Assignment 4

1. Sphere Tracing

1.3. Ray sampling

Visualization of torus:

2. Optimizing a Neural SDF

Input Image:

Predicted Image:

The MLP that I used for this question is similar to the model used in the previous homework. This model includes 6 layers witth 128 hidden layers with skip input at layer 3. I also remove the ReLU at the end beacause an SDF can have values less than 0.

The Eikonal loss basically enforces the gradient to be equal to 1. So for my implementation, I took the norm squared of the gradients and used that as the loss.

3. VolSDF

alpha: Controls the density at the surface and decreases near the object's boundary. This is due to the fact that density in this case models a homogeneous surface.

beta: Determines how much the density at the surface gets smoothed or basically the rate at which alpha gets decreased.

How does high beta bias your learned SDF? What about low beta?

A small beta leads to a large change in density while a large beta leads to a smoother change in density which can overall affect accurate sampling.
This will result in a sharper image for a low beta and a more blurred image for a high beta.

Would an SDF be easier to train with volume rendering and low beta or high beta? Why?

Although beta is a learnable parameter, it would be easier to train volume rendering with a higher beta because the model would better learn a larger range of values for the SDF.

Would you be more likely to learn an accurate surface with high beta or low beta? Why?

I think it depends on the frequency of the surface. If the goal is to reconstruct a surface with complex geometry,
then I would likey learn a more accurate surface with a low beta and vice-versa for a smooth surface with a high beta. But for most cases, for beter reconstructions, a low beta would be better.

For the color prediction, I chose a 2 layer network with a skip input at layer 3. This is followed by another linear layer and a sigmoid to get valid color ranges. For SDF to density, I used alpha = 10.0 and beta = 0.05.

Lego bulldozer Geometry:

Lego bulldozer Reconstructed:

4. Neural Surface Extras

4.2. Fewer Training Views

(Left: VolSDF, Right: NeRF - 100 Training Views):

(Left: VolSDF, Right: NeRF - 20 Training Views):

I trained both VolSDF and NeRF with 20 training views and compared the results to training with 100 views. It appears that NeRF results in a sharper reconstructed image compared to VolSDF in both cases. However, NeRF exhibits more artificats away from the surface.

4.3. Alternate SDF to Density Conversions

(Left: VolSDF, Right: NeuS)

(Left: VolSDF, Right: NeuS - Geometry)

I implemented the naive solution from the NeuS Paper with s = 100 which showed the best results.