What is a Gaussian Lobe concerning BRDF and NDF?

Question

I was reading a paper about voxel cone tracing and came across this term. So, what is a Gaussian Lobe or a Lobe in general?

I am asking in the context of Computer Graphics, BRDF and NDF.

Answer 1

In a BRDF or NDF this most likely refers to a spherical Gaussian distribution, also known as a von Mises–Fisher distribution.

It's similar to the usual Gaussian, but using the dot product between two unit vectors as a measure of distance to the mean:

$$ p(\hat v) = a \exp(\lambda(\hat \mu \cdot \hat v))$$

where $a$ is a normalization factor, $\lambda$ is a number controlling the width of the lobe (roughly corresponds to $1/2\sigma^2$ in the usual Gaussian), $\hat \mu$ is the unit direction vector for the mean of the lobe, and $\hat v$ is the unit direction vector at which we're evaluating the lobe.

(You'll also sometimes see it defined with $\lambda(\hat \mu \cdot \hat v - 1)$ in the exponent, but the $-1$ term can be pulled out and rolled into the normalization factor.)

Answer 2

Having a gaussian lobe usually means that your ndf is one or a weighted sum of gaussian distributions. Instead of your ndf, you can define your brdf in the same way.

A gaussian distribution is a function of the form $$\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

It has a hat shape common to many ndfs; this is good because it mimicks how normals align in most real world materials