Volume Rendering

Continuous

$$C(\mathbf{r})={\int }_{t_n}^{t_f}T(t)\sigma (\mathbf{r}(t))\mathbf{c}(\mathbf{r}(t),\mathbf{d})dt,T(t)=\exp \left(-{\int }_{t_n}^t\sigma (\mathbf{r}(s))ds\right),\mathbf{r}(t)=\mathbf{o}+t\mathbf{d}$$

Discrete (alpha compositing)

$$\hat{C}(\mathbf{r})=\sum _{i=1}^N{T}_i{\alpha }_i{\mathbf{c}}_i,{\alpha }_i=1-\exp (-{\sigma }_i{\delta }_i),T_i=\prod _{j=1}^{i-1}(1-{\alpha }_j)=\exp \left(-\sum _{j=1}^{i-1}{\sigma }_j{\delta }_j\right),{\delta }_i=t_{i+1}-t_i$$

Symbol	Meaning
$\sigma$	volume density: $\sigma (t)dt$ = probability that the ray stops in $[t,t+dt]$
$T$	transmittance: probability of reaching $t$ without a hit (visibility)
$T\sigma$	PDF of the stopping position; $1-T$ is the CDF
${\alpha }_i$	probability to stop within segment $i$ (opacity)
$\mathbf{c}$	(view-dependent) color

Intuition

Hits are independent, so survival probabilities multiply → exponential. A point contributes only if it is dense and visible; occlusion comes for free. The rendering is differentiable, so a scene can be learned from 2D images.

In 3D Gaussian Splatting the same compositing is used, but the elements are depth-sorted Gaussians and ${\alpha }_i$ = learned opacity × projected 2D Gaussian.

Appears in

• Lecture 7, Volume rendering
• Lecture 7, Transmittance
• Lecture 7, Discrete formula
• Lecture 7, NeRF vs. 3DGS
• Lecture 7, UniSurf: volume → surface rendering
• Lecture 8, Image formation: 3DGS uses the same compositing

Related concepts

• Gaussian Splatting
• Neural Field
• Positional Encoding
• Inverse Rendering
• Occupancy Field