Synopsis

Three-dimensional (3D) MRI plays an important role in many clinical applications due to its ability to provide the full geometry of the targeted region of the body. However, speed limitation remains the key challenge to 3D MRI. Here, we present a compressed sensing reconstruction scheme using the 3DMDTV regularization. The experiments demonstrate that for high acceleration factors, the proposed method has better performance than other schemes by providing more accurately recovered images with more subtle details preserved.

Purpose

Three-dimensional (3D) MRI has become increasingly important in many clinical applications such as cardiac, gastro-intestinal, vocal tract, lung imaging and etc. 3D MRI is able to provide the full geometry of the region of interest, and thus it is desirable to obtain images with both high spatial resolution and high volume coverage. However, acquisition speed, which remains the main challenge for 3D MRI, limits its clinical application. Clinicians have to compromise between spatial resolution and scan time, which leads to sub-optimal performance. In order to accelerate 3D MRI, researchers have proposed schemes exploiting the sparsity property of the MRI dataset by using compressed sensing algorithms [1],[2]. We have recently introduced a generalized higher degree total variation (HDTV) scheme to reconstruct images from the noisy and undersampled measurements [3]. The generalized HDTV penalties are defined as the $$$L_1$$$-$$$L_p$$$ norm of the $$$n$$$th degree directional image derivatives. By taking into consideration of all rotations of the directional derivative operators, HDTV minimizes the staircasing and patchy artifacts inherent with TV, while better preserves the edges and detailed features in an image compared with TV. However, since HDTV focuses on derivatives of a single degree, it still has some limitations in accurately recovering an image. Inspired by the total generalized variation (TGV)[4], which concerns the 1st and 2nd order derivatives of an image, we have proposed a multiple degree TV (MDTV) regularization [5], which provides improved image recovery performance. In this work, we demonstrate an algorithm for reconstructing high resolution 3D MRI dataset by using the 3DMDTV penalty.

Methods

The 3D MRI measurements are acquired using a Fourier sampling operator on a specified grid. The acquisition of the noisy and undersampled MRI measurements can be modeled as $$$\mathbf b={\cal A}(\mathbf \Gamma)+\mathbf n$$$. Here, $$${\mathbf \Gamma}$$$ is the MRI dataset to be recovered, $$${\cal A}$$$ is the Fourier domain undersampling operator, $$$\mathbf n$$$ is the additive noise. The problem of recovering the MRI dataset $$$\mathbf \Gamma$$$ can thus be formulated as a constrained optimization function:

$$\mathbf\Gamma^* =\min\limits_\mathbf\Gamma \|{\cal A}(\mathbf\Gamma)-\mathbf b \|^2+\lambda_1\underbrace{\int_{{\cal S}}|\mathbf s_1^H(\mathbf u){\cal D}_1(\mathbf \Gamma)|d\mathbf u}_{\mbox{1DTV}}+\lambda_2\underbrace{\int_{{\cal S}}|\mathbf s_2^H(\mathbf u){\cal D}_2(\mathbf\Gamma)|d\mathbf u}_{\mbox{2DTV}}$$(1)

Here $$$\mathbf s_n^H(\mathbf u){\cal D}_n(\mathbf \Gamma)$$$ is the $$$n$$$th degree directional derivative of $$$\mathbf \Gamma$$$, $$${\cal D}_n(\mathbf \Gamma)$$$ is the differential operator which provides the $$$n$$$th degree partial derivatives of $$$\mathbf \Gamma$$$, and $$$\mathbf s_n(\mathbf u)$$$ is the vector of polynomials in the components of the unit directions $$$\mathbf u \in {\cal S}$$$, where $$${\cal S}=SO(3)$$$ is the 3D rotation group, defined with the unit sphere. The optimization function includes the linear combination of the data consistency term, the 1st degree TV (1DTV), and the 2nd degree TV (2DTV) terms. We use a fast majorize minimize scheme to decouple (1) into subproblems which can be easier solved. Specifically, we approximate each of the $$$L_1$$$ norm with the Huber function, parameterized by a $$$\beta_i(i=1,2)$$$ parameter. This results in two subproblems, which can be solved exactly using one shrinkage step, and one subproblem which can be solved in one step using discrete Fourier transforms. In addition, we rely on continuation strategy on $$$\beta_i$$$. Specifically, we start with an initial value of $$$\beta_i$$$ and increase the value at each iteration by a constant factor. The algorithm stops when the relative error between two successive iterations is within a specified threshold.

Results and discussion

Conclusion

Acknowledgements

References

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[2] Kim D, Trzasko J, Smelyanskiy M, Haider C, Dubey P, and Manduca A. High-performance 3d compressive sensing MRI reconstruction using many-core architectures. Journal of Biomedical Imaging. 2011;2:1-11.

[3] Hu Y, Ongie G, Ramani S, and Jacob M. Generalized higher degree total variation (HDTV) regularization. IEEE Transactions on Image Processing. 2014;23(6):2423–2435.

[4] Bredies K, Kunisch K, and Pock T. Total generalized variation. SIAM Journal on Imaging Sciences. 2010;3(3):492–526.

[5] Hu Y, Lu X, and Jacob M. Multiple degree total variation (MDTV) regularization for image restoration. 2016 IEEE International Conference on Image Processing (ICIP).

[6] Samples of MR Images [Online]. Available:http://physionet.org/physiobank/database/images/.

[7] Example 4DMRI [Online]. Available:http://www.vision.ethz.ch/4dmri.