The Set Increment with Limited Views Encoding Ratio (SILVER) Method for Optimizing Radial Sampling of Dynamic MRI

Purpose To present and assess a method for choosing a fixed increment between spokes in radially sampled MRI that results in higher SNR than other radial sampling increments. Theory and Methods Sampling uniformity contributes to image SNR when reconstructed using linear methods. Thus, for a radial trajectory, uniformly spaced sampling is ideal. However, uniform sampling lacks post-acquisition reconstruction flexibility, which is often needed in dynamic imaging. Golden ratio-based increments are often used for this purpose. The method presented here, Set Increment with Limited Views Encoding Ratio (SILVER), optimizes sampling uniformity when a limited number of temporal resolutions are required. With SILVER, an optimization algorithm finds the angular increment that provides the highest uniformity for a pre-defined set of reconstruction window sizes. SILVER was tested over multiple sets and assessed in terms of uniformity, noise amplification, and SNR both in simulations and in acquisitions of a phantom and healthy volunteers. Results The proposed algorithm produced trajectories that, for the optimized window sizes, had higher uniformity and lower image noise than golden ratio sampling both in a simulated single-coil system and in a multi-coil system, assessed using simulation, phantom, and in vivo experiments. The noise in SILVER optimized trajectories was comparable to uniformly distributed spokes whilst retaining flexibility in reconstruction at multiple temporal resolutions. In a resting state fMRI experiment, tSNR increases at different spatial and temporal resolutions ranged from 21-72%. Conclusion SILVER is a simple addition to any sequence currently using golden ratio sampling and significantly increases sampling efficiency and SNR.


Introduction
Many recent MRI methods, especially highly accelerated and/or dynamic imaging, use radial k-space sampling because of its benefits related to reduced sensitivity to motion (1), and spatially incoherent aliasing when sampled below the Nyquist limit (2).
In some dynamic MR experiments, it is useful to be able to retrospectively reconstruct the same data with different temporal resolutions. Examples of this include generating high temporal resolution navigators for motion correction of lower temporal resolution data (3,4) and retrospective binning of data into frames of unequal length based on respiratory or cardiac phase (5,6). In these cases, the radial golden ratio (GR) method (7) is often used. In the GR method, the direction of each k-space sampling spoke is determined as a set angle increment from the previous spoke, such that each new spoke intersects the largest gap in k-space by the golden ratio. This approach results in relatively high uniformity for any number of subsequently acquired spokes and allows for complete flexibility in reconstruction at multiple temporal resolutions. It also does not repeat the exact same spoke during an experiment, eliminating the risk that retrospectively binned data will contain repeats of the same spoke.
However, the most SNR efficient 2D radial sampling method in terms of minimizing the gaps in kspace for a fixed number of spokes is uniformly distributed spokes. The cost is that uniform radial sampling allows for reconstruction of the data at only a fixed temporal resolution and requires complete a-priori knowledge of how many k-space spokes will be combined to form each frame. It does therefore not have the same flexibility as GR sampling.
In this paper, an alternative method of sampling, which allows for a tradeoff between the flexibility of GR and the SNR efficiency of uniformly distributed spokes, is presented. The idea behind this method is that GR is often unnecessarily general for imaging experiments that will be reconstructed at only a small number of temporal resolutions. The proposed method uses numerical optimization of the angular increment between subsequent spokes for a restricted set of window sizes (number of spokes per frame) with the aim of maximizing the uniformity of sampling within that set. By relaxing the requirement for 'near-uniformity' to only apply to a specific set of window sizes, it was hypothesized that higher sampling efficiency could be achieved whilst maintaining the favorable properties provided by the GR methods. In this paper a procedure for choosing the fixed angular increment between subsequent spokes is presented and the resulting sampling patterns compared with both the GR method and with radially uniform sampling in simulations, phantoms, and in vivo. We call the proposed method the Set Increment with Limited Views Encoding Ratio (SILVER) method.

