Low-Rank O-Space Reconstruction
Haifeng Wang1, Emre Kopanoglu1, R. Todd Constable1,2, and Gigi Galiana1

1Department of Radiology and Biomedical Imaging, Yale University, New Haven, CT, United States, 2Department of Neurosurgery, Yale University, New Haven, CT, United States

Synopsis

Low-Rank O-Space presents a scheme to incorporate O-Space imaging with Low-Rank matrix recovery. The Low-Rank reconstruction based on iterative nonlinear conjugate gradient algorithm is applied to substitute the previous Kaczmarz and Compressed Sensing (CS) reconstructions to recover highly undersampled O-Space data. The simulations and experiments illustrate the proposed scheme can remove artifacts and noise in O-Space imaging at high reduction factors, compared to results recovered by Kaczmarz and CS. Moreover, the proposed method does not need to modify the conventional O-Space pulse sequences, and reconstruction results are better than those in radial imaging recovered by Kaczmarz, CS, or Low-Rank methods.

Audience

Researchers interested in parallel imaging, nonlinear gradient encoding, or low-rank matrix recovery

Purpose

Nonlinear spatial encoding magnetic fields (SEMs), such as those used in O-Space imaging [1,2], have been shown to improve image reconstructions under high acceleration factors. Low-Rank reconstruction [3-7], based on the development of Low-Rank matrix completion in Compressed Sensing (CS) theory [8], has been shown to provide excellent image recovery from reduced data sets when applied to appropriate sampling in k-space. In this paper, we present a scheme to incorporate O-Space imaging with Low-Rank matrix recovery. The simulations and phantom experiments illustrate that the proposed scheme can greatly remove artifacts and noise in O-Space imaging at high reduction factors, compared to Kaczmarz [1,2] and CS reconstructions [9,10].

Theory

Neglecting relaxation effects, the signal $$$s_{q}$$$ from the $$$q$$$-th RF channel can be expressed as:$$ s_{q}=\int_{\omega}^{}m({\bf x}){\it C_{q}}({\bf x})e^{-i\phi({\bf x},t)}d{\bf x}, $$ where $$${\it m}({\bf x})$$$ is the magnetization at location $$$ {\bf x}=(x,y,z) $$$, $$${\it C_{q}}({\bf x})$$$ is the sensitivity of $$$q$$$-th coil, and the integral is over $$$\omega$$$, which is the region of interest; $$$\phi({\bf x},t)$$$ is the spatially dependent encoding phase. For the O-Space echo corresponding to the $$$l$$$-th center placement (CP) at $$$ (x_{l},y_{l}) $$$, the spatially dependent encoding phase $$$\phi({\bf x},t)$$$ of the signal equation becomes, which is $$ \phi_{l}({\bf x},t)=k_{x}(t)x+k_{y}(t)y-\frac{1}{2}k_{z2}(t)((x-x_{l})^{2}+(y-y_{l})^{2}), $$ where $$$ k_{x}(t)=\gamma\int_{0}^{t} G_{x}(\tau)d\tau $$$, $$$ k_{y}(t)=\gamma\int_{0}^{t} G_{y}(\tau)d\tau $$$, and $$$ k_{z2}(t)=\gamma\int_{0}^{t} G_{z2}(\tau)d\tau $$$. $$$ G_{x}(t) $$$, $$$ G_{y}(t) $$$ and $$$ G_{z2}(t) $$$ are gradients waveforms on X, Y and Z2 directions; $$$ \gamma $$$ is the gyromagnetic ratio. To further improve image quality, we replace our standard Kaczmarz reconstruction or CS algorithm with Low-Rank matrix recovery. Similar to the previous work on CS reconstruction for O-Space imaging [9,10], we apply the Low-Rank reconstruction with O-Space imaging. Assuming a desired image in matrix form $$${\bf S} \in C^{n \times m}$$$ in O-Space imaging, $$$\bf s $$$ is the vectorized version of the desired image by row concatenation, $$${\bf s} = {\it vet}({\bf S}) $$$, and this convex optimization may be written as: $$ {\bf s} = {\it argmin}(\lambda_{1}TV({\bf S})+\lambda_{2}\parallel{\bf S}\parallel_{*}+\parallel{\bf f-Es}\parallel_{2}^2)), $$ where $$$ \bf f $$$ is the measured signal; $$$\parallel{∙}\parallel_{2}$$$ and $$$\parallel{∙}\parallel_{*} $$$ are ℓ₂ and nuclear norm; $$$ {\it vet}(∙) $$$ is the vectorization function; $$$ TV(∙) $$$ is total variation function; $$$\lambda_{1}$$$ and $$$\lambda_{2}$$$ are the relaxation convergence parameters and is typically set for strongly under-relaxed reconstructions for gradual convergence; $$$\bf E $$$ is the encoding matrix, with its inverse calculated by the Kaczmarz iterative algebraic reconstruction [9]. The iterative nonlinear conjugate gradient (NCG) method is applied to optimize the above problem.

