Photon Primitives C++ · Research

Let ε measure distance from a troublesome configuration. Suppose the complete sample weight grows as |W| ≈ Aε^−α, while the probability density of that event behaves as p(ε) ≈ Bε^β, with β > −1. Then the local q-th absolute moment is

E[|W|^q] ∝ ∫₀^ε₀ ε^{β − qα} dε  ·  finite exactly when qα < β + 1.

Equality gives logarithmic divergence. A stronger inequality in the other direction gives power-law divergence. Thus the order of 1/J alone is insufficient: acceptance, sampled density, transport factors and cancellations all belong in W and p. A degeneracy reached only through a small set of sampled configurations can behave very differently from one reached with substantial density.

“Each sphere has a singular silhouette” is a statement about its spatial support. “A particular pixel samples that silhouette” is a statement about random primitives. They use different coordinates and measures. A continuous silhouette also does not imply that a discrete pixel lands on an exact infinity; it usually samples a large but finite near-tangent value.

An exact singularity can have probability zero while its neighborhood still gives an infinite second moment. Conversely, an algorithm that places positive probability on a genuinely undefined or infinite contribution has changed the sampling problem. That requires an explicit treatment of the degenerate event; sharing otherwise identical samples does not create it by itself.

Scope. These are local asymptotic models. They assume a nonzero leading coefficient and no other dominant singularity. Numerical guards and finite precision truncate the implemented tails; clamping or discarding contributions can also change the target integral.

For a sphere with sampled radius t and camera-ray distance δ > 0 from its center, the two roots are s = s₀ ± √(t² − δ²). Its projected Jacobian is J = t√(t² − δ²). At t = δ + ε, two roots merge and W grows as ε^−½, provided the transport numerator remains nonzero. A regular radius density therefore gives a finite mean but a logarithmically divergent ideal second moment.

A cone has the same generic fold exponent when its visible roots merge. Whether that fold is actually reached depends on the sampled nappe and the camera segment. Its apex and polar limits are separate degeneracies. RadianceLab’s directional disk is a meridian half-plane swept about the light axis: its ordinary crossings have no quadratic fold. It can instead become ill-conditioned near the axis; sampling an area emitter can make those configurations statistically significant.

An exact illustrative projection. A uniform, unit-energy thin sphere shell viewed orthographically has radial intensity

I_R(ρ) = 1 / [2πR√(R² − ρ²)],   0 ≤ ρ < R.

The outer 1% of its radius carries 14.1% of its energy in 1.99% of its projected area. This shows why a thin bright rim can dominate perception despite finite total energy. It is a shell-projection model, not a RadianceLab measurement.

At fixed distance from that rim, its coefficient falls as R^−3/2, while projected area grows as R². A bigger sphere can therefore cost more without having a stronger singularity. Perspective, camera containment, free-flight sampling and the phase function alter the practical result. Cones and half-planes can also cover most of the frame.

Write F_j(π) for one primitive’s complete contribution to pixel j: zero on a miss, and the sum of all supported crossings on a hit. A shared batch gives L̂_j = (1/N)Σ_iF_j(π_i). Independent per-pixel batches replace π_i with π_ij.

With the same primitive distribution, N independent draws, fixed camera rays and matching weights, both constructions have the same per-pixel distribution. Their expected ordinary image MSE is therefore equal at that budget, when it is finite. Shared samples instead add

Cov(e_j, e_k) = Cov(F_j(π), F_k(π)) / N.

Broad coherent errors affect denoising, image averages and perception differently from pixel noise. Correlation can also help differences between neighboring pixels; it is not universally harmful. Independent shared batches across P passes reduce covariance by 1/P, but do not generally turn its normalized spatial pattern into white noise.

Freeze one finite primitive pool forever and additional passes only refine that pool’s conditional image. Camera jitter can improve pixel integration, but cannot remove the fixed pool’s photon-sampling error. Refreshing the pool introduces new photon information.

One extra tradeoff: if a pass uses one camera ray for N primitives, a fixed total budget B = NP buys only P independent camera samples. Under independent camera and photon sampling, the finite-variance decomposition is

Var(L̂) = (N/B) Var_r(E_π[F | r]) + (1/B) E_r(Var_π[F | r]).

The first term grows with batch size: silhouettes, depth of field and narrow caustics may need more camera samples even when primitive reuse is cheap. Equal passes, equal primitive counts and equal elapsed time answer different questions.

Compare densities for the same contributing path in the same measure. For a one-sample mixture with strategy probabilities a_i, the balance heuristic gives

[f / (a_ip_i)] × [a_ip_i / Σ_ka_kp_k] = f / Σ_ka_kp_k.

