Variable Density Poisson Disk (VDPD) sampling is currently the de facto standard for undersampling in Parallel Imaging Compressed Sensing (PICS). VDPD undersampling patterns contain both structural and random undersampling¹, making them ideally suited for PICS, which exploits both redundancy in the coil sensitivities and sparsity in the image.

The variable sampling density is often modelled with a sampling probability that decays with a certain power of the distance to the center of k-space, where a higher power results in a denser sampling in the center of k-space: $$$P(x) = (1-x)^d$$$ with $$$-1<x<1$$$, where $$$x$$$ is the distance to the center of k-space. In this work we refer to this power $$$d$$$ as the sampling density. Currently there is no proven way to choose this sampling density optimally. In addition, many authors seem to choose the sampling density ad hoc as they do not describe how they chose their sampling density or do not mention the sampling density altogether.

In this study we investigated the effect of sampling density and randomness in VDPD undersampling patterns on CS reconstruction errors and provide recommendations on how to prevent suboptimal PICS reconstructions due to suboptimal choices of the sampling density.

We obtained 3 different datasets of fully sampled MRI scans. Firstly, we used the fully sampled scans provided on mridata.org². This dataset consisted of PD-weighted 3D SE knee scans of 20 subjects (matrix size 320x320x256; resolution 0.5x0.5x0.6 mm; TE/TR 25/1550 ms; 8 coils; spherical shutter) at 3 Tesla (GE Healthcare, Milwaukee, WI). Secondly, we acquired T1-weighted 3D SSFP prostate scans of 4 patients who underwent low-dose-rate brachytherapy (matrix 292x376x96; resolution 1.2 mm isotropic; TE/TR 2.7/4.6 ms; 24 coils; spherical shutter) at 3 Tesla (Ingenia, Philips, Best, The Netherlands). Finally, we acquired T1-weighted 3D SSFP brain scans of 2 healthy volunteers (matrix 240x192x266; resolution 1 mm isotropic; TE/TR 3.7/20 ms; 16 coils) at 1.5 Tesla (Ingenia, Philips, Best, The Netherlands).

For each dataset we generated 110 random VDPD 8-fold undersampling patterns with the sampling density varying from 0 to 2.5 with steps of 0.25. We retrospectively undersampled each scan with each of the 110 patterns and performed a PICS reconstruction using the ESPIRiT method³, as implemented in the Berkeley Advanced Reconstruction Toolbox⁴. Additionally, we undersampled and reconstructed each scan using a 2x2 structural undersampling pattern with the resolution restricted to achieve 8-fold acceleration. This can be considered a baseline for Parallel Imaging acceleration. Reconstruction errors were measured using the Normalized Root Mean Squared Error (NRMSE) metric with a fully sampled reconstruction as reference.

Figure 1 shows the NRMSE reconstruction errors for each of the 110 VDPD patterns and each of the datasets. The density at which the lowest errors occurred differed per dataset: 2 for the knee dataset, 2.5 for the prostate dataset, and 0.5 for the brain dataset. The influence of randomness within each sampling density was very small.

Figures 2-4 show CS reconstructions (B) for each dataset, using the undersampling pattern with the lowest mean NRMSE for the dataset.

We performed an exhaustive analysis of the reconstruction errors resulting from using different VDPD undersampling patterns in PICS reconstruction. Across 3 datasets, we found major differences in NRMSE errors when using different sampling densities. The worst case errors were 20-40% higher than the errors at the optimal sampling density. This clearly shows that ad hoc choices of VDPD sampling densities can result in significantly worse PICS reconstructions.

Because the optimal sampling densities were dataset dependent, reference scans from the dataset are needed to choose the optimal sampling density for future accelerated scans of the same anatomy and with the same pulse sequence. We recommend acquiring at least one fully sampled reference scan. This scan can be retrospectively undersampled and reconstructed to measure the reconstruction errors resulting from different VDPD sampling densities. Since the influence of randomness on the reconstruction errors appeared to be small, only one random pattern per sampling density needs to be evaluated. The undersampling pattern with the lowest reconstruction error on the reference scan should then be used for future scans, which should avoid most suboptimalities due to poorly chosen VDPD sampling densities.