Photometric Stereo (shape-from-shading with multiple light directions) L1-098
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
K images are captured under known directional light L_k, same camera pose. Under Lambertian reflectance, I_k(u,v) = rho(u,v) * max(0, n(u,v) . L_k); stacking gives I_stack = N * L^T where N = rho * n is the albedo-modulated normal. Least-squares gives N_hat; Poisson integration converts normals to a height field.
L-DAG
Well-posedness W
- Existence:
- true
- Uniqueness:
- true
- Stability:
- conditional
- κ:
- 400
With K >= 3 non-coplanar light directions and Lambertian BRDF, the linear system has rank 3 and is well-conditioned. Degrades for specular highlights (outlier fraction), cast and attached shadows (reduced effective K), non-Lambertian materials; robust methods (RANSAC, rank-3 SVD, learned PS) handle outliers up to ~30% specular fraction.
Solvability C
- Solver class:
- Woodham least-squares, robust PCA rank-3, SVD / SFS, learned (PS-FCN, CNN-PS, DeepPS)
- Convergence rate q:
- 2
- Complexity:
- O(H * W * K) for linear; O(N_iter * H * W * K) for robust PS