Properties of set increment sampling
In this study radial sampling trajectories with a pre-set, or fixed, increment between spokes are considered. Set increment sampling is easy to design and implement as one only needs to decide on a single angle increment, θ, that is added to the projection angle of the previous spoke. For radial spokes that span the full diameter of a circular k-space, θ can be defined using a sampling ratio, 0 < α < 1, such that = × 180°.
When α is a rational fraction, the exact same spoke will eventually be repeated, whereas if the step is irrational, or rational with a very large denominator in its simplest form (practically irrational), no two spokes acquired within the duration of the experiment will be the same. If multiple temporal resolutions are required, it is beneficial to use irrational sampling to avoid duplicate spokes in the different window sizes, especially if data is combined across all frames to construct a temporal average image, for example for coil sensitivity map estimation. Also, in imaging experiments where the length of a frame is unknown a-priori, practically irrational increments are preferred to avoid acquiring the same spoke again within each frame. Setting α to the reciprocal golden ratio which is often referred to as the most irrational number (8), is therefore commonly done in order to sample without repeats.
Regardless of whether α is rational or irrational, acquiring data in k-space with a set angular increment generates data that can be flexibly reconstructed, as any grouping of N subsequently acquired spokes will have the same k-space sampling pattern, only rotated ( Figure 1). This allows for sliding window (9) and view sharing (10) reconstructions. shows the effect of an irrational α; Non-overlapping subsequent frames have no repeat spokes, and overlapping frames have some of the same spokes. In both (B) and (C) a subsequent frame (whether overlapping or not) is simply a rotaion of the previous frame and will thus have the same level of uniformity in each case.
In this study, all sampling patterns that are studied use set increment sampling. Uniform radial sampling with full width spokes can be achieved with set increment sampling by choosing the angular increment between subsequently acquired spokes to be & = $ ' , where N is the number of spokes used to reconstruct one frame. In GR sampling, on the other hand, the step is instead approximately 111.24°, with () = √"#$ % ≈ 0.6180. The aim of SILVER is thus to find a more optimal increment, 0 < α < 1, when near uniformity is only required for a certain set of window sizes.

Sampling and SNR
With non-uniform sampling density, denser areas of sampling (e.g. near the centre of k-space or where two radial sampling spokes are nearly overlapping) have to be down-weighted. If each sample is correctly weighted the signal can be accurately reconstructed. However, noise measured in each sample is averaged with noise from all other samples when using a Fourier transform, and noise averaging is most effective with equal weighting of each noise sample. Sampling with non-uniform density thus leads to non-ideal averaging, and noise amplification in MRI (11,12). The proposed method thus aims to improve uniformity of sampling to improve SNR.
Another way of considering the relationship between sampling and SNR is to consider the linear model governing the acquisition and reconstruction methods. When sampling k-space in a nonuniform manner, off the Cartesian grid, reconstruction generally involves taking the pseudo-inverse of the acquisition operator, E, either directly or through iterative methods. E is defined by the forward model of MRI, y = Ex + n, where y is the data recorded by the scanner, x is the object that is being imaged and n is thermal noise. The operator E is generally composed of a coil sensitivity operator that models how each receive channel weights the object x spatially, and a sampled Fourier transform that models how the data is acquired using spatial encoding along a specific trajectory. In a pseudo-inverse reconstruction (E'E) -1 must be calculated or iteratively estimated, and its diagonal entries (assuming pre-whitened noise covariance) will govern noise amplification (13). Therefore, if the spokes are not well distributed, some of the columns/rows of E will be similar, leading to poor conditioning of E and greater noise amplification. Strictly speaking, the noise amplification depends on the interactions between both the sampling trajectory and the coil sensitivities. However, other sampling methods, such as CAIPI (14), have been shown to reduce noise amplification robustly without requiring consideration of specific coil sensitivity profiles.

