19-EFT.WP.Methods.SynthData v1.0 | Chapter 6 — Physics/Simulation-Driven Synthesis & Scene Graphs (Domain Randomization / SCM)

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Chapter 6 — Physics/Simulation-Driven Synthesis & Scene Graphs (Domain Randomization / SCM)

I. Scope & Targets

1. Goals
   • Build controllable, auditable, scene-level synthetic data grounded in physics/simulation and structural causal models (SCM). Unify geometry, dynamics, materials, illumination, camera/sensor chains, and time base within a scene graph G=(V,E).
   • Use domain randomization and SCM do(•)-style interventions to generate representative samples while preserving physical constraints, units and dimensions, and enforcing T_arr and tau_mono/ts consistency.
2. Inputs
   Canonical schema SRef, reference set D_ref, scene/physics specification SceneSpec, SCM graph and equation family F, randomization policies and bounds, sensor model SensorSpec, contracts and SLOs.
3. Outputs
   Scene graph G, parameter sample set Theta, synthetic data D_syn, metrics & audit reports, and manifest.synth.scene.*.
4. Applicability
   Scenes with strong physical constraints, geometric precision, and modelable sensor chains. For purely high-dimensional texture fitting, combine with Chapter 5 (deep generators).

II. Terms & Symbols

• Scene & physics: G=(V,E), v ∈ V, state(v), theta_phys, u (disturbances/exogenous inputs), C (constraint set), Pi_C(•) (constraint projection).
• Simulator: S(theta,u) -> x (world state/intermediates), H(x) -> y (sensor readout).
• Randomization: theta ~ p(theta | bounds, policy), u ~ p(u), coverage covg_hcube.
• Causality: SCM = { X_j = f_j( PA_j, U_j ) }, intervention do(X_k = a).
• Sensors: PSF/OTF/MTF, noise eta, readout model y = H(x) + eta.
• Time & paths: tau_mono, ts, gamma(ell), T_arr, delta_form, offset/skew/J.
• Metrology: unit(x), dim(x), check_dim(expr).

*III. Axioms P406- **

• P406-1 (Physical Traceability): Every observable must arise from a declared physics/simulation pathway S ∘ H, with unit/dim stated.
• P406-2 (Graphical Consistency): The scene graph G and the SCM parent sets PA_j must agree and be acyclic (or supply an explicit dynamic solve order).
• P406-3 (Explicit Constraints): Register geometric/dynamic/energy/topology constraints in C; every sample is corrected by Pi_C or rejected.
• P406-4 (Randomization Coverage): The support and sampling coverage of p(theta) must be audited and persisted.
• P406-5 (Time Base First): Evaluate internally on tau_mono, publish via ts mapping; record offset/skew/J.
• P406-6 (Dual Arrival Forms): Whenever path propagation is involved, log both formulations and delta_form.
• P406-7 (Sensor Closure): H must include PSF/OTF/MTF and a noise family, aligned with Methods.Imaging.
• P406-8 (Reproducibility): Persist seed/rng/solver, timestep, and tolerances; declare non-determinism sources and variance.
• P406-9 (Synthesis Accountability): Any post-processing (textures, shading, stylization) must not violate unit/dim or physical constraints.
• P406-10 (Compliance Mode): For sensitive entities (people/vehicles/scenes), declare masking/abstraction policies at the SCM or policy layer.

*IV. Minimal Equations S406- **

• S406-1 (Randomized Sampling)
  theta ~ p(theta | bounds, policy)，u ~ p(u)，x = S(theta,u)，y = H(x) + eta，eta ~ p_eta。
• S406-2 (Constraint Projection)
  x' = Pi_C(x) = argmin_{z ∈ C} d(z, x); residual res_cons = d( x', x ).
• S406-3 (Conservation Check)
  Linear conservation: res_lin = || A x' - b ||_2; require res_lin ≤ tol_cons.
• S406-4 (Pose/Chain Kinematics)
  Rigid chain composition T_world,obj = ( ∏_k T_k ); project to image plane p = K [R|t] P.
• S406-5 (Sensor Readout)
  y = ( PSF * radiance )(•) + eta, with PSF convolution and eta comprising shot/read/quantization components.
• S406-6 (Dual Arrival Forms)
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ),
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |。
• S406-7 (Causal Intervention Synthesis)
  X_j := f_j( PA_j, U_j ); intervention do(X_k=a) replaces the corresponding equation and generates from the modified graph.
• S406-8 (Alignment Objective)
  min_{policy} D( p_syn(y; policy) || p_ref(y) )，D ∈ {W1, KL, MMD}。