Methods

The simulations used a geometric phantom with the 64×64 resolution to study the normalized mean-square-error (NMSE) of reconstruction results at reduction factors of 4, 8, 16 and 32. Experiments were performed on a SIEMENS MAGNETOM 3.0T Trio scanner (Erlangen, Germany). The Z2 SEM gradient inserts [9] were built by Resonance Research, Inc. (Billerica, MA), which of 38cm diameter can run a maximum current of 625 Ampere giving Z2 strength of 0.94 Gauss/cm2. Another SIEMENS 8-channel head coil was used inside the gradient coil. Images were reconstructed with the 128×128 resolution.

Results

Figure 1 shows simulation results, including reference, radial and O-Space imaging at reduction factors of 8 and 16. Few improvements to reduce aliasing artifacts are observed if using the CS reconstruction in radial and O-Space imaging, but applying Low-Rank reconstruction clearly reduces undersampling artifacts, particularly for the O-Space encoded images. Figure 2 summarizes these results for a range of reduction factors and reconstruction methods applied to both radial and O-Space images. The proposed Low-Rank method improves NMSE over CS reconstruction, especially at high reduction factors, and the improvement is greater for O-Space encoded images. In Figure 3, experimental phantom results also show the proposed Low-Rank O-Space method reduces artifacts and recovers more detail (red arrows) than the either Kaczmarz or CS reconstruction of O-Space with pseudo-random disturbance. Moreover it is better than the best image attainable from radial encoding.

Discussion and Conclusion

In summary, the proposed method applies Low-Rank reconstruction to the problem of image reconstruction when imaging with nonlinear spatial encoding methods. The Low-Rank O-Space approaches can eliminate aliasing artifacts caused by undersampling in O-Space imaging. Moreover, the images are better than those achieved with radial data using either Kaczmarz, CS or Low-Rank reconstruction. It should also be noted that this method does not require modification of the O-Space acquisition strategy [9,10]. In the future, it may be beneficial to apply Low-Rank reconstruction to other nonlinear spatial encoding methods.

Acknowledgements

Support from the NIH, grant numbers R01-EB012289, R01-EB016978 and K01-CA168977 are gratefully acknowledged.

References

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[9]. Tam LK, Galiana G, Stockmann JP, Tagare H, Peters DC, Constable RT. Pseudo-random center placement O-space imaging for improved incoherence compressed sensing parallel MRI. Magn Reson Med. 2015 Jun;73(6):2212-24.

[10]. Tam LK, Galiana G, Wang H, Kopanoglu E, Dewdney A, Peters DC, Constable RT. Incoherence Parameter Analysis for Optimized Compressed Sensing with Nonlinear Encoding Gradients. Proc. of the 22nd Annual Meeting of ISMRM, Milan, Italy. 2014; Abstract: 1559.

Figures

Figure 1: Simulations of geometric phantom with the 64×64 resolution including reference, radial imaging, O-Space imaging and their different images with the referent images at reduction factors of 8 and 16.

Figure 2: NMSE (Normalized Mean Square Error) of simulations (64×64) at different reduction factors including corresponding reconstruction methods of radial and O-Space imaging.

Figure 3. Experimental results with the 128×128 resolution including reference, radial imaging, and O-Space imaging at a reduction factor of 8.



Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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