16-889 Assignment 4

Luyuan Wang (luyuanw@andrew.cmu.edu)

1. Sphere Tracing

Result:

part_1

In Sphere Tracing, we walk along a ray until we are close to the surface, or exceed the maximum steps. If the condition of f(p)≤ϵ is true, we need to stop the loop. It easy to do so if we use nested loops to check every points, but is not efficient. To vectorize the code, I create a mask tensor to mask out the points that already meet the condition and don't need to be updated anymore.

2. Optimizing a Neural SDF

The rendered mesh:

part_2_backup

The MLP has a similar structure with NeRF. The input is encoded with a Harmonic embeding layer. The MLP has 9 linear layers, and ReLU is used as the nonlinear function. There is a skip connection from the embedding to the 5th linear layer. The output is normalized with a sigmoid function.

The Eikonal loss is defined as the following equation:

Eikonal Loss=(||gradient||−1)2

3. VolSDF

I increased the number of points per ray from 128 to 256. With more sample points, the network should be able to learn a more detailed surface structure.

part_3_geometry_backup3part_3_backup2

In the VolSDF paper, the alpha and beta should be two learnable parameters. But at here, we used fixed numbers. alpha is just a scaling factor of the calculated density. beta controls the scale of the Laplace distribution. If beta approach to zero, the density function converges to the indicator function of the occupied space.

  1. A high beta makes the surface blurry. A lower beta produces a clear surface.
  2. A high beta is easier to train. A low beta gives a more strict constrain, that the network may not converge at the begining.
  3. We are more likely to learn an accurate surface with a low beta, as a lower beta make the spread of the Laplace dirstribution smaller.

After I increase the beta to 0.25 (5x greater), the surface is not very sharp:

part_3_b025

After I decrease the beta to 0.01 (5x less), the network doesn't converge, and only outputs black images:

part_3_b001

4. Neural Surface Extras

4.3 Alternate SDF to Density Conversions

I tried the solution in the NeuS. The SDF to Density conversion is defined as the following equation:

ϕs(x)=se−sx/(1+e−sx)2

I got a relative good result with s = 10, but it seems not as good as our previous implementation.

part_3_Neus