Measuring sampling efficiency
Because SNR intrinsically depends on the uniformity of the local k-space sampling density, different methods for estimating the local sampling density have been proposed. For 2D radial sampling in particular, Winkelmann et al. defined sampling density by the inverse of the average azimuthal distance between adjacent spokes (7). For spiral imaging the sampling density has been estimated using the speed of spiraling around k-space (15), or using voronoi cells (16). For 3D radial sampling numerically defined approaches using voronoi cells on spheres have also been used (17). An alternative to these methods of considering density distributions, specific to radial sampling, is to consider a physical model based on electrostatic force or potential between charges placed on the spoke tips. This constitutes a version of the Thomson problem (how charges would naturally be distributed on a sphere with minimal potential energy) where each charge has a pair on the opposite side of the sphere (13,14). Methods like these are extensively used to choose diffusion sampling directions in diffusion tensor imaging (20,21). Electrostatic methods, although more commonly used in 3D sampling, can also be easily applied to 2D sampling problems. An electrostatic potential minimizing model was used in this version of SILVER because of its easy extension to 3D in the future, and its high penalty for overlapping spokes.
In the electrostatic potential model, the radial sampling pattern is treated as an ensemble of unit charges placed on both ends of each spoke spanning the diameter of the unit circle. The total energy of the system is thus defined as: Where rij is the distance between the i th and j th points, and N is the number of spokes (2N is therefore the number of spoke tips).
Sampling efficiency compared to radially uniform sampling, η, was then defined as the ratio of total electrostatic potential stored in the system of point charges, U, to a system with the same number of spokes in the lowest possible energy state (uniformly distributed spokes), Uref: Uref is included in the efficiency metric, η, as a normalizing factor, so that comparison of efficiencies at different N is possible. For sampling with a set increment, U reduces to a function of only α, and the number of spokes, N. For a uniform radial distribution α = 1/N, so Uref is simply a function of N.

Optimization
The SILVER method was formulated as an optimization problem: to maximize the minimum efficiency, η, for a pre-defined set of window sizes, S = {N1, N2, N3, …}. The objective function was therefore defined as: Where N is the number of spokes in the window, α is the fixed sampling ratio (as defined in section An additional, optional, criterion was included in the optimization algorithm to check for Nyquist coverage at some maximum number of spokes to enable coil sensitivity estimation from a fully sampled, time averaged, set of k-space data. A flowchart for this additional step in the algorithm is shown in Figure 2b.

Simulating and measuring noise amplification
To compare SILVER optimized imaging more directly with GR and uniform radial sampling, noise amplification in images acquired with the three methods was predicted using Monte Carlo simulations and compared to the predicted efficiency using the uniformity metric, η. Gaussian complex noise was generated and reconstructed linearly using an iterative conjugate gradient reconstruction for the trajectory being examined. The reconstruction algorithm was implemented in MATLAB using the built-in preconditioned conjugate gradients method, pcg, with 100 iterations. A final filtering step was performed which involved setting the unsampled corners of (Cartesian) k-space to zero to avoid signal in regions unconstrained by the circular sampling window. This was done by taking the final image output from the iterative reconstruction, applying a Cartesian fast Fourier transform, using a circular mask to set the corners of k-space to zero and then transforming back to image space using a 2D inverse fast Fourier transform. The reconstruction was run 100 times for the same trajectory with different instances of noise, and the resulting pixelwise standard deviation across instances was measured. The resulting noise standard deviation maps were compared using a Wilcoxon signed-rank test at a significance level of 0.005 to control for multiple comparisons. Next, the effect of trajectory choice with a fixed set of coils was explored to determine whether optimizing for the trajectory alone can reduce noise amplification in a more complex parallel imaging setting. The same simulation as for the single coil case was performed, but with 10 spokes and eight coil sensitivity maps (compressed from a 32-channel phantom measurement using singular value decomposition (22)). The uniform sampling angle was thus 18º, and SILVER was optimized for S1 = {5, 10}, S2 = {10, 15}, and S3 = {9, 10, 11}, giving optimized angles 32.759º, 126.936º, and 98.680º, respectively. Additionally, the experiment was repeated 179 more times with the starting angle shifted by 1º. This was to further study how the trajectory and coil sensitivity maps interacted. The noise means of each angle for each trajectory were also compared using the Wilcoxon signed-rank test at a significance level of 0.005 to control for multiple comparisons.

