Introduction

While there are many approaches towards fulfillment of convex optimization with non-smooth regularizers, a flexible yet easy and comprehensive to realize method is always beneficial. One of the algorithms is proposed in [1], though it’s useful, yet few people ever get familiar to it. Here, we demonstrate that this algorithm can be easily adapted to many reconstruction problems in MRI. The original mechanisms and rigorous math can be retrieved from [1], and an accelerated version combining the idea of FISTA [2] is presented below. It’s an iterative algorithm which aims at minimizing a sum of functions by successive evaluations of their gradients or proximity operators using a proximal algorithm, in which only first-order information of the functions is exploited. Fenchel–Rockafellar conjugate is involved for proximal operator, which satisfies the useful Moreau identity. Since the proximity operator of an indicator function is simply the projection onto the set, proximal algorithms can be viewed as generalizations of algorithms to find an element in the intersection of convex sets by successive projections (POCS). The major merits include ability for high dimensional large-scale optimization, single loop in efficiency, and simple explicit expressions in coding. Essentially, the method will find $$$x=\arg min_{x}{f(x)+g(x)+\sum_{m=1}^Mh_{m}(L_{m}(x))}$$$, where the first term can be data consistence in MRI reconstruction, the second term will be an indicator function, and the multiple (M) regularizers can be included in the third term. Choose the parameters τ > 0, σ > 0, ρ > 0, t=1 and the initial estimates x⁽⁰⁾, u₁⁽⁰⁾, …, u_M⁽⁰⁾, then iterate, for i = 0, 1, . . ., the optimization can be performed in the steps of Fig. 1. Where L_m* is the adjoint operator of L_m, and $$$pro{x_{\sigma h_m^*}}\left( u \right) = u - \sigma \;pro{x_{h_m^{}/\sigma }}\left( {u/\sigma } \right)$$$

Methods

We have used the above algorithm to GRASP/XDGRASP [3, 4], BCS [5], suppression of MRI truncation artifact [6], and Single STEP QSM [7] with successes. We will only explain our approach towards GRASP/XDGRASP and MRI truncation artifact without details on their original concepts. In GRASP/XDGRASP, regularization of temporal total variations (TV_t:= $$${h_m}\left( {{L_m}x} \right)$$$ ) is applied on the image time series in the reconstruction, L_m and L_m* are temporal differentiation and its adjoint, and h_m and h_m* are l₁-norm and its adjoint. prox_τg is simply a projection function, and f is the data consistence in K space. For GRASP m equals 1, and for XDGRASP m equals 2. The two TV_t of XDGRASP work on heart and breath phase as described in [4]. In suppression of MRI truncation artifact, regularization of mixture of 1st and 2nd order spatial total variations (TV_s:= $$${h_m}\left( {{L_m}x} \right)$$$ ) is applied on the image in the restoration, L_m=1 and L_m=1* are 1st order differentiation and its adjoint, L_m=2 and L_m=2* are 2nd order differentiation and its adjoint, and h_m and h_m* are l1-norm and its adjoint. prox_τg is a projection function, and f is the data consistence in the confidential region of Fourier domain, the optimization aims to extrapolate the Fourier domain beyond the confidential region with the later unchanged [6]. For GRASP/XDGRASP, we used the DCE data and pre-processing code provided by the authors [3, 4] on their website. For suppression of MRI truncation artifact, we have tested the method on clinical images acquired with various sequences, as will be demonstrated below.

Results

Overall using the current method could earn 3-5 fold acceleration comparing to their original Matlab implementations [3-5, 7]. Without fine tuning of parameters, our reconstruction results of GRASP/XDGRASP are comparable to [3, 4] as shown in Fig. 2, with 3 and 5 fold speeding up for each. Fig. 3 is the results of truncation artifact suppression on various part of body in clinical images, and Fig. 4 are the comparison of one K-space truncated spin image.

Discussion and Conclusion

We have tested the algorithm on various MRI reconstruction problems for dynamic imaging, static imaging, as well as parameter mapping. Our results demonstrated the algorithm is efficient in fulfill those tasks, yet its coding is simple and explicit, the main loop is usually around 10 to 20 lines plus auxiliary functions of regularizations and their adjoints, as well as proximal operators, which are usually straightly available in the literature. While we haven’t make fine tuning of parameters (except in truncation artifact) in the implementation, we expect better performances would be of sure if we do so. Lastly, we believe the algorithm would be able to help other reconstruction problems straightly, or with minor efforts.

Acknowledgements

No acknowledgement found.

References

[1] Laurent Condat, J Optim Theory Appl (2013) 158:460–479, [2] Amir Beck and Marc Teboulle, SIAM J. IMAGING SCIENCES Vol. 2, No. 1, pp. 183–202, [3] Feng L, et al. Magn Reson Med. 2014; 72:707–717, [4] Feng L, et al. Magn Reson Med. 2016 February ; 75(2): 775–788, [5] S.G.Lingala, M.Jacob, , IEEE TMI, pp 1132-1145, vol.32(6), June 2013, [6] Block, K.T., et al., Int. J. Biomed. Imaging, 2008, 1–8, [7] I. Chatnuntawech, et al., NMR in Biomedicine, 2016.