Appendix

The introduction of convolution and attention to the space of rays in 3D required additional geometric representations for which there was no space in the main paper to elaborate. We will introduce here all the necessary notations and definitions. We have accompanied this presentation with examples of specific groups to elucidate the abstract concepts needed in the definitions. Figure 10: The visualization of Plücker coordinates: A ray xcan be denoted as (d,m)where x is any point on the ray x, and dis the direction of the ray x. mis defined as x d. Given the action of the group G on a homogeneous space X, and given x0 as the origin of X, the stabilizer group H of x0 in G is the group that leaves x0 intact, i.e., H = {h G|hx0 = x0}. The group, G, can be partitioned into the quotient space (the set of left cosets) G/H and X is isomorphic to G/H since all group elements in the same coset transform x0 to the same element in X, that is, for any element g gH we have g x0 = gx0. Example 1. SE(3) acting on the ray space R: Take SE(3) as the acting group and the ray space R as its homogeneous space. We use Plücker coordinates to parameterize the ray space R: any x R can be denoted as (d,m), where d S2 is the direction of the ray, and m = x d where x is any point on the ray, as shown in figure 10. R is the quotient space SE(3)/(SO(2) R)up to isomorphism. Example 2. SE(3) acting on the 3DEuclidean space R3: R3 is isomorphic to SE(3)/SO(3). Consider another case when SE(3) acts on the homogeneous space R3; for any g = (R,t) SE(3) and x R3, gx = Rx+t. If the fixed origin is [0,0,0]T, the stabilizer subgroup is H = SO(3) since any rotation g = (R,0)leaves [0,0,0]T unchanged. The last example is SO(3) acting on the homogeneous space sphere S2. Given the fixed origin point as [0,0,1]T, the stabilizer group is SO(2).