Beyond directions: Rotation sets for triaxial diffusion encoding by geometric filter optimization (GFO)

TL;DR

The paper tackles the problem of obtaining accurate powder-averaged signals in diffusion MRI when using triaxial, non-axisymmetric diffusion encoding. It introduces Geometric Filter Optimization (GFO), which designs rotation sets by optimizing a sampling filter on SO(3) to approximate uniform sampling without increasing scan time. GFO demonstrably reduces spectral leakage and improves the precision (and to some extent the accuracy) of powder averages and higher-order rotational invariants, particularly at modest numbers of rotations and moderate b-values, with caveats at high b and large N. The approach is practical (offline optimization ≈ 30 seconds), scalable to triaxial encodings, and has potential to enhance advanced diffusion models and multi-shape encoding strategies in diffusion MRI.

Abstract

Purpose: We aim to improve the accuracy of the diffusion-weighted powder average signal for diffusion encoding with arbitrary shape. This enables a categorical improvement in all quantification based on, for example, tensor-valued diffusion encoding at no additional cost to acquisition time. Methods: We propose a method to generate optimal rotation sets that are applied to the diffusion encoding gradient waveform to yield powder averages with maximal accuracy. The method, termed ``Geometric Filter Optimization'' (GFO), amounts to designing an appropriate sampling filter which is approximately flat in the relevant part of the associated frequency space. We characterize the filter properties and benchmark the performance in terms of the accuracy and precision of powder averages and higher order rotational invariants. Results: GFO filters were found to have much smaller spectral leakage than other designs. We found that GFO leads to marked improvements in precision and accuracy in powder averaging over generic diffusion encoding objects, and similarly in higher order rotational invariants, although for sufficiently high $b$ and $N$, accuracy, but not precision, deteriorated compared to electrostatic repulsion. Conclusion: GFO provides an efficient recipe for obtaining orientations for powder averaging of signals with non-axisymmetric diffusion encoding. It places no additional demands on gradient system performance and can be used to shorten scan time.

Figures (6)

• Figure 1: The landscape of b-tensors includes all possible combinations of eigenvalues and can be visualized as glyphs in a triangle wherein the corners are the only axi-symmetric cases: linear, planar, and spherical. The interior of this triangle (pink area) indicates all other b-tensor shapes, each with three unique eigenvalues, i.e., the b-tensors are triaxial.
• Figure 2: Relative band amplitudes as a function of $l$-band for 100 random diffusion tensors show that the power is dominated by even-$l$ bands and that power dissipates rapidly with $l$. The blue shaded region indicates the range over the distribution and circles show the mean. The red dashed line shows the Sobolev form $(1+(l(l+1))/\kappa^2)^{-s}$ fit to the data for even $l$, giving $\kappa \simeq 6.3$ and $s\simeq 10.7$ for $b = 1$^2/ (a), and $\kappa \simeq 7.4$ and $s\simeq 7.9$ for $b = 3$^2/ (b).
• Figure 3: The impact of hyper parameters $s$ and $\kappa$ in the GFO+$D_2$ optimization is visualized in terms of the coefficient of variation (CV) of the signal powder average over 2601 $L=6$ Euler grid rotations of the diffusion tensor. The underlying GFO set consisted of 24 rotations and was designed with $L = 8$. Two different b-values ($b=1.5$ m^2/ms left and $b=3$ m^2/ms right) are shown.
• Figure 4: The spectral profiles of the GFO filters show the best approximations to the ideal. The filter band powers $E_l(f)$ across $l$ for different $N$ and all schemes. The ideal filter has $E_l(f) = \delta_{l0}$ and is shown left most. Note the logarithmic color scale.
• Figure 5: The top plot shows that GFO has the best performance across the seven methods in terms of the coefficient of variation (CV) in the signal powder average (lower is better). Bottom left plot shows a representative histogram of powder averages at $N=28$ wherein GFO produces a tighter distribution compared to electrostatic repulsion. The bottom right plot shows that all methods have negligible signal bias, but electrostatic repulsion and uniform sampling have the worst overall accuracy.