Collision, track length & ratio tracking

Open View → Optics & primitive laboratory → Volume estimators. This is a native scalar transport experiment. It samples real collision histories and running estimates; the plotted dots are the events actually used by the last sample. Hidden tabs stop sampling. Pause retains the accumulated state; Reset returns to deterministic independent random streams.

The target on a finite ray segment is

I = integral from 0 to D of q(t) T(t) dt
T(t) = exp(-integral from 0 to t of sigma(s) ds).

q is a nonnegative scalar source, sigma is positive extinction, and D is a known boundary. Constant, smooth heterogeneous and layered extinction profiles have analytically known optical depths. A separately evaluated, refined Simpson quadrature supplies the reference integral. The panel reports its refinement delta as well as the Monte Carlo estimates. Spatially varying q prevents an otherwise constant-weight example from hiding important terms.

Collision: the extinction belongs to the sampling density

Delta tracking samples the first real collision by thinning Poisson candidates of rate M >= max sigma. A rejected candidate continues in the same medium. The real-collision density is p(t) = sigma(t) T(t), with an escape atom of mass T(D). A collision contributes q(t)/sigma(t); an escape contributes zero.

This is the concrete f/p cancellation: T cancels because the sampling law contains precisely that same transmittance. The weight is constant only if q/sigma is constant. Even then the finite-domain escape event generally leaves variance. Multiplying another transmittance after this cancellation changes the expected result. Introducing more path coordinates alone supplies no such proof.

Track length: score the surviving interval

Sample the same first-collision law independently, but score

integral from 0 to min(tau,D) of q(t) dt.

The expectation of its indicator is P(tau > t) = T(t), so interchanging the integral and expectation recovers the same I. The lab integrates its source profile analytically along the surviving interval. It does not quadrature-sample that interval or quietly omit escaped paths. In a thin homogeneous medium this often has far less variance than scoring rare collisions; it is not universally better in optically thick media.

Ratio tracking: retain every null-event factor

Sample U uniformly from [0,D]. Draw Poisson candidates on [0,U], form

R(U) = product over candidates i of [1 - sigma(t_i)/M],
score = D q(U) R(U).

The Poisson product identity gives E[R(U) | U] = T(U). Integrating over U again gives I. The factors lie in [0,1] because the supplied majorant bounds the entire analytic profile. Raising it changes event cost and variance, not the target. There is no biased iteration cap or early product cutoff. This is ordinary ratio tracking, not residual-ratio tracking or a general signed-density method. The violet staircase depicts the successive product weights.

For the reference definitions and null-collision context, see PBRT's transmittance chapter and its volume-transport references. The three scalar expectation arguments above specialize those ideas to the explicit target implemented here.

Read the error and cost together

Each method has an independent random stream and running mean, variance and standard error. The displayed 1.96 SE is an asymptotic interval scale, not a finite-sample guarantee or proof that a renderer is unbiased. Density calls per sample and measured CPU sampling time distinguish work from sample count. The reference has its own displayed quadrature refinement error.

Suggested experiments:

• Set homogeneous extinction to 0.07, length 4 and source variation 0. Rare collision scores contrast with almost-full surviving track lengths.
• Raise extinction to 1.5. Nearly every ray collides; constant collision scores become competitive while track lengths still vary.
• Choose layered fog and source variation 0.8. The changing q/sigma exposes why a flat-weight argument cannot simply be transferred to heterogeneity.
• Raise the majorant from 1 to 4 times its required bound. The target stays fixed while null-event counts and ratio-product behavior change.

PrismVolumeEstimatorTest compares means with the common quadrature, checks homogeneous closed-form mean/variance and escape mass, checks optical depth against independent integration, and rejects an invalid majorant. It also tests the ratio product at a fixed endpoint. These tests cover this controlled lab, not all scene transport.

Scene studies and current renderer boundaries

For 105–106 the colours differ, so equal radiance values would not be equal luminance-weighted flux. The scene normalizes both source area and the colour's linear RGB luminance. These are useful spatial comparisons, while moving the camera or editing one source enables further study. Positions and colours are different across the three sources; this is not a variance ranking from an isolated single-source ablation. The earlier 80–82 size/profile matrices remain available.

.\Showcases.ps1 -Scene 103
.\Showcases.ps1 -Scene 104 -Research
.\Showcases.ps1 -Scene 105
.\Showcases.ps1 -Scene 106 -Strategy 13

The 103–104 GPU scenes demonstrate the existing spatial density implementation. They do not certify its null-collision weighting: the support audit records incomplete candidate continuation, homogeneous shadow attenuation, and a volume-NEE density-measure issue. The new lab contains the complete bounded scalar tracking loops and provides an independent teaching/test reference; it does not change those GPU or UPBP call paths. PP31 still requires a common homogeneous medium, so the launcher does not pair it with the heterogeneous chamber. CPU throughput and GPU capability are separate questions.

A production transport fix needs the same survival tests within each actual backend, plus heterogeneous shadow paths, spectral weighting and matched source measures. Full manifold exploration remains a separate, low-priority solver project: the existing mode 8 is a specular-aware mutation prototype, not a constraint Newton solve.