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Steps: For each direction in the ray, we bigin at the origial point. For each point, we get the distance to the object from the implicit function, then move the starting point towards the direction of ray by that much distance. Then calculate the new distance from the new starting point, and repeat the above process until either the distance is smaller than 1e-5 which is the threshold(hit the surface), or the distance between the new starting point and the original is larger than 5(not hit the surface).
Output
Input 
Q: How does high beta bias your learned SDF? What about low beta?
High beta makes the density less sensitive, which makes the plot more blurred. Low beta makes the density changes more fierce given different distance, which makes the plot sharper.
Q: Would an SDF be easier to train with volume rendering and low beta or high beta? Why?
High: it makes the density less sensitive towards the update of distance. So it leads to smoother gradient in training, which makes training easier.
Q: Would you be more likely to learn an accurate surface with high beta or low beta? Why?
Low: It makes the density changes more fierce given different distance(The derivative of density over distance is higher for lower beta), which makes the surface sharper and more accurate.
Comment on the settings you chose, and why they seem to work well.
The most important setting to my understanding is the HarmonicEmbedding. If we directly use the xzy as input, the result is much worse, or even lead to nan during training. It shows the HarmonicEmbedding to helps to capture more fine-grained densities and features. I used 8 mlp layers with 256 nodes for each layer and relu activation. Then an extra layer used for distance prediction and two layers with a sigmoid function to predict features. But I found generally the number of layers or number of nodes for each layer is not that important unless too small. Other hyperparameters are directly imported from the config file. I also tried different settings of alpha and beta, and seems the default setting(alpha=10, beta=0.05) leads to the best performance as below.
