3D Gaussian Splatting (3DGS) — explicit real-time rendering L1-083
Unclaimed Principle — open for contribution
This Principle is declared in the catalog but has no reference solver, no pinned dataset, and is not registered on-chain. There is no reward pool. Submitting a cert against this Principle today will record the cert for reproducibility but pay zero PWM.
To claim it as a Bounty #7 contribution: open a PR adding (1) a reference solver, (2) ≥1 dataset pinned to IPFS, (3) updates to the L3 manifest with dataset CIDs. After verifier-agent triple-review, the founders' 3-of-5 multisig signs PWMRegistry.register() and the Principle becomes mineable.
Forward model E
C(r) = \sum_{i=1}^{N_visible} T_i \cdot alpha_i_2d \cdot c_i, T_i = prod_{j<i} (1 - alpha_j_2d), alpha_i_2d = alpha_i \cdot exp(-0.5 (r - mu_proj_i)^T Sigma_proj_i^{-1} (r - mu_proj_i))3DGS represents a scene as a set of N anisotropic 3D Gaussians parameterised by center mu_i, covariance Sigma_i, RGB color c_i (or spherical-harmonic coefficients), and opacity alpha_i. Rendering projects each Gaussian onto the image plane as a 2D Gaussian, sorts by depth, and alpha-composites per pixel. The explicit primitive set supports real-time rasterization and direct differentiation.
L-DAG
Well-posedness W
- Existence:
- true
- Uniqueness:
- false
- Stability:
- conditional
- κ:
- 4000
Non-convex parameter fitting. Unique 3D Gaussian cloud only up to primitive-reparameterization (splitting one Gaussian into two overlapping ones yields identical rendering). Stability risks: pose error introduces baked-in floaters; aggressive densification produces over-opaque pancake Gaussians (mitigated by Mip-Splatting / 2DGS).
Solvability C
- Solver class:
- Adam on Gaussian parameters with custom CUDA rasterizer; gradient-driven densify + prune; follow-ups: Mip-Splatting (anti-aliased), 2DGS (surface-aligned disks), SuGaR (mesh extraction)
- Convergence rate q:
- 1.8
- Complexity:
- O(N_projected * N_rays) per training iteration, ~100-1000x faster rendering than NeRF