If one strategy’s density vanishes at a tangency while another has density bounded away from zero there, a bounded numerator no longer produces that pole. Merely being nonzero at each nearby point is weaker than having a positive lower bound. Shared zeros, missing path support, unbounded transport factors or singular configurations elsewhere can leave heavy tails. Counts replace selection probabilities appropriately in the multi-sample formulation; rejected draws still belong in the normalization.

A concrete cancellation test. Integrate f(x) = 1 on (0, 1). With p₁(x) = 2x and an equally likely uniform partner p₂(x) = 1, the mixture density q(x) = x + ½ keeps the sample weight 1/q below 2. But if both partners use 2x, then q(x) = 2x: the mean is still 1 while E[W²] = ∫₀¹ 1/(2x) dx diverges. The limiting rates and shared zeros settle the question; writing an informal “∞/∞” does not.

The published derivation, §6, places photon-surface intersections in standard path space and derives both discrete and continuous MIS. Its limitations, §9, explicitly show residual surface-to-medium singularities when the available strategies do not cover each other adequately.

Statistical cancellation does not eliminate root solves, alternate density evaluations or visibility queries. A low MIS weight may be known only after that work. For a finite-variance estimator, equal-time error depends on variance and cost per useful sample; for heavy-tailed cases, report multiple independent runs and convergence curves instead of trusting one apparently stable variance number.

Deng et al., Photon Surfaces (2019) provides the change-of-variables framework and MIS derivations. Figure 5 compares estimators at equal time and notes that MIS overhead can outweigh its variance improvement. Section 9 specifically identifies inefficient full-sphere work under strongly anisotropic scattering, proposing spherical sectors to concentrate effort. These are concrete foundations for the footprint hypothesis.

This follow-up separates those foundations into testable mechanisms: singularity order together with event density; per-pixel error versus spatial covariance; camera samples versus primitive samples; and useful crossings versus screen coverage. The shell projection and local moment thresholds above are explanatory derivations. They do not establish which mechanism dominates a particular native run—that requires matched settings and measured work.

For single scattering from a quad, use z = (s, u, v): distance s along the camera ray and normalized coordinates (u, v) on the emitter. Each exact technique supplies a crossing density P_i(z) per ds du dv. This includes emitter selection, sampling probabilities, Jacobian, valid branches and support.

RadianceLab’s directional densities start conditional on the sampled light point. Their conversion is P_i = A · p_select · p_area · p_i(s | y). For a uniformly sampled quad, A · p_area = 1. Plane densities already use the joint measure. Omitting this factor can make light size or source selection quietly change the estimator.

For allocations a_i ≥ 0 summing to one, write q(z) = Σ_ia_iP_i(z). Sample counts replace allocation fractions in a multi-sample balance weight. Do not renormalize each density over successful crossings: failed draws still count, and a sphere or cone can contribute more than one crossing.

Photon Surfaces, §6, derives the common-path MIS construction. Veach’s chapter on MIS separates the one-sample mixture from the multi-sample estimator; their variance formulas and sampling schedules should not be interchanged.

Fix a supported light/scatter pair (y, x), let ω = (x − y)/t and let d be the unit camera direction. Away from coordinate poles, each exact member has the form P_i = c_i|v_i · d|, where c_i contains its actual positive density factors.

Build the weighted Gram matrix B = Σ_i(a_ic_i)²v_iv_i^T. Then

q(d) ≥ √(d^TBd) ≥ √λ_min(B).

Rank below three exposes a viewing direction where every selected density vanishes at this event. Full rank provides a local density floor. Keeping the real coefficients matters: unit-normalizing the rows reveals geometric independence but hides an exponentially unlikely free-flight distance or a tiny emitter probability.

UT, VT and the arbitrary plane all have v perpendicular to ω. Adding another orientation can improve balance within that plane but does not fill its missing direction. Sphere, disk and cone span three orthogonal directions away from the axis; this explains a genuine geometric strength of their trio.

This is a local statement. The floor can approach zero as the light point, distance or support changes. A single well-conditioned event is not a proof of finite image-wide variance.

Any pair of these dot-product densities has rank at most two. Neither pair covers every possible camera direction. The useful question is where its missing direction falls relative to the scene.

For a light axis a = n, sphere + disk has a common zero along (n · ω)ω − n—the direction supplied by the cone. Sphere + UV instead has a common zero along ω × n—the direction supplied by the disk. These statements assume nonparallel n and ω and positive sampling factors. A camera can favor either pair.

