This is part of my journey of learning NeRF.

## 2.3. Differentiable Forward Maps

### Differentiable rendering

Volume rendering can render fogs. Sphere rendering only render the solid surface, and needs ground truth supervision.? Neural renderer combines the two.

### Differentiation itself

Design a neural network with higher order derivatives constraints and therefore directly use its derivative.

For example the Eikonal equation forces the neural network has a derivative as 1. Adding the eikonal loss then promises the neural network valid.

Generally, this kind of problems are: the solutions are constrained by its partial derivatives.

### Special: Identity Operator

$\text{Reconstruction} \rightarrow \hat 1()\rightarrow \text{Sensor domain}\\ \text{Reconstruction} == \text{Sensor domain}$

Q&A:

• Can we obtain a neural network in just one forward, without optimization?
• Can we design special forward maps for specific downstream tasks, eg., classification? Absolutely yes. We can design it to represent a compact representation as the sensor domain. The key idea is to get a differentiable function to map your specific recon and sensor domain.