V. Metrology Flow M40-6 (Physics/Scene Closed Loop)

• Ready
  Specify G and the SCM, constraint set C, p(theta), p(u), SensorSpec, metrics, SLOs, and compliance manifest keys.
• Binding
  Validate unit/dim via SRef; assemble the S ∘ H pipeline; register tau_mono and the ts mapping.
• Sampling
  Use LHS/Sobol or similar coverage designs to draw Theta; record seed/rng and coverage covg_hcube.
• Simulation
  Compute x = S(theta,u), y_raw = H(x); write time/path fields and both T_arr forms.
• Constraints & Checks
  Compute x' = Pi_C(x); evaluate res_cons/res_geom; reject or reproject non-compliant samples.
• Alignment & Reweighting
  balance_distribution( ref, y_raw ) -> map; apply to samples or update the randomization policy.
• Noise & Sensor Closure
  Inject eta ~ p_eta to obtain y; verify PSF/OTF/MTF and readout consistency.
• Evaluation
  Report physical residuals, geometric errors, W1/MMD, coverage, and task utility; record offset/skew/J and delta_form.
• Persist & Publish
  Emit manifest.synth.scene, sign and freeze; archive failed samples and complete audit trail.

VI. Contracts & Assertions C40-6xx

• C40-601 (Geometric Closure): res_geom ≤ tol_geom; no self-intersections or illegal collisions in joints/chains.
• C40-602 (Conservation Residuals): res_lin ≤ tol_cons; conservation of energy/mass/charge meets thresholds.
• C40-603 (Sensor Consistency): MTF(f) ≥ mtf_min(f) over the specified band; noise_params within calibrated ranges.
• C40-604 (Coverage): covg_hcube ≥ covg_min; parameter extreme frequencies ≤ freq_max.
• C40-605 (Causal Consistency): SCM is acyclic or has a stable solver; do(•) samples move distributions in the expected direction (monotone or counterfactual checks pass).
• C40-606 (Time/Arrival): non_decreasing(tau_mono), J ≤ J_max, delta_form ≤ tol_Tarr.
• C40-607 (Units & Dimensions): check_dim(expr)=true.
• C40-608 (Performance & Stability): latency_ms_p99 ≤ SLO_sim, oom_rate ≤ oom_max.
• C40-609 (Reproducibility): seed/rng/solver/steps persisted and replayable.

VII. Implementation Bindings I40-6*

• design_scene_graph(SceneSpec) -> G
• define_scm(graph, equations) -> SCM
• sample_theta(prior, bounds, policy, n, seed) -> Theta
• simulate_scene(G, theta, u, solver, dt) -> x
• apply_constraints(x, rules) -> x'
• sensor_render(x', SensorSpec) -> y_raw
• inject_noise(y_raw, noise_spec) -> y
• align_timepath(y, ref) -> y' (write offset/skew/J, T_arr, delta_form)
• balance_distribution(ref, y', method) -> map
• evaluate_physics(y', contracts) -> report
• emit_scene_manifest(artifacts) -> manifest.synth.scene
• Invariants: reproducible(seed); sum(weights)/N ≈ 1 (if reweighting); delta_form ≤ tol_Tarr; unit/dim conserved; latency_p99 meets SLO.

VIII. Cross-References

• This volume: Chapter 3 (schema binding & units/dimensions), Chapter 5 (deep generation synergy), Chapter 12 (fidelity/utility evaluation), Chapter 13 (release freeze).
• Methods.Imaging v1.0: Chapters 5/6/7/8/9/13 (PSF/OTF/MTF, sampling & noise, geometric calibration, time/path gating).
• Methods.Cleaning v1.0: Chapters 5/6/10 (time base, arrival time, contract-driven release).
• Methods.CrossStats v1.0: Chapters 7/9/14 (drift alignment, calibration transfer, SLO & audit).

IX. Quality Metrics & Risk Control

1. Core SLIs
   res_geom, res_cons, W1/MMD, covg_hcube, latency_ms_p99, oom_rate, delta_form, J, noise_fit_R2.
2. Common risks & mitigations
   • Insufficient coverage → expand bounds/policies; adopt adaptive randomization or importance sampling.
   • Geometric/dynamic instability → shorten dt, switch to stable solvers, strengthen constraint projection.
   • Sensor mismatch → re-calibrate PSF/MTF and noise family, or apply calibration transfer after H.
   • Arrival/time drift → re-run align_timepath, constrain J, and review delta_form.
   • Compute bottlenecks/OOM → tiled/hierarchical rendering, cache intermediates, lazy-load textures/geometry.
   • Causal distortion → audit SCM structure; add holdout interventions or counterfactual checks.

Summary

• This chapter defines a physics/SCM-driven synthesis loop on G=(V,E): axioms P406-*, equations S406-*, flow M40-6, contracts C40-6xx, and bindings I40-6*.
• Deliverables include the scene graph, parameter samples, synthetic data, quality & audit reports, and manifest.synth.scene, providing a reproducible, traceable, and physically consistent foundation for downstream evaluation and release.