17-EFT.WP.Methods.Imaging v1.0 | Chapter 13 Time/Path Gating and Arrival-Time Harmonization

Chapter 13 Time/Path Gating and Arrival-Time Harmonization

One-Sentence Goal
Define unified time gating G(t) and path gating A_path(•) conventions and—together with the two-form arrival-time consistency T_arr—align multi-device/multi-modality data on tau_mono while preserving consistent physical path semantics.

I. Scope & Targets

1. Inputs
   • Timing & sync: ts, tau_mono, offset, skew, J, exposure/trigger metadata { t_open, t_close, fps, rolling }.
   • Path & medium: gamma(ell), n_eff(x), c_ref, path gate A_path(θ, λ, pol) composed from aperture / FOV / polarization / wavelength.
   • Imaging kernel & noise: h/OTF, S_n, and optional motion field v(t).
2. Outputs
   • Standardized gate descriptors: normalized G(t) and A_path(•) with parameter sets.
   • Arrival-time report: both T_arr forms and their difference delta_form, plus threshold contracts.
   • Alignment & cropping artifacts: time-aligned frame/event sequences and path-consistent FOV crops / weighting maps.
3. Applicability
   • Supports global/rolling shutters, pulsed lighting / delay gates, ToF / streak / event cameras.
   • Supports space-invariant or blockwise variant PSFs; for strongly scattering media, declare spatiotemporal dependence of n_eff and downgrade to approximations.

II. Terms & Variables

1. Time gating
   • G(t): gate function (rectangular / weighted / pulse train); T_exp = ( ∫ G(t) dt ); G_norm(t) = G(t) / T_exp.
   • Row timing: t_row(i) = t_start + i * dt_row (rolling shutter), with row readout period dt_row.
2. Path gating
   • A_path(θ, λ, pol) ∈ [0,1]: acceptance function driven by NA / filtering / polarization / occlusion;
   • FOV and occlusion mask mask_path(x).
3. Arrival time & two forms
   • T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) (constant-outside form).
   • T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ) (general form).
   • Difference:
     delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |.
4. Shape matching & consistency
   • Time-shape similarity: eta_G = ( ∫ G_a G_b dt ) / sqrt( ( ∫ G_a^2 dt ) ( ∫ G_b^2 dt ) ).
   • Path overlap: eta_path = ( ∫ A_a A_b dΩ ) / sqrt( ( ∫ A_a^2 dΩ ) ( ∫ A_b^2 dΩ ) ).
   • Data-consistency residuals: res_time = | t_ref - t_meas |; res_path for FOV boundary mismatch.

III. Axioms P213- (Gating & Arrival Baseline)*

• P213-1 (Linear domain): Apply all gating and arrival-time calculations in the linear radiometric domain (Chapter 4); never estimate in a non-linear rendering domain.
  P213-2 (Time-base first): Define G(t) and t_row(i) on tau_mono; publish on ts (Methods.Cleaning v1.0, Ch. 5).
• P213-3 (Two forms in parallel): Compute T_arr using both forms and record delta_form.
• P213-4 (Explicit paths): Any arrival-time analysis must state gamma(ell), the measure d ell, and the domain of A_path(•).
• P213-5 (Physical conservation): Gate energy conservation ( ∫ G_norm(t) dt ) = 1; path gate must be consistent with the PSF’s effective NA.
• P213-6 (Rolling correction): For rolling shutters provide t_row(i) and row-level offset/skew; compensate in reconstruction/registration.
• P213-7 (Causality & triggering): Record the relative phase of active illumination and exposure; reproducible to within ≤ tol_phase.
• P213-8 (Auditable): Gate shapes, parameters, calibration evidence, and hashes go to manifest.imaging.gate with signature.

IV. Minimal Equations S213-*

• S213-1 (Gated imaging forward model)
  y(r) = ( ∫ G(t) [ h ⊗ x(•, t) ](r) dt ) + n(r).
• S213-2 (Motion blur via temporal PSF)
  Define temporal PSF p_t(t) = G_norm(t); the effective spatial blur kernel is
  h_eff = ( ∫ p_t(t) * warp(h, v(t)) dt ).
• S213-3 (Rolling-shutter timestamp)
  ts(i,j) = t_row(i) + dt_pix * j, where dt_pix is per-column readout increment.
• S213-4 (ToF round-trip arrival time)
  T_arr^2w = ( 2 / c_ref ) * ( ∫_{gamma} n_eff d ell ); with the general form, replace the integrand by ( 2 n_eff / c_ref ).
• S213-5 (Gate-shape similarity)
  eta_G = ( ∫ G_a G_b dt ) / sqrt( ( ∫ G_a^2 dt ) ( ∫ G_b^2 dt ) ), with contract eta_G ≥ tol_gate_shape.
• S213-6 (Path overlap)
  eta_path = ( ∫ A_a A_b dΩ ) / sqrt( ( ∫ A_a^2 dΩ ) ( ∫ A_b^2 dΩ ) ), with contract eta_path ≥ tol_path_shape.
• S213-7 (Two-form gap threshold)
  delta_form ≤ tol_Tarr, with thresholds derived from medium variability and propagation distance.
• S213-8 (Alignment criteria)
  | offset | ≤ tol_offset, | skew | ≤ tol_skew, J ≤ tol_jitter, and res_time ≤ tol_time_res.

