Beyond Directions: Symmetry-Aware Rotation Sets for Triaxial Diffusion Encoding by Geometric Filter Optimization

To improve the accuracy of diffusion-weighted powder average signals for diffusion encoding with arbitrary b-tensors.

We identify an intrinsic dihedral ( $D_2$ ) symmetry of diffusion signals for arbitrary diffusion encoding, which defines their natural signal space (a quotient of 3D rotations). Based on this, 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 a sampling filter that is approximately flat over the relevant part of the associated frequency space. We characterize the filter properties and benchmark performance in terms of the accuracy and precision of powder averages and higher-order rotational invariants, including comparison with spherical designs and electrostatic-repulsion-based designs defined on the same space.

We found that GFO leads to marked improvements in precision and accuracy in powder averaging over diffusion encoding b-tensors, including axisymmetric and triaxial configurations. For higher-order rotational invariants, the performance was more nuanced, with GFO, electrostatic repulsion, and spherical designs exhibiting different trade-offs in bias and precision depending on $b$ and $N$ .

A fundamental $D_2$ -symmetry of tensor-valued diffusion encoding was shown to constrain its rotational structure and guide the design of optimal rotation sets. This yielded GFO, which provides an efficient recipe for obtaining orientations for powder averaging of signals with axisymmetric and triaxial diffusion encoding. It places no additional demands on gradient system performance and can be used to shorten scan time.