Rendering

Point-based rendering with $\alpha$-blending, NeRF-style volumetric rendering, and Gaussian Splatting share several aspects of the image-compositing process. In all these cases, a pixel's color can be obtained by combining the contributions of the primitives intersected by the ray associated with that pixel. The final color is obtained as a weighted sum of the contributions along the ray, with weights determined by local opacity and accumulated transmittance.

The color $C$ of a pixel can be obtained by integrating along the optical ray $\mathbf{r} = \mathbf{o} + t\mathbf{d}$ the radiometric contributions weighted by density and transmittance, over the interval $[t_n,t_f]$:

$$C = \int_{t_n}^{t_f} T(t) \sigma \left( \mathbf{r}(t) \right) \mathbf{c} \left( \mathbf{r}(t), \mathbf{d} \right) dt$$ (10.111)

where

$$T(t) = \exp \left( - \int_{t_n}^{t} \sigma \left( \mathbf{r}(s) \right) ds \right)$$ (10.112)

The function $T(t)$ denotes the transmittance accumulated along the ray from $t_n$ to $t$, namely, the fraction of radiation that has not been attenuated by the scene before reaching position $t$.

The integral can be approximated by discretizing the ray, yielding the following summation:

$$C = \sum_{i=1}^{N} T_i \left( 1 - \exp( - \sigma_i \delta_i) \right) \mathbf{c}_i$$ (10.113)

where $T_i = \exp \left( - \sum_{j=1}^{i-1} \sigma_j \delta_j \right)$ where $\delta_i = t_{i+1} - t_{i}$.

This representation reduces to classical $\alpha$-blending by using $\alpha_i = 1 - \exp \left( - \sigma_i \delta_i \right)$. The transmittance can then also be written as $T_i = \prod_{j=1}^{i-1} (1 - \alpha_j)$.

In typical neural point-based approaches, the color $C$ of a pixel is obtained by blending the $\mathcal{N}$ ordered points that underlie the pixel:

$$C = \sum_{i \in \mathcal{N} } c_i \alpha_i \prod_{j=1}^{i-1} \left( 1 - \alpha_j \right)$$ (10.114)