Photon primitives / a connected statistical schematic

Where the bright rims, noise and work come from.

The same estimator can draw a rim—or a cloud of noise.

Both images sample the same random centers and radii. The left reuses each complete sphere shell across every pixel; the right draws independent shells at each pixel. Misses contribute zero and still count in the average.

Complete samples / pixelRadius variationShared display exposureGrazing cutoff |cos θ|

The displayed scalar is an orthographic projection of unit energy spread uniformly over a sphere. Its bright rim comes from the Jacobian, not from a BSDF. Correlation changes the spatial pattern of error; it does not change the per-pixel expected value.

—Shared image · area sum ≈

—Independent image · area sum ≈

—Continuous retained energy / draw

Model, equations and interpretation

Iₜ(ρ) = 1 / [2πt √(t² − ρ²)],  0 ≤ ρ < t
∫ Iₜ(ρ) 2πρ dρ = 1

The nonzero cutoff discards |cos θ| below the selected value. A complete sphere then retains exactly 1 − cutoff energy; it is not renormalized. The two image sums are finite pixel quadratures, so they fluctuate and can miss a thin rim. Exposure is a common display transform. These are projected shell samples, not native radiance renders.

A near-tangent ray squeezes area into a bright rim.

Move a camera ray toward tangency. The two sphere crossings approach each other while the projected Jacobian collapses. A cutoff limits the displayed tail by dropping a real part of the estimator.

Sphere radius tCamera-ray offset δGrazing cutoff |cos θ|

The geometry is a meridian of a three-dimensional sphere. The weight plot uses relative impact δ/t and a logarithmic vertical scale. The gold marker is the chosen ray; the dashed region is the explicitly rejected grazing band.

—Projected Jacobian J

—Raw shell intensity

—Selected ray / cutoff

Model, equations and interpretation

J(t, δ) = t √(t² − δ²)
t = δ + ε, δ > 0  ⇒  1/J ∼ ε⁻¹ᐟ²

The square-root tail describes nondegenerate tangency at fixed δ > 0. At δ = 0, shrinking t instead gives J = t², a different limit. Misses are zero; an exactly tangent ray is a measure-zero boundary where the ideal pointwise shell density diverges. A finite pixel footprint and a pointwise cutoff are different regularizations.

The bound finds candidates. The primitive decides the hits.

This discrete work ledger separates support construction, bounding-box candidates and actual disk hits. Compare constructing a support once for the view with constructing it again for every camera query.

Retained supportsProjected radius · pixelsQuery-grid scale

Each circle is one sphere’s projected support on the query grid. Warm cells entered a bounding box but missed the true disk; mint cells hit. Overlap adds work because one query may encounter several supports. Larger footprints increase candidate work even when the number of supports is unchanged.

—Support constructions

—Bounding-box candidates

—True support hits

Model, equations and interpretation

Q = grid width × grid height
Retained construction count = N
Per-query reconstruction count = NQ

These are exact operation counts for the stated toy schedules, not renderer timings, GPU occupancy or a prediction of native speedup. Both schedules inspect the same supports. Bounds are axis-aligned boxes; a different hierarchy, batching scheme, cache scope or query distribution changes this ledger. Native algebraic conditioning and radiance variance are separate costs.

A complementary density covers the troublesome region.

Integrate g(x) = 1 over [0,1]. Draw one sample from an equal mixture q = ½(p₁ + p₂) and return 1/q(x). A uniform partner covers the endpoint; a second density with the same zero leaves it exposed.

Density power kTail test: weight exponent αTail test: density exponent β

Density plot: gold p₁ = (k+1)xᵏ, mint p₂, white mixture q. Weight traces: gold flat and mint balance on exactly the same x values and technique choices. Switching weighting changes the readouts, never the samples. The separate α/β test describes a general endpoint power law.

—Exact expectation · both

—Selected weights · second moment

—Frozen samples · selected mean

Model, equations and interpretation

Equal technique probability αᵢ = ½; q = ½(p₁ + p₂)
Wflat = 1 / pchosen(x); Wbalance = 1 / q(x)
E[Wbalance²] = ∫ 1 / q(x) dx
E[Wflat²] = ½∫1/p₁ dx + ½∫1/p₂ dx
p(ε) ∝ εᵝ, W(ε) ∼ ε⁻ᵅ
Mean finite: β − α > −1
Second moment finite: β − 2α > −1

These are two unbiased one-sample estimators on a common dx measure. Flat weighting averages independently unbiased strategy estimates; balance uses the mixture denominator. With a uniform partner q ≥ ½, balance weights are bounded by 2, but flat weighting still has an infinite second moment for k ≥ 1. With p₂ = p₁, both retain that endpoint divergence. This is not an unweighted sum of two stratified estimates, and MIS is not a universal cure when all densities share the same bad region.