Rational Approximation of Golden Angles: Accelerated Reconstructions for Radial MRI

TL;DR

RAGA addresses the need for flexible, dynamic MRI sampling by approximating golden-angle trajectories with rational angles, enabling equidistant angular reordering and precomputable reconstruction components. It defines a base angle $\phi_i^N = \pi / G_i^N$ and an increment $G_{i-1}^1$, such that the full frame length is $S = G_i^N$ and the temporal order is captured by ind$_t = (t\cdot G_{i-1}^1) \bmod S$, preserving bijectivity. Numerical analyses show RAGA closely matches the PSF and SPR of golden-angle schemes while offering machine-precision reproducibility and reduced data management, thanks to repeating frames and the possibility of Toeplitz/GROG preprocessing. Phantom and in vivo experiments demonstrate comparable image quality to golden-angle sampling across retrospective temporal resolutions, with significantly simplified data handling and reconstruction workflows for dynamic and quantitative MRI.

Abstract

Purpose: To develop a generic radial sampling scheme that combines the advantages of golden ratio sampling with simplicity of equidistant angular patterns. The irrational angle between consecutive spokes in golden ratio based sampling schemes enables a flexible retrospective choice of temporal resolution, while preserving good coverage of k-space for each individual bin. Nevertheless, irrational increments prohibit precomputation of the point-spread function (PSF), can lead to numerical problems, and require more complex processing steps. To avoid these problems, a new sampling scheme based on a rational approximation of golden angles (RAGA) is developed. Methods: The theoretical properties of RAGA sampling are mathematically derived. Sidelobe-to-peak ratios (SPR) are numerically computed and compared to the corresponding golden ratio sampling schemes. The sampling scheme is implemented in the BART toolbox and in a radial gradient-echo sequence. Feasibility is shown for quantitative imaging in a phantom and a cardiac scan of a healthy volunteer. Results: RAGA sampling can accurately approximate golden ratio sampling and has almost identical PSF and SPR. In contrast to golden ratio sampling, each frame can be reconstructed with the same equidistant trajectory using different sampling masks, and the angle of each acquired spoke can be encoded as a small index, which simplifies processing of the acquired data. Conclusion: RAGA sampling provides the advantages of golden ratio sampling while simplifying data processing, rendering it a valuable tool for dynamic and quantitative MRI.

Figures (7)

• Figure 1: $~$Comparison of a golden ratio (left) and a RAGA (right) sampling scheme. The golden ratio sampling scheme uses the golden angle $\Phi$. The first four spokes of the trajectory are highlighted. Because the angle is irrational the spoke angles never repeat and new projections are acquired each time. The RAGA sampling scheme approximates $\Phi$ with an approximation order of $i=5$. It acquires the same set of spokes as an equidistant angular pattern with 13 spokes, but reordered so that the angle between temporally consecutive spokes approximates an angle of $\Phi$. The pattern repeats after all 13 spokes of the equidistant pattern. All spokes are acquired exactly once. Note that a low number of 13 spokes was used for illustration only, practical RAGA trajectories would use a higher number of spokes corresponding to the Nyquist limit.
• Figure 2: A: An example for an equidistant angular sampling scheme with ${\color{blue}G_5^1} =$ 5 spokes defined over a half-circle (left) and the corresponding RAGA sampling that approximates $\psi^1$ with $\psi_5^1$ (right). The sampling order in RAGA corresponds to a golden ratio sampling scheme and its temporal evolution is marked with the time index $t$. The corresponding indices in the equidistant angular pattern $\text{ind}_t$ are calculated with the RAGA increment $G_{i-1}^1=$${\color[rgb]{1,0.5,0}G_4^1}=$$3$ and Equation \ref{['Eq::raga_increment']}. Extending RAGA to the full circle leads to flipped readout directions relative to the equidistant sampling defined over a half-circle. This encoding ambiguity can be avoided by either using an extended index space or by directly covering a full circle using doubled golden ratio angles. B: RAGA sampling using an extended space of indices. Golden ration sampling with $\psi^1$ is approximated with RAGA with $\psi_5^1$ by sampling a full frame with an even number $S = 2 G_i^N$ of spokes and using the increment $G_{i-1}^1$. C: RAGA sampling approximating the doubled golden ratio angle $2 \psi^1$ using an odd number of spokes $S$.
• Figure 3: Data storage and processing for RAGA and golden ratio based sampling. Note that the small approximation order of $i=5$ is chosen only for illustration purposes. A: Golden ratio based data is stored in a time series of individual spokes and knowledge about $\psi^1$ is required for a reconstruction. RAGA sampling can be stored in an equidistant angular order which yields a natural decomposition of the time series into full frames. This allows reconstruction of each full frame even without any additional information. With only knowledge about the index increment $G^1_4$ corresponding to the underlying base angle (here: $\psi^1_5$), the temporal order of all spokes in the RAGA dataset can be recovered and frames with arbitrary temporal footprint can be constructed by binning as in golden ratio sampling. B: In RAGA sampling, segments consisting of full frames can be extracted and processed without the need to keep track of their position in the original data set. Furthermore, all full frames have identical spokes which allows sharing of precomputed data for image reconstruction. When extracting data from a non-repeating trajectory, the position $t_n$ needs be known for each fragment to be able to recompute the original trajectory and no precomputed data can be shared.
• Figure 4: A: Sidelobe-to-peak ratio (SPR) for sampling schemes based on the golden ratio angles $\psi^1$, $\psi^7$, and $2\psi^7$, their rational approximations with the RAGA sampling $\psi_{13}^1$, $\psi_{10}^7$, and $2\psi_{10}^7$, and an equidistant scheme using a base angle of $\phi$. The SPR is shown for various window sizes. Golden ratio angles are plotted with solid lines for better differentiation from their rational approximations that are shown with colored dots. Elements of the generalized Fibonacci series are marked with dotted vertical lines. B: Various sampling schemes for the rational approximations $\psi_{13}^1$, $\psi_{10}^7$, and $2\psi_{10}^7$ as well as equidistant sampling with $\phi$. The time index is color-coded and the calculated SPR of each sampling scheme is shown. C: Calculated SPR for sampling schemes based on the single golden ratio angles $\psi^1$ and $\psi^7$ and similar doubled golden ratio angles $2\psi^2$ and $2\psi^{14}$. The SPR is shown for various window sizes. Single golden ratio angles are plotted with dotted lines for better differentiation from their doubled alternatives that are shown with solid lines.
• Figure 5: A: Reconstruction times and memory usage of an inverse nuFFT averaged over 10 runs with and without Toeplitz embedding applied to data of the Shepp Logan phantom simulated for trajectories with golden ratio and RAGA sampling scheme. B: Calibration and gridding times for data preprocessing using GROG of simulated multi-coil data averaged over 10 runs for golden ratio and RAGA trajectories.

Theorems & Definitions (3)

• Theorem 1
• Lemma 1
• Lemma 2