volsdf.md

Notes on the up-sampling algorithm and error bound of VolSDF

In VolSDF, they prove a error bound of the discontinuous Riemann Sum's approximations in the opacity's calculation, and derive a ray point up-sampling algorithm to control the error bound to keep smaller than manually set episilon, which is set to 0.1.

1. up sampling algorithm's visualization in tensorboard when training

	@0k	@4k	@10k	@200k
up sample iterations until converged
beta heat map

2. up sampling algorithm single ray testing

• network's beta = 0.001
• eps = 0.1
• To try it yourself, run:

python -m debug_tools.test_volsdf_algo

0-th iteration

• 128 uniform sample points.
• If use the network's beta, then error_bound.max=inf, which does not satisfy <eps=0.1.
• for beta+=0.431, the plots are

1-st iteration

• 256 sampling points.
• If use the network's beta, then error_bound.max=inf, which does not satisfy <eps=0.1.
• for beta+=0.049, the plots are

2-nd iteration

• 384 sampling points
• If use the network's beta, then error_bound.max = 1.3e5, which does not satisfy <eps=0.1.
• for beta+=0.023, the plots are

3-rd iteraion

• 512 sampling points
• If use the network's beta, then error_bound.max = 0.570, which does not satisfy <eps=0.1.
• for beta+=0.013, the plots are

4-th iteration

• 640 sampling points
• If use the network's beta, then error_bound.max = 0.116, which does not satisfy <eps=0.1.
• for beta+=0.001, the plots are

5-th iteration

• 768 sampling points
• If use the network's beta, then error_bound.max = 0.0220, which satisfies <eps=0.1. The plots are: