatomy being the same, the mutual information in multi-contrast data can be exploited to improve image reconstruction, especially in accelerated acquisition schemes such as Compressive Sensing (CS). This study proposes a CS-reconstruction algorithm that uses four regularization functions; joint L1-sparsity and TV-regularization terms to exploit the mutual information, and individual L1-sparsity and TV-regularization terms to recover unique features in each image. The proposed method is shown to be robust against leakage-of-features across contrasts, and is demonstrated using simulations and in-vivo experiments.

Compressive Sensing (CS) uses randomly undersampled
acquisitions to yield incoherent aliasing artifacts, and a nonlinear
reconstruction to suppress these artifacts^{1-7}. In many MRI applications,
multi-contrast images are acquired together to maximize diagnostic information,
since different contrast mechanisms provide complementary information. With the
underlying anatomy being the same, previous studies have demonstrated
advantages of joint reconstruction of these multi-contrast images^{5-7}. While earlier methods enforced
sparsity individually on each acquisition, recent reconstruction have leveraged
joint-sparsity terms across acquisitions to improve image quality^{5,7}.

In this study, we present an adaptation of the
Alternating Direction Method-of-Multipliers^{8} for joint reconstruction of
multi-contrast images. As opposed to previous CS-algorithms, the
proposed method uses four regularization functions; L1-sparsity and
TV-regularization for individual contrasts, and Color-TV^{9} and Group-L1-sparsity^{5} imposed on all contrasts
simultaneously. Joint sparsity/TV terms recover information shared across
acquisitions while individual terms recover features unique to each contrast.

The proposed method uses Color-TV^{9} and Group-L1-sparsity^{5} (Eqs. [3]-[4]) to enhance
correlated features and individual TV and L1 functions (Eqs. [5]-[6]) to
facilitate recovery of individual features via the following optimization
framework.

Solve:

$$\min_x\alpha CTV(x)+\beta GSp(x)+\sum_i\gamma TV\left(x^{(i)}\right)+\sum_i\theta Sp\left(x^{(i)}\right)$$

subject to

$$\Vert Ax_i-y\Vert_2<\epsilon_i$$

where

$$CTV(x)=\sum_n\sqrt{\sum_{i=1}^k\left(\nabla_1\vert x^{(i)}[n]\vert\right)^2+\left(\nabla_2\vert x^{(i)}[n] \vert\right)^2}$$

$$GSp(x)=\sum_n\sqrt{\sum_{i=1}^k\left( x^{(i)}[n]\right)^2}$$

$$TV(x^{(i)})=\sum_n\sqrt{\left(\nabla_1\vert x^{(i)}[n]\vert\right)^2+\left(\nabla_2\vert x^{(i)}[n]\vert\right)^2}$$

$$Sp(x^{(i)})=\sum_n\vert x^{(i)}[n]\vert$$

where,
$$$ \alpha,\beta,\gamma,\theta $$$ denote regularization weight parameters. $$$\epsilon_i^2$$$ (noise energy for contrast *i*)
can be calculated in practice from
data acquired via a rapid excitation-less acquisition.

Two different versions of the above framework
were implemented for individual (ASEL-CS-indiv) and joint (ASEL-CS-j) reconstruction
of multi-contrast images. Both were compared to the following
state-of-the-art CS reconstructions: SparseMRI^{1}, recPF^{3}, TVCMRI^{2}, GSMRI^{5}, FCSA^{4}, FCSA-MT^{7}. The undersampling masks were
generated using probability-density-functions that decay with the third-order
of the linear-distance/radius (for one-/two-dimensional undersampling) in
k-space. One-eighth of the central k-space was fully-sampled. The same set of
masks were used in all methods, but different masks were generated for
each contrast. Image-quality metrics used were; structural-similarity-index
(SSIM), peak signal-to-noise-ratio (pSNR), normalized-root-mean-squared-error
(nRMSE), mean-magnitude-error (mmE).

**Simulation 1: **Same settings as Ref^{7}; real and noiseless images (SRI-24
atlas^{10}), 100 iterations, $$$\alpha$$$=0.01,$$$\beta$$$=0.035 for all methods, $$$\gamma=\theta$$$=0 (no individual
terms for ASEL-CS-j^{11}).

**Simulation 2:** Complex and noisy images (Ref^{12}: segmented brain, 11 tissues), 250
iterations. Regularization parameters were optimized for each method (interval-search
algorithm seeking maximum SSIM). For ASEL-CS-j, $$$\alpha,\beta$$$ were optimized
as above for $$$\gamma=\theta$$$=0, and $$$\gamma,\theta$$$ were manually tuned afterwards.

**Simulation 3: **A potential pitfall in joint
reconstruction is leakage of distinct features among images. ASEL-CS-j was
tested against this pitfall by introducing distinct artificial tissues to the
images.

**In-vivo Experiment: **Experiments were conducted on a 3T scanner (Siemens Healthcare,
Erlangen). Full k-space data (32-channel receiver) were acquired, subsampled retrospectively,
reconstructed separately for each channel for each method using optimized
parameters, and combined using the algorithm given in Ref^{13}. The images were normalized to the range
[0, 255] for each channel and each contrast to avoid parameter mismatch. The channel-combination
algorithm^{13} automatically handled the
amplification of signals from farther channels due to this normalization.

The proposed method uses four distinct regularization functions: individual L1-sparsity and TV-regularization, and joint L1-sparsity and TV-regularization. In contrast, GSMRI uses only group L1-sparsity and FCSA-MT uses only the joint terms. In ASEL-CS-j, joint penalty terms exploit the correlated features across images, while individual terms enable recovery of distinct features in each image.

The proposed method outperformed state-of-the-art CS reconstructions in all datasets reported here. Furthermore, our results indicate that ASEL-CS-j is robust against leakage-of-features across images. We observed that omission of individual penalty terms led to leakage-of-features across contrasts, minor albeit visible. Yet, the simultaneous use of both individual and joint terms suppressed these artifacts effectively.

To ensure that similar penalty weights and
reconstruction parameters work well, *in-vivo *data were
normalized to the same intensity range in the image domain. Although this led
to different noise-floors across contrasts, it was observed that this normalization is critical to improve convergence behavior across all methods implemented here.

The authors would like to thank the authors of
the methods SparseMRI^{1}, recPF^{3}, TVCMRI^{2}, GSMRI^{5}, FCSA^{4} and FCSA-MT^{7} for making the source codes of
their algorithms available online.

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