Researchers interested in parallel imaging, nonlinear gradient encoding, or low-rank matrix recoveryNonlinear 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].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.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.

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.

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.