Why init beta needs sqrt?

[fishfishson]

Hi,

Thx for your excellent repo. In volsdf rendering part, I find you add extra sqrt when initializing beta which does not show in volsdf paper's equation. Could you pls tell me why you use sqrt here?

[ventusff]

Hi,
This is a good question :) This part is my biggest confusion when implementing VolSDF.

It is mainly about how to deal with alpha and beta simultaneously.

In VolSDF's paper, the upsample algorithm choose a suitable initial beta that promises the error bound, which is written as:

However, the alpha in the numerator on the right side is actually undefined in the paper and is unclear. I've tried with using the network's alpha (which is self.forward_ab()[0]), but that will cause the upsampling algorithm completely fail to produce reasonable results.

Considering that this expression is derived assuming corresponding alpha and beta (by "corresponding", I mean alpha=1/beta, which is stated in Sec3.5. in VolSDF's paper), the expression above could be further derived as follow:

Hence produce the sqrt operation.

[fishfishson]

@ventusff Great answer! I got it.