Sebastian Rosenzweig^{1,2} and Martin Uecker^{1,2}

^{1}Diagnostic and Interventional Radiology, University Medical Center Göttingen, Göttingen, Germany, ^{2}Partner Site Göttingen, German Centre for Cardiovascular Research (DZHK), Göttingen, Germany

SSA-FARY SE is a simple yet powerful novel method to estimate a suitable temporal basis for subspace reconstructions of time-series data from a potentially very small auto-calibration region. Here, we first describe the general strategy of subspace constraint time-series reconstruction. Then, we show how SSA-FARY SE can be used to estimate a suiteable temporal basis. Finally, we demonstrate its functionality by estimating temporal basis functions from the DC-components of radial spokes in single-slice and Simultaneous Multi-Slice free-breathing cardiac MRI acquisitions.

The minimization problem for time-resolved MRI with temporal subspace constraint is given by [1,2]:

$$$x=\underset{x}{\text{argmin}}||y-P\mathcal{F}CTx ||_2^2+\lambda||x||_2^2$$$,

with $$$T$$$ the basis for the temporal subspace, $$$C$$$ the coil sensitivities, $$$\mathcal{F}$$$ the Fourier operator and $$$P$$$ a projection onto the k-space trajectory. $$$x$$$ are the image coefficients to be estimated and $$$y$$$ is the acquired data. The full time-resolved movie can then be composed by $$$Tx$$$. Given a suitable $$$T$$$, the minimization can be performed efficiently by precomputing the sampling kernel as described in [5,6].

This work focusses on the estimation of the temporal basis $$$T$$$ using SSA-FARY SE. Compared to SSA-FARY for self-gating [4], SSA-FARY SE omits the zero-padding step, thus only two basic mathematical operations need to be performed to determine the temporal basis (FIG_1).

A suitable AC-region $$$\boldsymbol{X}$$$ is Hankelized $$$\mathcal{H}$$$ by sliding a window of size $$$W$$$ through the temporal domain of each channel of the AC-region.

$$$\boldsymbol{A}=\mathcal{H}\boldsymbol{X}$$$.

The resulting Block-Hankel matrix $$$A$$$ is decomposed using an SVD

$$$\boldsymbol{A}=USV^H$$$.

The temporal basis $$$T$$$ is obtained by selecting the $$$N_\text{b}$$$ most significant singular vectors of $$$\boldsymbol{A}$$$ given by the columns of $$$U$$$ (FIG_1).

The AC-region should contain rich information about the motion during data-acquisition to allow for the estimation of a temporal basis able to accurately describe the temporal process of each pixel. Therefore, a large AC-region, e.g. realized by low-res reconstructions or navigator lines [1], is desirable. This, however, comes at the crucial cost of increased measurement or reconstruction time.

In [4] we have demonstrated that SSA-FARY is capable of detecting motion signals from very small AC-regions through the use of Hankelization (time-delayed embedding). Here, we show that the SSA-FARY approach also improves the estimation of a suitable temporal subspace (SSA-FARY SE).

FIG_4 depicts the six most significant (complex) basis functions estimated with PCA (left) and SSA-FARY SE (right), together with the corresponding complex image coefficients. Note the cleaner basis functions for SSA-FARY SE and the corresponding reduced amount of streakings in e.g. the coefficient for $$$T 6$$$.

FIG_5 shows a 1.8s snippet of the full SMS movie for three time-consistent slices.

[1] Z. Liang "Spatiotemporal imaging with partially separable functions." IEEE Biomed. Imag., vol. 4, pp. 988-991, 2007.

[2] S. Poddar et al. "Free-breathing cardiac MRI using bandlimited manifold modelling." arXiv:1802.08909, (2018).

[3] A. Christodoulou et al. "High-resolution cardiac MRI using partially separable functions and weighted spatial smoothness regularization." In Proc. IEEE Eng. Med. Biol. Soc., pp. 871-874, 2010.

[4] S. Rosenzweig, et al. "Cardiac and Respiratory Self-Gating in Radial MRI Using an Adapted Singular Spectrum Analysis (SSA-FARY)." IEEE Trans. Med. Imag., vol. 39, no. 10, pp. 3029-3041, 2020.

[5] M. Mani et al. "Fast iterative algorithm for the reconstruction of multishot non-cartesian diffusion data." Magn. Reson. Med. 2015, vol. 74, pp. 1086–1094, 2015.

[6] J. Tamir et al. "T2 shuffling: sharp, multicontrast, volumetric fast spin‐echo imaging." Magn. Reson. Med., vol 77, pp. 180-195, 2017.

[7] S. Wundrak et al. "Golden ratio sparse MRI using tiny golden angles." Magn. Reson. Med., vol 75, pp. 2372-2378, 2016.

[8] M. Uecker et al. "Berkeley advanced reconstruction toolbox." In Proc. Intl. Soc. Mag. Reson. Med., vol. 23, p. 2486, 2015.

[9] S. Rosenzweig et al. “Simple auto-calibrated gradient delay estimation from few spokes using radial intersections (RING).” Magn. Reson. Med., vol. 81, no. 3, pp. 1898–1906, 2019.

[10] M. Uecker et al. “Nonlinear inverse reconstruction for real-time MRI of the human heart using undersampled radial FLASH.” Magn. Reson. Med., vol. 63, no. 6, p. 1456–1462, 2010.

[11] S. Rosenzweig et al. “Simultaneous Multi-Slice Real-Time Imaging with Radial Multi-Band FLASH and Nonlinear Inverse Reconstruction.” In Proc. Intl. Soc. Mag. Reson. Med., vol. 24, p. 0518, 2017.

FIG_1: Schematic of SSA-FARY SE. The complex auto-calibration (AC-)region is Hankelized $$$\mathcal{H}$$$ using a window of size $$$W$$$, which yields the Block-Hankel matrix $$$A$$$. This matrix is decomposed using a Singular Value Decomposition (SVD). The columns of the resulting matrix $$$U$$$, often called Empirical Orthogonal Functions, are ordered according to their significance, specified by the singular values that make up the diagonal of $$$S$$$. The most significant $$$N_\text{b}$$$ singular vectors are used as a basis for the temporal subspace.

FIG_2: Comparison of the single-slice temporal subspace reconstruction with a rank $$$N_\text{b}=30$$$ subspace estimated using PCA (left) and SSA-FARY SE (right). The movie shows a 1.8s snippet of the full time-series. The snippet corresponds to the time-frames highlighted by the orange box in FIG_3. (The movie of the full time-series could not be included here due to file-size limits.)

FIG_3: Temporal evolution of a horizontal line extracted from the single-slice time-series reconstructed with a rank $$$N_\text{b}=30$$$ temporal subspace, which was estimated using PCA (left) and SSA-FARY SE (right). The white arrows highlight a region where SSA-FARY SE can resolve details better than PCA. The orange box marks the time-frames shown in the movie-snippet FIG_2.

FIG_4: Real- and imaginary part of the six most significant temporal basis functions $$$T\,1\;-\;T\,6$$$ and the corresponding complex image coefficients for PCA (left) and SSA-FARY SE (right). The temporal basis functions are provided by the columns of the matrix $$$U$$$, see *Step 2* of the theory section.

FIG_5: Temporal subspace reconstruction of the 3-slice SMS-FLASH acquisition using $$$N_\text{b}=30$$$ basis functions estimated with SSA-FARY SE. The movie shows a 1.8s snippet of the full time-series. (The movie of the full time-series could not be included here due to file-size limits.)