Lesson 03

The forward pass: volumetric compositing

How millions of blobs become a pixel — and why it's the same equation as NeRF.

To render, 3DGS projects each Gaussian onto the screen (the EWA math from lesson 02, reused as a layer), sorts them by depth, and composites them front-to-back. A pixel's color is:

$$C \;=\; \sum_{i} c_i\,\alpha_i' \prod_{j<i} \big(1-\alpha_j'\big), \qquad \alpha_i' = \alpha_i\,\exp\!\Big(-\tfrac12 (\mathbf x-\boldsymbol\mu_i')^{\!\top} \Sigma'^{-1} (\mathbf x-\boldsymbol\mu_i')\Big)$$

The product $\prod_{j<i}(1-\alpha_j')$ is the transmittance $T_i$: how much light "survives" everything in front. Each Gaussian contributes with weight $w_i = \alpha_i' \, T_i$: a lot if it is opaque and unoccluded, little otherwise.

● α₁ 0.75 ● α₂ 0.55 ● α₃ 0.85

Fig. 1 — interactive. A ray passes through three Gaussians. Top: the scene (drag the blobs to reorder them!). Middle: the transmittance $T(s)$ collapsing at each blob. Bottom: the color accumulating along the ray. Note the asymmetry of occlusion: bring the blue blob in front of the others and raise its α — the rest all but vanish from the final color.

The bridge to NeRF — this is exactly NeRF's volume rendering equation, discretized. NeRF and 3DGS optimize the same functional, with two opposite discretizations of the radiance field: NeRF is Eulerian (a continuous field in the weights of an MLP; you query space with ray marching — hundreds of queries per pixel), 3DGS is Lagrangian (the field is carried by explicit particles: one sort + blending, real-time). All the practical differences follow from this choice of coordinates: speed, editability, and the ability to do LoD/streaming per region of space.

Why it matters for training

Every operation here — Gaussian footprint, product of transmittances, weighted sum — is differentiable in the parameters $(\mu_i, \Sigma_i, \alpha_i, c_i)$. This is what allows lesson 04 to exist: the gradient of the photometric loss can flow back from the pixel to every single Gaussian, through this formula.