Example use case -TURBINE fMRI
After the simulation experiments, SILVER was assessed in both in a phantom and in vivo in two healthy volunteers. The in vivo data was acquired with approval by the local ethics committee.
SILVER was applied to a multi-shot 3D fMRI experiment, with a 3D radial-Cartesian sequence consisting of rotating EPI blades, TURBINE (4). In multi-shot 3D fMRI experiments, there are SNR gains to be made relative to 2D multi-slice methods, but the acquisitions are more sensitive to motion.
Therefore, reconstructing a fast navigator can be useful for estimating motion which can be used to correct the full reconstruction. SILVER can thus be used to optimize TURBINE to two temporal resolutions simultaneously: the fast navigator and the full imaging temporal resolution.
Each TURBINE-EPI blade was fully sampled along the phase encoding direction (z-direction), which allowed 2D radial k-space data to be generated for each slice separately, after Nyquist ghost correction and an inverse Fourier transform along the z-direction. The sequence parameters were TE In vivo, two additional, more realistic, resting state fMRI scans were acquired without repeats, with αSILVER,no-Fib, and αGR, for 5 min 6 s each. For these, tSNR was directly compared between GR and SILVER. Again, statistical significance was measured using a Wilcoxon signed-rank test at a significance level 0.025.
The data were reconstructed slice-by-slice after Nyquist ghost correction, global phase correction, and an inverse Fourier transform across the fully sampled phase-encoding direction (superior-inferior), with the same iterative reconstruction method as had been used for the Monte-Carlo noise simulations described in section 3.2. The short acquisitions that had two repeated acquisitions were phase aligned by multiplying the k-space data from the first acquisition by a global phase factor, e iφ , on a spoke-byspoke basis, which minimizes the L2 distance between the spokes (complex data vectors) in the first and second acquisition. A complex subtraction of the repeats then yielded pure noise images that were studied by taking the pixel-wise standard deviations to measure noise levels. Additional predictions of noise were performed for the measured coil sensitivities and coil covariances. In this case, noise was not only predicted for the first N spokes, but simulated noise data was reconstructed using continuously sampled spokes leading to a rotating trajectory that varies frame-by-frame, such that the expected temporal noise could be estimated as the standard deviation across time.
The longer resting state fMRI scans were reconstructed using the same linear reconstruction framework, but only a single repeat was acquired. This was done to demonstrate the tSNR benefit even in the presence of signal fluctuations. These data were examined by measuring the pixelwise tSNR taking the temporal mean divided by the temporal standard deviation. Additional compressed sensing reconstructions were performed, with sparsity enforced in the spatial wavelet domain (4-level Daubechies db2) to assess whether the benefit from SILVER persisted for non-linear reconstructions.
The compressed sensing reconstruction replaced the iterative conjugate gradient method with the fast iterative shrinkage thresholding algorithm (FISTA (23)) with a step size of 10 -4 for low resolution reconstruction with 10 spokes/frame and 10 -6 for high resolution reconstruction with 46 spokes/frame.
The regularisation factor was set to 10 -6 , and the number of iterations was 50. These parameters were chosen empirically.
The coil sensitivity maps for all reconstructions were estimated using the adaptive combine method, using coil images reconstructed from all the data from the GR acquisition, and used for all reconstructions. The SILVER acquisitions also produced fully sampled temporal mean images when combined across all frames that could have been used for coil sensitivity estimation. However, the GR sensitivity map was used for all to reduce sources of variance in the results.

