24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 12 — Noise, Cleaning, and Robustness

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Chapter 12 — Noise, Cleaning, and Robustness

One-Sentence Goal
Establish an end-to-end convention—from signal-domain cleaning → complex/filtration robustification → stable invariant publication—so the Topological Atlas remains stable, auditable, and replayable under noise and outliers.

I. Scope and Objects

1. Inputs
   • Raw data: scalar / vector / orientation fields f(x), v(x), n(x) (n ∈ S^1/S^2); trajectories/worldlines Γ; event streams.
   • Noise & outlier models: noise ∈ { Gaussian, Poisson, speckle, salt&pepper, bias, drift }, with outliers.
   • Existing products: K(τ), D (persistence diagrams), Atlas with T_ab, ψ_a (Chs. 7–9).
2. Outputs
   • Cleaned data & robust complexes: f' , n' , Γ' , K'(τ) , D'; robust parameters σ, λ, TV, iters.
   • Robustness evaluation: d_B(D, D'), η_stab (perturbation response), u_clean.
   • Manifest: manifest.topo.cleaning (methods, hyperparameters, evidence, two-form gaps, signatures).
3. Boundaries & constraints
   • Orientation fields preserve unit norm: ||n'|| = 1; enforce non_decreasing(τ); keep Σ_a ψ_a ≡ 1 (Ch. 9).
   • Parallel forms: clean→build vs build→simplify must be compared and recorded.

II. Terms and Variables

• Noise scale & metrics: σ (strength), SNR/PSNR, ε (perturbation radius), p_out (outlier rate).
• Cleaning operators: Gaussian G_σ, heat ∂_t f = Δ f, anisotropic diffusion ∂_t f = div(g(|∇f|) ∇f), TV denoising, bilateral B_σ, NLM, graph Laplacian L = D − W.
• Circular/spherical means: circmean on S^1, projmean on S^2 (average then renormalize).
• Robust losses: ρ_Huber(r;δ), ρ_Cauchy(r;c); weights w = ψ(r).
• Two-form gap: delta_form_clean = dist_intervals( D(clean(data)) , simplify(D(raw)) ).
• Reference conditions: RefCond = { σ, model, TV, λ, iters, grid, τ_policy }.

III. Axioms P912-*

• P912-1 (Orientation preservation) — Clean S^1 / S^2 fields on-manifold (circular/spherical means or exponential maps); enforce unit norm.
• P912-2 (Topology-first) — Cleaning must not change rank(H_k) beyond allowed gates; protect topology by persistence-thresholding (Ch. 8).
• P912-3 (Parallel forms) — Compute formA = clean→build and formB = build→simplify and record delta_form_clean.
• P912-4 (Atlas consistency) — On overlaps, use consistent parameters or rescale via T_ab; keep Σ_a ψ_a intact.
• P912-5 (Auditability) — Persist filters, weights, masks, iteration trajectories; fix randomness with seed.
• P912-6 (Units) — check_dim(all params) = "[1]" (except space/time steps).

IV. Minimal Equations S912-*

1. Signal-domain cleaning

• S912-1 (isotropic kernel): f' = G_σ * f, where G_σ(x) = (1/( (2πσ^2)^{d/2} )) exp( −|x|^2/(2σ^2) ).
• S912-2 (heat diffusion): ∂_t f = Δ f, discrete form f^{t+1} = (I − αL) f^t, α > 0.
• S912-3 (anisotropic): ∂_t f = div( g(|∇f|) ∇f ), e.g., g(s) = 1/(1+(s/κ)^2).
• S912-4 (TV-ROF): min_u ( (1/2) ||u − f||_2^2 + λ ||∇u||_1 ).
• S912-5 (bilateral/NLM): u(x) = ( Σ_y f(y) w_s(x,y) w_r(f(x),f(y)) ) / ( Σ_y w_s w_r ).

2. Orientation-field robustification

• S912-6 (S^1 fields): complex phase z = exp(iθ), z' = (Σ_y w(x,y) z(y))/|Σ_y w(x,y) z(y)|, θ' = arg z'.
• S912-7 (S^2 fields): n' = normalize( Σ_y w(x,y) n(y) ).

3. Complex & filtration robustification

• S912-8 (robust thresholds): τ' = r(τ; σ) = a τ + b minimizing d_B(D, D'), a > 0.
• S912-9 (persistence simplification): remove pairs with pers < τ_pers, with τ_pers derived from σ and stability theorems.
• S912-10 (graph-Laplacian smoothing): K'(τ) = smooth(K(τ), L_K, iters), preserving k-adjacency.

4. Outliers & robust fitting

• S912-11 (Huber): ρ_Huber(r) = { (1/2) r^2, |r| ≤ δ ; δ(|r| − δ/2), |r| > δ }, weight w = ψ(r) = min(1, δ/|r|).
• S912-12 (RANSAC): for tracks Γ, inlier set I* = argmax_I |I| with residual < ε.

