Train and render a 3D Gaussian Splatting scene

A small red Gaussian sits close to the camera, a bigger blue one sits farther back and slightly off to the side. Drag the query pixel across the scene: right where the red one projects, the pixel is almost pure red; drift toward the blue one's projected center and it shifts, tempered by however much red is still bleeding through in front of it.

The full pipeline, chained: project (ch. 2's camera, applied to each Gaussian's center, plus ch. 5's world-to-screen spread shrink), then composite (ch. 4's falloff-weighted opacity, ch. 5's nearest-first ordering) —

• $\mu$ — a Gaussian's world-space center; $(x, y)$ — where it projects to on screen.
• $\sigma_{\text{world}}, \sigma_{\text{2D}}$ — a Gaussian's spread before and after projection.
• $T_i$ — transmittance remaining before Gaussian $i$; $T_{\text{final}}$ — what's left after every Gaussian, painted with the background.

1. Project both Gaussians through the same pinhole camera as chapter 2

   • The near, on-axis Gaussian ($\mu = (0,0,5)$) projects to the principal point $(50, 50)$ with $\sigma_{\text{2D}} = 0.1 \times 100/5 = 2$.
   • The farther, off-axis one ($\mu = (0.3, 0, 10)$) projects to $(53, 50)$ with $\sigma_{\text{2D}} = 0.3 \times 100/10 = 3$.
2. Composite at the near Gaussian's own projected center, (50, 50)

   Weight is exactly 1 there, so the near Gaussian's contribution is just its opacity: $0.6$. Transmittance remaining: $0.4$.

3. The far Gaussian and the background split what's left

   3 units away in a spread-3 blob, the far Gaussian's weight is $e^{-0.5} \approx 0.6065$, giving alpha $\approx 0.9\times0.6065\approx0.5459$ and contribution $\approx 0.4 \times 0.5459 \approx 0.2183$ (all in blue, since the far Gaussian is pure blue). The remaining transmittance $\approx 0.4\times(1-0.5459)\approx0.1816$ goes to the white background.

   Summing each channel — near Gaussian's contribution, far Gaussian's contribution, background's contribution:

   • Red: $0.6 + 0 + 0.1816 \approx 0.782$
   • Green: $0 + 0 + 0.1816 \approx 0.182$
   • Blue: $0 + 0.2183 + 0.1816 \approx 0.400$

   Final pixel $\approx (0.782, 0.182, 0.400)$ — mostly red, tinted by blue, softened by background white.

Same scene, formula values spelled out live: watch $\sigma_{\text{2D}}$ shrink for the farther Gaussian (it's already smaller on screen despite being the physically larger one, $\sigma_{\text{world}} = 0.3$ vs. $0.1$) and watch each Gaussian's contribution bar move as the query pixel crosses between them.

Two more query pixels through the same rendered scene:

1. Query pixel (53, 50) — the far Gaussian's own projected center

   Now the far Gaussian's weight is 1 there instead: its raw alpha is $0.9$, but it composites second (nearest-first), so its actual contribution depends on how much the near Gaussian (weaker here, off its own center) left behind.

2. Query pixel (60, 50) — past both projected centers

   Both Gaussians' weights have decayed further from their falloffs; the background's white starts to dominate the composited pixel, since neither Gaussian has much opacity left to contribute out there.