Sphere + UT has another missing direction: the component of the chosen emitter edge perpendicular to ω. Changing the edge orientation moves that weakness. UV’s area factor and free-flight density, and UT/VT’s edge factors, additionally change the comparison with emitter size. The phase function and visibility decide whether the uncovered directions carry significant energy.

Different integrated coordinates are a design clue, not a quality score. Complementary Jacobians, adequate probability where the integrand matters, and affordable evaluation are the three tests. The paper’s single-scattering comparisons likewise change with light size and show that adding strategies can cost more than it saves.

Choose a small, fixed set of camera directions, emitter positions and ray distances. Evaluate all supported densities at each event, with quadrature weights ω_ℓ and an explicit transport payload f_ℓ. Before estimating a score, report any region with nonzero payload but zero combined density. A coarse grid cannot certify that no such region exists: also probe the analytic common-zero directions.

H(q) ≈ Σ_ℓ ω_ℓ f_ℓ² / q_ℓ  ·  I ≈ Σ_ℓ ω_ℓ f_ℓ.

For an ordinary one-sample importance estimator, H is its second moment and H − I² is its variance. For native multi-root primitives, H is only the diagonal second-moment term: paired roots and shared-prefix covariance also contribute. Use it as a density-based screening score, not a renderer benchmark.

There is a useful analytical addition test. Add a candidate density p with a small allocation ε, so q_ε = (1 − ε)q + εp. Define the normalized error-weight measure ν(z) ∝ f(z)²/q(z). Where these integrals and derivatives exist,

H′(0) = H(q) [1 − E_ν(p/q)].

The candidate improves this score when it supplies density in the places that currently dominate it. A large global distance between two PDFs says less: they may differ mostly in dark regions.

If the mean work changes as C_ε = (1 − ε)C + εc, the proxy C_εH(q_ε) initially improves exactly when E_ν(p/q) > c/C. This assumes no new shared overhead; real MIS adds density evaluations, branching and memory traffic. Include those costs in a measured model before predicting elapsed-time gains.

Check limits as well as grid values. If q ≈ ε^γ, f ≈ ε^η and the base integration measure contributes ε^βdε, the local second-moment integral is finite only when 2η − γ + β > −1. Equality is logarithmic divergence. A finite pilot can miss such a tail even when every sampled score looks healthy.

These curves sweep one photon coordinate and use a finite-width kernel to reach the camera ray. Their tangent factors are |t̂_i × d|, rather than the surface factors |n̂_i · d|. Different tangent directions can protect one another from grazing configurations, but density factors, curvature and kernel width remain part of the comparison.

At finite radius, different curves can deposit at different nearby positions and have different biases. Sharing the same radius cancels a common normalization in some weights; it does not make their blurred integrands identical. A finite-radius combination should therefore retain an explicit bias qualification.

A controlled baseline is a fixed blend of an exact-family MIS estimate and the existing curve-family estimate: L̂ = (1 − λ)L̂_exact + λL̂_curve,r. Its bias is λb_curve,r. It can converge to the unblurred target as the radius shrinks with sufficient samples, provided both groups cover the same source and path class. This is a finite-radius composite; it does not claim cross-group MIS weighting.

VCM’s implementation derivation includes the merge-acceptance area πr² and sample counts in the path-density bookkeeping. UPBP’s extended MIS, §6, goes further: it combines ordinary path estimators with extended-space estimators that can have different finite-radius biases.

Let x be the common path and y the technique’s extra kernel-offset variable. UPBP’s extended balance weights use w_i(x) ∝ n_i/κ_i(x), with

κ_exact(x) = f(x)²/p_i(x),   κ_kernel(x) = ∫ f_i(x,y)²/p_i(x,y) dy.

The kernel integral accounts for extra sampling variance. Defining the common path, offset map, branches, support and joint density is the work required before curved primitives can enter that construction. The theorem retains bias bounds; it does not turn a finite merging radius into an unbiased renderer.

A useful approximation to derive and test: if transport, density and geometry vary slowly across a locally straight kernel, κ can simplify using V_eff = 1/∫K². For a normalized uniform one-dimensional box, V_eff = 2r. This motivates a tube-acceptance factor in a cross-family weight. Curvature and tangency are precisely where that approximation needs scrutiny.

All cross-evaluated weights must refer to the same canonical camera/light path. Evaluating each member’s weights at its own displaced curve point is a different construction. A practical extended-MIS implementation can use approximations, but should identify them, compare shrinking-radius limits and preserve full support rather than assuming that equal kernel widths settle the derivation.