To show that randomly perturbing the golden angles in a k-t radial acquisition scheme significantly reduces peak side lobes in the x-f point spread function, with little penalty in spatial encoding efficiency.

Radial encoding schemes based on the golden ratio (or the golden angle) are popular for their near-optimal spatial encoding efficiencies¹ and favorable spatial point-spread function (PSF) properties² when combining an arbitrary number of projections to form an image. The latter in particular makes radial acquisitions desirable for image reconstruction using compressed sensing due reduced coherent aliasing artefacts (i.e. low side-lobe energy in the PSF³), and accelerated spatio-temporal acquisitions employing k-t reconstructions also use the golden angle radial acquisition scheme⁴. Intuitively, the quasi-random nature of this sampling might be expected to have beneficial incoherence properties, but to our knowledge the effect of the fixed angular increment on the overall x-f PSF⁵ has not been considered.

We show that the x-f PSFs of a golden-angle acquisition possess considerable coherence along the frequency domain, which we can dampen by perturbing the golden angles with a normally-distributed random variable. This results in a more noise-like and incoherent PSF, and can be achieved with only small penalties to the spatial encoding efficiency when low numbers of projections are combined. The impact of reduction in x-f PSF coherence is demonstrated in a rank-constrained k-t reconstruction of FMRI data⁶, showing clear reduction of high-frequency artefacts.

The standard golden-angle radial scheme is defined as $$$\theta_n=180n/\phi$$$, where $$$\phi=(1=\sqrt{5})/2$$$ (hereby denoted “$$$\theta_n$$$”). While this distribution of projections exhibits quasi-random spatial behavior, it has high temporal regularity due to its fixed increment. Here, we modify the golden-angle acquisition to include a random perturbation, defined as $$$\tilde{\theta}_n=180n/\phi+\tau$$$, where $$$\tau\sim N(0,\sigma)$$$ denotes a zero-mean, $$$\sigma$$$º standard deviation normally distributed random variable (hereby denoted “$$$\tilde{\theta}_n(\sigma)$$$”). This results in a set of projections that is on average centered on golden angle increments (Fig. 1a), inheriting its favorable spatial properties (Fig. 1b), but with a disrupted temporal regularity (increased incoherence).

We compute x-f PSFs from k-t radial sampling masks for reconstructions with 8–19 projections/image, using the NUFFT⁷, followed by a temporal Fourier transform. A synthetic artefact voxel was superimposed on a resting FMRI dataset (64x64 matrix, 500 time points) and retrospectively sampled using the $$$\theta_n$$$ and $$$\tilde{\theta}_n(2)$$$ schemes. Two subjects were scanned at 3T under a 5-minute resting FMRI condition, using the $$$\theta_n$$$ and $$$\tilde{\theta}_n(2)$$$ schemes in a 3D hybrid radial-Cartesian trajectory⁸, which is similar to a stack-of-stars, except the Cartesian direction is traversed in a single shot using an EPI readout. Slice-by-slice radial reconstruction was performed onto a 100x100 spatial matrix with 300 time-points (20 projections/image, R=5 relative to Cartesian equivalent sampling, and TR=1 s) using an iterative hard thresholding and matrix shrinkage algorithm⁶ with a rank constraint of 32.

Figure 2 shows the peak side-lobe amplitude of the x-f PSF, between the $$$\theta_n$$$ and $$$\tilde{\theta}_n(2)$$$ schemes, for a 10-projection reconstruction, indicating the reduction in PSF coherence across both space and frequency. Figure 3 shows a central slice through the x-f PSF (at y=0) for both schemes for reconstructions combining across different number of projections, highlighting the considerable variability and spectral coherence in the $$$\theta_n$$$ scheme, while the $$$\tilde{\theta}_n(2)$$$ scheme shows less coherent, noise-like properties, with much less dependence on the number of projections included. In Figure 4, the retrospective sampling experiment shows the impact of the artefact voxel on the resultant image reconstructions, with much higher normalized RMSE in the images derived from $$$\theta_n$$$. Figure 5 shows spectra averaged over the entire brain in representative slices for each subject. In this data, clear reductions of certain spectral aliasing artefacts are observed with the perturbed golden angles (black arrows), although some residual aliasing is still evident.We have presented a simple way of modifying golden angle acquisition schemes to reduce temporal coherence, while still achieving high spatial efficiencies at small temporal windows. This approach demonstrates a reduction, although not complete elimination of coherent artefacts in low-rank k-t image reconstruction, and highlights the importance of minimizing temporal coherence in sparsity or rank-based non-linear reconstruction strategies. Further optimization of this approach, by designing optimized pertubations or exhaustive search can lead to further improvements in the incoherence of spatio-temporal radial sampling under near-arbitrary projection combinations.