Results
The SILVER optimization succeeded in improving the minimum efficiency within all the targeted sets compared to GR. As shown in Figure 3a, for the trialed sets, the improvement varied from 0.067% (for SILVER optimized for five Fibonacci numbers) to 4.2% (for SILVER optimized for the set S = {4, 8}). 13 out of the 16 trialed sets achieved improvements larger than 1%. Figure 3b shows an example of efficiencies for different window sizes sampled with GR and two of the SILVER optimized increments. Within the targeted window sizes, the two SILVER optimizations consistently outperformed GR, even reaching higher efficiency than GR at its peaks at Fibonacci numbers. show the window sizes that the two SILVER increments were optimized for respectively. Both optimizations reach higher efficiency for their optimized window sizes than GR, which has characteristic peaks at Fibonacci numbers 13, 21, and 34.
Noise levels in the single coil simulations (Figure 4) followed the predicted order with lowest noise in uniformly sampled data and highest noise in GR sampled data, with the SILVER optimized trajectories having noise levels in between GR and uniform. All differences were statistically significant except the two SILVER trajectories optimized for S = {50, 60}, and S = {60, 70}. In single coil simulations with other angle increments noise levels correlated with efficiency estimated using the electrostatic potential metric, especially near optimal efficiency (Supporting Information Figure   S1). In multi-coil simulations ( Figure 5), the same trend was observed, with highest noise in GR and lowest in the uniform sampling case. However, as the trajectories interacts with the coils, the relative order of SILVER sets did not directly correspond to the estimates based on efficiency of the trajectory alone. The effect of the trajectory interacting with the coil sensitivities was also noticeable when the trajectory was rotated with respect to the coils (Figure 6), although these fluctuations were of smaller magnitude than the differences between the methods. One of the SILVER sets, S = {5, 10}, produced noise that on average was similar to GR with a non-significant difference, whereas the other two sets had significantly lower average noise.   Figure 5a as the trajectory is rotated with respect to the coil sensitivity maps. Uniform sampling has an expected period of 18 degrees because that is the spoke spacing.
The acquired phantom data corresponded closely to the noise predicted from the coil sensitivities, noise covariance, and the performed trajectory, shown in Figure 7. For all four window sizes, SILVER had a performance with noise levels lower than GR. The difference in median noise level between uniform and GR sampling was +26%, +48%, +25%, and +5%, for window sizes 8, 10, 46, and 55. The difference between uniform and SILVER was much smaller: 0%, -1%, +9%, and +1% for the same window sizes. The noise maps, normalized to the uniform sampling median value, are shown in Figure 7a, and the corresponding violin plots of measured and predicted noise values shown in 7b. Similar results to the phantom results were found in vivo ( Figure 8). However, a small discrepancy between predicted and measured noise was found due to additional signal fluctuations arising from physiological noise and any residual aliasing. The difference in median noise level between uniform and GR sampling for subject B shown in Figure 8 was in this case +27%, +48%, +18%, and -2%, for window sizes 8, 10, 46, and 55. The difference between uniform and SILVER was +4%, +22%, +11%, and -1% for the same window sizes. Similar results were measured in the other subject (Supporting Information Figure S2). The localised noise increases around the sagittal sinus in the measured data are probably due to pulsatility.
The longer fMRI experiment also showed improvements in tSNR for SILVER compared to GR.
Using a linear reconstruction, the tSNR for the fast navigator (10 spokes/frame) improved from 10.46 to 17.98 (72%) by using SILVER instead of GR, and for the imaging resolution (46 spokes/frame) the tSNR improved from 16.40 to 19.83 (21%). Figure 9 shows tSNR maps in an example slice from subject A and tSNR histograms across all slices. The results in subject B were comparable (Supporting Information Figure S3). Also, when using a more sophisticated reconstruction (wavelet regularized compressed sensing), SILVER continued to demonstrate higher tSNR than GR ( Figure 10 shows an example slice of subject A). The difference was somewhat smaller than in the linear case (35% improvement for the low-resolution navigator and 15% for high-resolution imaging), but nonetheless substantial. The results from subject B were similar and are attached in the Supporting Information ( Figure S4).

