Left-Invariant Diffusion on the Motion Group in terms of the Irreducible Representations of SO(3)

In this work we study the formulation of convection/diffusion equations on the 3D motion group SE(3) in terms of the irreducible representations of SO(3). Therefore, the left-invariant vector-fields on SE(3) are expressed as linear operators, that are differential forms in the translation coordinate and algebraic in the rotation. In the context of 3D image processing this approach avoids the explicit discretization of SO(3) or $S_2$, respectively. This is particular important for SO(3), where a direct discretization is infeasible due to the enormous memory consumption. We show two applications of the framework: one in the context of diffusion-weighted magnetic resonance imaging and one in the context of object detection.

💡 Research Summary

The paper presents a novel framework for formulating convection‑diffusion equations on the three‑dimensional motion group SE(3) using the irreducible representations of the rotation group SO(3). The authors start by recalling that SE(3) is the semidirect product of translations ℝ³ and rotations SO(3). Its Lie algebra is generated by six left‑invariant vector fields: three translations (Tₓ,T_y,T_z) and three rotations (Jₓ,J_y,J_z). By switching to a complex basis (~T,~J) the rotational part can be expressed directly through the Wigner D‑matrices D⁽ʲ⁾ₙₘ(α,β,γ), which constitute the unitary irreducible representations (UIR) of SO(3).

A key observation is that any square‑integrable function φ on SE(3) can be expanded in the Wigner basis as
φ(g) = Σ_{j=0}^{∞} Σ_{n,m=-j}^{j} (2j+1)/(8π²) D⁽ʲ⁾ₙₘ(R) f⁽ʲ⁾ₙₘ(x),
where g = (x,R) ∈ ℝ³×SO(3) and the coefficients f⁽ʲ⁾ₙₘ(x) depend only on the spatial coordinates. This separates the spatial differential operators from the angular part, which becomes purely algebraic.

The rotational vector fields act on the Wigner basis via well‑known angular‑momentum ladder operators:
J_z D⁽ʲ⁾ₙₘ = i m D⁽ʲ⁾ₙₘ,
J_{±1} D⁽ʲ⁾ₙₘ = i √