<p>In section 4.2 of this paper:</p>
<p><a href="http://cseweb.ucsd.edu/%7Eravir/274/15/papers/a209-xu.pdf" rel="nofollow noreferrer">http://cseweb.ucsd.edu/~ravir/274/15/papers/a209-xu.pdf</a></p>
<p>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.</p>
<p>EDIT:</p>
<p>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 <span class="math-container">$\bar z$</span></p>

CC BY-SA 4.0
2020-06-23T21:42:48.000Z
"ASGs is another ASG" Because the product of 2 Gaussian distributions in the euclidean case is another Gaussian distribution. I know that this does not guarantee that it will be the case for the spherical case too but that's why I had the hunch.
Makogan
CC BY-SA 4.0
2020-06-23T23:04:54.603Z
$v\cdot v = ||v||^2 = 1$ btw, remember these are vectors not scalars.

More importantly, numerical simulations and a visualization of the result strongly suggest that the product of 2 ASG's is either an ASG or something very very close to an ASG. Consider the figures in this question:
https://math.stackexchange.com/questions/3704580/finding-a-new-lobe-in-the-product-of-2-anisotropic-gaussians
Makogan
<rendering><geometry><hemisphere>
Has anyone found an analytic solution to the product of spherical gaussians from Xu et al?
Makogan
# Has anyone found an analytic solution to the product of spherical gaussians from Xu et al?

(Asked by: username_0 on 2020-06-05)

In section 4.2 of this paper:

[http://cseweb.ucsd.edu/~ravir/274/15/papers/a209-xu.pdf](http://cseweb.ucsd.edu/%7Eravir/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 $\bar z$

**Tags:** rendering, geometry, hemisphere

> **Comments:**
> **username_0**: "ASGs is another ASG" Because the product of 2 Gaussian distributions in the euclidean case is another Gaussian distribution. I know that this does not guarantee that it will be the case for the spherical case too but that's why I had the hunch.  
> **username_0**: $v\cdot v = ||v||^2 = 1$ btw, remember these are vectors not scalars.  More importantly, numerical simulations and a visualization of the result strongly suggest that the product of 2 ASG's is either an ASG or something very very close to an ASG. Consider the figures in this question: https://math.stackexchange.com/questions/3704580/finding-a-new-lobe-in-the-product-of-2-anisotropic-gaussians  

---

## 0 answers