5. Two-form gap & stability

• S912-13: delta_form_clean = d_B( D_{clean→build}, D_{build→simplify} ).
• S912-14 (perturbation response): η_stab(ε) = sup_{||δ||≤ε} d_B( D(f+δ), D(f) ).

6. Units & dimensions

• S912-15: check_dim(σ) = unit(x) (dimensionless if in pixels), check_dim(λ) = "[1]", check_dim(d_B) = unit(τ).

V. Metrology Workflow M90-12

• Ready: estimate noise/outliers (residual spectra / Mahalanobis / spatial coherence), set σ / λ / iters and τ_pers initial values, lock RefCond.
• Clean signals/orientations: apply G_σ / TV / anisotropic to f, manifold means to n, and robust fitting to Γ; persist masks & weights.
• Robustify complexes: rescale filtrations by r(τ;σ) on the Atlas, simplify persistence, and keep k_max.
• Parallel forms: compute D_{clean→build} and D_{build→simplify}; obtain delta_form_clean.
• Robust fits & outliers: run Huber/RANSAC on Γ / events; tag removed samples and persist evidence.
• Assess & tune: grid-search or Bayesian optimize over d_B / η_stab / SNR until contracts hold.
• Persist:
  manifest.topo.cleaning = { method, params, masks, τ_pers, r(•), d_B, η_stab, delta_form_clean, RefCond, algo.ver, seed, evidence }, signed.

VI. Contracts & Assertions C90-121x (suggested gates)

• C90-12101 (Topology preservation): | β_k' − β_k |_p95 ≤ tol_beta (default tol_beta = 1).
• C90-12102 (Two-form coherence): delta_form_clean_p95 ≤ tol_clean (suggest 0.02 * scale_τ).
• C90-12103 (Stability gate): η_stab(ε) ≤ ε + tol_stab (suggest tol_stab = 0.05).
• C90-12104 (Atlas coherence): sup_x | Σ_a ψ_a − 1 | ≤ 1e-6; no degradation in Δ_cyc (Ch. 9).
• C90-12105 (Orientation preservation): ||n'|| = 1 with numerical error ≤ 1e-6; track continuous branches across phase jumps.
• C90-12106 (Resources / SLO): cleaning runtime and memory within quotas; otherwise trigger reduced-order strategies.

VII. Implementation Bindings I90-12*

• I90-121 estimate_noise(data, policy) -> { σ, model, SNR }
• I90-122 denoise_scalar(field, method, params) -> f' (methods { Gaussian, Heat, Aniso, TV, Bilateral, NLM, Graph })
• I90-123 denoise_orientation_S1(theta, W) -> theta'
• I90-124 denoise_orientation_S2(n, W) -> n'
• I90-125 robust_tracks(Γ, loss, eps) -> Γ', tags (loss { Huber, Cauchy, RANSAC })
• I90-126 stabilize_filtration(K(τ), σ, policy) -> K'(τ), r(τ)
• I90-127 simplify_persistence(D, τ_pers) -> D'
• I90-128 compare_clean_forms(D_clean_build, D_build_simplify) -> delta_form_clean
• I90-129 evaluate_stability(data, ε_grid) -> η_stab
• I90-12A assert_cleaning_contracts(ds, rules) -> report
• I90-12B emit_cleaning_manifest(results, policy) -> manifest.topo.cleaning

Invariants: non_decreasing(τ); Σ ψ ≡ 1; ||n'|| = 1; delta_form_clean ≤ tol_clean; all check_dim pass.

VIII. Cross-References

• Complex construction & filtration: Chapter 7; stability & distances: Chapter 8.
• Atlas & transition coherence: Chapter 9 (T_ab, ψ_a, Δ_cyc).
• Invariant metrology: Chapter 10 (pre/post Inv two-form comparisons).
• Incremental maintenance: Chapter 11 (submit cleaning-parameter changes as patches); error propagation: Appendix E.

IX. Quality & Risk Control

• SLOs: delta_form_clean_p95 ≤ tol_clean; η_stab gate pass-rate ≥ 99%; orientation-preservation failure ≤ 1e-6.
• Fallback path: TV / anisotropic → heat → Gaussian → persistence-only simplification → H0-only, relaxing constraints and tagging fallback.stage.
• Audit: persist kernels/weight fields, masks, iteration curves, removed persistence pairs, outlier IDs, seeds, and resource traces; produce before/after snapshots & hashes for key regions.

Summary
This chapter codifies the engineering loop Noise → Cleaning → Robustness as P912 / S912 / M90-12 / C90-121x / I90-12*. By signal cleaning + robust complexes + two-form comparison + contract gates, the Topological Atlas stays stable, usable, and auditable under noise and anomalies, with results published uniformly as manifest.topo.cleaning.*.