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High resolution 3D MRI reconstruction using 3DMDTV regularization
Yue Hu1, Xin Lu1, Kuangshi Zhao2, and Mathews Jacob3

1Harbin Institute of Technology, Harbin, People's Republic of China, 2CSIC 703 Institute, 3University of Iowa, Iowa City, IA, United States

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

In Fig.1, we compare the 3DMDTV scheme with 3DTV and 3DHDTV methods. We retrospectively undersample a fully sampled contrast enhancement MR Angiography (CE-MRA) dataset (512 × 512 × 76) [6]. Each frame of the dataset is undersampled using 60 uniformly spaced radial lines in k-space, which are randomly rotated for different frames. The acceleration factor is around 9.3. We observe that 3DTV reconstruction has patchy artifacts which blur some details in the image. While the 3DMDTV method preserves more subtle features and provides more accurate reconstructions, compared with either 3DHDTV or 3DTV. In Fig.2, we compare the reconstructions of a 3D abdominal MRI dataset (190 × 160× 25) [7] using different schemes. The acceleration factor is around 6.5. The results show that 3DMDTV is able to recover the fine details of the image. In contrast, we observe blurring in the 3DTV and 3DHDTV reconstructions. We use the mean square error (MSE) metric to quantitatively evaluate the results.

Conclusion

We proposed a 3D MRI dataset reconstruction algorithm based on 3DMDTV regularization. Our experiments demonstrate that under high acceleration, this scheme is able to provide more natural-looking recovered images with fine details preserved.

Acknowledgements

Natural Science Foundation of China No. 61501146, No.61401123, Natural Science Foundation of Heilongjiang Province No. F2016018.

References

[1] Bhave S, Lingala SG, Newell JD, Nagle SK, and Jacob M. Blind compressed sensing enables 3-dimensional dynamic free breathing magnetic resonance imaging of lung volumes and diaphragm motion. Investigative radiology. 2016;51(6):387-399.

[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.

Figures

Compressed sensing recovery of 9.3-fold under-sampled CE-MRA dataset. The figure shows comparisons of reconstructions using direct IFFT, 3DTV, 3DHDTV, and the proposed 3DMDTV schemes. The first row shows the maximum intensity projection (MIP) images. The middle row shows one frame of the 3D dataset for each of the method. The first image in the last row indicates the undersampling pattern for one frame, and the rest images show the error images with respect to the full sampled data. The arrows highlight regions where 3DMDTV preserves fine details better compared with 3DTV and 3DHDTV.

Comparisons of the proposed 3DMDTV method against 3DTV and 3DHDTV schemes. The acceleration factor is around 6.5. The first row shows the maximum intensity projection (MIP) images. The middle row shows one frame of the dataset for each of the method. The first image in the last row shows the radial undersampling pattern for one frame, and the rest images show the error images with respect to the full sampled data. The arrows indicate that 3DMDTV has better performance in recovering the subtle features compared with 3DTV and 3DHDTV.

Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)
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