Has anyone found an analytic solution to the product of spherical gaussians from Xu et al? In section 4.2 of this paper:
http://cseweb.ucsd.edu/~ravir/274/15/papers/a209-xu.pdf
Xu et al approximate the product of 2 anisotropic Gaussian distributions. I want to know if someone has found not an approximation but THE solution to what the resulting ASG distribution is after taking the product.
EDIT:
The goal is to either show (or prove it's not possible) that given 2 arbitrary ASG's their product is a third ASG with specified lobe $\bar z$
Consider $f(v) := S(v;z_1)\cdot S(v;z_2)$ with $z_1=z_2=&lt;1,0,0&gt;$. Then
$$f(v) = \max(v_1,0)\cdot\max(v_1,0) = 0, \qquad \text{ if } v_1&lt;0, $$
$$f(v) = \max(v_1,0)\cdot\max(v_1,0) = v_1^2 \qquad \text{ if } v_1\ge 0. $$
We want to find a $\tilde z$ such that $f(v) \stackrel{!}{=} S(v;\tilde z) = \max(v\cdot\tilde z,0)$ for all unit vectors $v$. But $v=&lt;0,\pm 1,0&gt;$ implies $\tilde z_2 = 0$ and similarly $v=&lt;0,0,\pm 1&gt;$ implies $\tilde z_3=0$. Consequently, $S(v;\tilde z) = \max(v_1 \tilde z_1, 0)$ and it is not possible to choose $\tilde z_1$ such that
$$\max(v_1\tilde z_1,0) = f(v)$$
for all unit vectors $v$.