Discussion and conclusions
We have presented a method of choosing an optimal angular increment for dynamic radial MRI when the set of window sizes to consider is constrained. We have shown that this method results in decreased noise compared to GR sampling with only a minor change required to the acquisition protocol: the single value that controls the angular increment.
We showed that small differences in uniformity measured using the electrostatic potential metric can lead to large differences in the amount of thermal noise in radial acquisitions. The results showed a max difference of ~4% in electrostatic potential, but up to ~70% difference in tSNR. The largest improvements in uniformity were for small sets including only two temporal resolutions, and in these cases improvements over GR at Fibonacci numbers was even possible. For larger sets, the improvement in uniformity was lower. However, even when no significant improvement was achieved, SILVER converged onto a trajectory with the same efficiency as GR (i.e. GR or a tiny golden angle (24)). So, there was no disadvantage to optimizing for SILVER-based increments on any set of window sizes of interest. SILVER is easy and fast to implement (for example, SILVER for S = {10, 46} took ~7 s to run on a MacBook Pro, 2.7 GHz Intel Core i7, without any optimization for speed) and will only be needed to be optimized once for every set of window sizes of interest.
SILVER is optimized without knowledge of the coil sensitivities, and the trajectory inevitably interacts with the sensitivity maps in a parallel imaging reconstruction. By trialling SILVER at different starting angles, the effect of this interaction could be assessed. With uniform sampling, the same trajectory with respect to the coils are acquired for each frame, but with GR or SILVER each frame has a rotated version of the trajectory and can thus have varying noise amplification in different frames. The temporal behaviour and periodicity for these methods should be studied further for applications where the temporal signal characteristics can affect image analysis, such as in fMRI.
The results showed good consistency between predicted and measured noise, especially in the phantom, but slight deviations were observed in vivo, likely due to signal changes originating in pulsatility and subtle motion. In the data presented, no motion correction was applied since the subjects were well behaved and not using motion correction avoids skewing tSNR measurements due to effects originating in the interpolation. SILVER showed an improvement in tSNR for both linear and non-linear reconstructions indicating that the method is not limited by a specific reconstruction approach.
The example application of SILVER presented in this paper was a multi-temporal resolution fMRI experiment, where one resolution was intended to be used as a navigator, the other as the imaging resolution. Examples of other applications of SILVER include retrospective gating for cardiac/respiratory motion, including XD-GRASP (25), or multiple temporal resolutions for simultaneous angiography and perfusion imaging (e.g. CAPRIA (26)).
Because conditioning of the acquisition operator depends on the coil sensitivities, as well as the sampling trajectory, optimizing for sampling uniformity alone cannot be guaranteed to always produce the optimal trajectory leading to the lowest noise. However, without knowledge of the specific coil sensitivity maps to be used, optimizing for uniformity alone is a tractable step in the right direction, similar to how in CAIPI (14) the conditioning of the parallel imaging problem is improved by changing the trajectory alone. In this paper, we have shown that optimizing the trajectory without knowledge of the coils in general leads to reduced noise as coils intended for parallel imaging are independently optimized by the coil manufacturers. In one case SILVER was observed to produce on average slightly higher noise than GR (when varying starting angles were trialed, Figure 6), but the difference was non-significant. Similarly, in the noise measurements for 55 spokes in-vivo and 8 spokes in the phantom (Figures 7 and 8), GR and/or SILVER produced lower noise than uniform sampling, which could be due to interaction with the coil sensitivities or simply due to variance in the measurements. Again, in these cases, the size of these effects was very small (less than 2% difference in noise standard deviation).
In future work, the SILVER framework could also be extended to optimize for properties other than trajectory uniformity. This could be done through design of alternative cost functions, including cost functions that consider the temporal incoherence of the operator for temporally regularized reconstructions. Additional constraints on the optimization could also be included to limit the SILVER angles to be small for less eddy current contamination, an issue that is currently addressed in GR using the tiny golden angle approach proposed by Wundrak et al. (24). However, unlike other work that optimize trajectories more generally (e.g. SPARKLING (27)), SILVER is focused on achieving as large improvements to image quality as possible with as small a change as possible to existing sequences for ease of adoption.
Both SILVER and GR only optimize for contiguously binned spokes, and there is no guarantee that binned data from multiple repeats of radial acquisitions are distributed optimally. Suggestions for overcoming this problem have been proposed for golden ratio sampling when the number of repeats are known a-priori by e.g. Fyrdahl et al. (28) and Song et al. (29), and these methods are equally applicable to SILVER.
Radial sampling does not have to be performed with a fixed increment and could instead be acquired with some other scheme for determining the spoke directions. However, this project focuses on fixed increment radial imaging because of the easy implementation, the consistency in the uniformity of the trajectory across frames, and the ability to use a sliding window for reconstruction if desired. MR sequences that currently use uniform or golden ratio increments could easily be modified to SILVER sampling by changing a single parameter, the increment angle, in the sequence implementation. The SILVER framework also has a potential to be extended to imaging with other trajectories that have previously been derived from the golden ratio, e.g. spirals (30,31), cones (32), and even Cartesian sampling (33).
In 3D, a set increment is harder to define than in 2D. Chan et al. (17) proposed the multi-dimensional golden means method, a 3D analogy of the golden ratio method with a constant azimuthal angle increment and a constant z-axis increment of the tip of the spoke constrained to the surface of a sphere. However, subsequent frames generated with this method are not simply a rotation of the spokes from the previous frame, and therefore not all image frames are guaranteed to have the same sampling efficiency. Because of the increased complexity of 3D radial sampling and the ambiguity of how to define a set increment, we constrained the scope of this paper to focus on the more commonly used 2D case, although extension of SILVER to 3D trajectories, where the electrostatic potential metric of uniformity should work as well, will be considered in future work.
In conclusion, SILVER is a method that generalizes set increment sampling beyond golden ratio derived methods when some knowledge of the number of spokes to combine in a frame is known apriori, as is commonly the case. Therefore, sometimes SILVER is better than gold, and when it is not, it is just as good.

Data availability statement
Code for SILVER optimization, all reconstructions, and analysis used in this paper are available on