A local algebraic test
Photon primitives / before the render
Which strategies cover each other’s blind spots?
Choose two or three exact primitives. Probe their actual density formulas on the same emitter and camera ray, then see which extra strategy covers the expensive regions.
Three selected. Remove one before adding another.
Payload covered · finite probes
Minimum q × ray length
Density second-moment proxy / I²
Supplied cost × proxy / I²
What would a small allocation to another strategy do?
Each candidate is evaluated where the current mixture contributes most to the error-weight integral. A value above 1 reduces this density proxy at equal work; the cost threshold changes when you supply different work estimates.
| Candidate | Error-weighted p / q | Cost threshold c / C | Local prediction |
|---|
Native densities, payload and what this test can establish
One common measure
The square source is x(u,v) = (L(u − ½), L(v − ½), 0), with u,v in [0,1]. The camera point is y(s), s in [−1.3,1.3]. Every density below is expressed in ds du dv. Source selection probability is 1 and uniform area sampling contributes A · pdfArea = 1.
r = y − x, t = |r|, ω = r/t, μ = n·ω pω = μ/π, pₜ = σ exp(−σt), A = L² Puv = A |n·d| pω pₜ / t² Put = |(eᵤ × ω)·d| pω / t² Pvt = |(eᵥ × ω)·d| pω / t² Parb = A/(2R) |(ê × ω)·d| pω / t² Psphere = pₜ |ω·d| Pcone = |μ(ω·d) − n·d| / t Pdisk = |(n × ω)·d| / (2πt(1 − μ²))
eᵤ and eᵥ are the source edges of length L; ê is the selected in-plane direction. The fixed arbitrary plane samples an offset uniformly over the circumscribing radius R = L/√2. The source is diffuse, the supported hemisphere has μ > 0, and there are no occluders. The exact disk axis is a coordinate singularity; it is reported if encountered. These are the exact-plane and directional native formulas; finite-width blurred curves have a different target and are excluded.
Useful evidence, with a defined target
f = A (μ/π) σ exp[−σ(t+s+1.3)] / t² I = ∫ f ds du dv q = Σ aᵢ Pᵢ; H = ∫ f²/q ds du dv ν ∝ f²/q dH/dε at 0 = H [1 − Eν(p/q)] Cε Hε decreases if Eν(p/q) > c/C
The payload is an explicitly scaled diffuse single-scatter model, with camera attenuation measured from the start of the segment. It is not native absolute radiometry. No density is renormalized over hits: misses remain part of the sampling process.
H is a diagonal density second-moment proxy. A native cone or sphere draw can produce two roots; their cross terms and shared-prefix covariance are not included. The finite midpoint quadrature is a diagnostic, not an infinite-variance test or a renderer benchmark. Changing the scene or payload can change the ranking.
At a fixed supported event, Pᵢ = cᵢ|vᵢ·d|. A rank-deficient set of normals has a shared-zero ray direction. A full-rank set gives a local bound q ≥ √λmin(B), with B = Σ(aᵢcᵢ)²vᵢvᵢᵀ. This lower bound applies at that event, not over all paths, support boundaries or emitter positions.