16-889 Assignment 4

Byeongjoo Ahn (bahn@andrew.cmu.edu)

1. Sphere Tracing (30pts)

The rendering from sphere tracing is shown below. Starting from the ray origin, we take the step based on the SDF value at the current point, and repeat the process until the points hits the surface. The SDF threshold used for checking if the point is on the surface is $10^{-5}$ and the maximum number of iterations is set to be $100$.

2. Optimizing a Neural SDF (30pts)

We use the following loss function for optimizing a neural SDF.

$|f(p)| + \|\|\nabla f(p)\|-1\|_2^2$,

where the first term enforces the network to output zero on the surface, and the second term regularizes the SDF using the Eikonal regularization that enforces the norm of the gradient to be one.

The input point cloud and the output surface from the trained neural SDF are shown below.

We used the 6 layers of MLP with 128 hidden units along with the harmonic embedding. The last layer does not have an activation function. We set eikonal weight to be $0.02$.

3. VolSDF (30 pts)

The geometric and rendering results from VolSDF are shown below.

Alpha determines the scale of the density, and beta determines the shape (smoothness) of the density function.

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

   → lower beta makes the surface shape sharper.

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

   → It would be easier to train with volumetric rendering with a higher beta as the variance of SDF near surface is smoother and the loss is not changed abruptly with respect to the shape update.

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

   → It would be easier to learn an accurate surface with a lower beta as the variance of SDF near surface is sharper and the small error in the surface results in larger error compared to a higher beta.

4. Neural Surface Extras

4.2 Fewer Training Views (10 pts)

The results with fewer training views (#view = 20) for both NeRF and VolSDF are shown below. We can see the floating artifacts in the background in the result of NeRF with fewer training views. VolSDF shows similar results in geometry even in the case of fewer training views, but the rendering shows slightly worse results though we will need higher resolution results for a better qualitative comparison.