V. Pipeline & Operational Flow M130-*

• M130-1 Readiness: From metadata, extract exposure/trigger/readout to build initial G_raw(t) and A_path_raw(•).
• M130-2 Standardization: Normalize G_norm(t) = G_raw / ( ∫ G_raw dt ); project A_path_raw onto the system NA/filter domain.
• M130-3 Time-base binding: Express all timestamps and gates on tau_mono; write offset/skew/J.
• M130-4 Rolling modeling: From t_row(i) and dt_pix synthesize per-row/column timestamp map ts(i,j).
• M130-5 Path modeling: Construct A_path(θ, λ, pol) and mask_path(x) from optics/occlusion/polarization/band.
• M130-6 Two-form arrival: Given gamma(ell) and n_eff, compute both T_arr forms and delta_form.
• M130-7 Gate matching: Compute eta_G and eta_path; if below thresholds, adjust exposure/sync or crop/resample to common FOV.
• M130-8 Rectification & compensation: Correct rolling/jitter time distortion; synthesize h_eff for subsequent reconstruction (Chapter 12).
• M130-9 Contract checks: Verify tol_Tarr, tol_gate_shape, tol_path_shape, tol_offset, tol_skew, tol_jitter.
• M130-10 Persistence & signature: Write manifest.imaging.gate (parameters for G(t) and A_path(•), alignment report, hashes, signature).

VI. Contracts & Assertions

• assert gate_energy: | ( ∫ G_norm dt ) - 1 | ≤ tol_gate_energy.
• assert gate_shape: eta_G ≥ tol_gate_shape and pulse width T_exp within tolerance.
• assert path_shape: eta_path ≥ tol_path_shape, FOV overlap ≥ tol_fov_overlap.
• assert arrival_consistency: delta_form ≤ tol_Tarr.
• assert timing_sync: | offset | ≤ tol_offset, | skew | ≤ tol_skew, J ≤ tol_jitter.
• assert rolling_rectified: row/column time-skew residual ≤ tol_row_skew.
• assert log_reproducible: hash_sha256(manifest.imaging.gate) matches the runtime log.

VII. Implementation Bindings I130-*

• I130-1 build_time_gate(meta) -> G(t), T_exp
• I130-2 build_path_gate(optics, filters, pol) -> A_path(•), mask_path(x)
• I130-3 bind_timebase_to_tau(G, ts_info) -> G_tau, { offset, skew, J }
• I130-4 model_rolling_timestamps(meta) -> ts(i,j), t_row(i), dt_pix
• I130-5 compute_arrival_forms(gamma, n_eff, c_ref) -> { T_arr_const, T_arr_general, delta_form }
• I130-6 match_gates(G_a, G_b, A_a, A_b) -> { eta_G, eta_path }
• I130-7 rectify_rolling_and_jitter(frames, ts_map) -> frames'
• I130-8 synthesize_temporal_psf(G, v(t), h) -> h_eff
• I130-9 emit_gate_manifest(params, metrics) -> manifest.imaging.gate

VIII. Cross-References

• Time axis & synchronization: Methods.Cleaning v1.0, Chapter 5 (tau_mono, offset/skew/J).
• Path & arrival-time: Methods.Cleaning v1.0, Chapter 6 (gamma(ell), two forms, delta_form).
• Optical kernels & FOV: this volume’s Chapter 5 (PSF/OTF/MTF) and Chapter 9 (geometry/registration).
• Noise & reconstruction: Chapters 7 and 12 for noise modeling and use of h_eff.
• Calibration & units: Chapter 4 for unit/dimension and energy conservation.
• Companion white papers: see EFT series, Vol. 2 S/P for two-form measures and thresholds.

IX. Quality Metrics & Risk Control

1. Timing: eta_G, offset, skew, J, res_time.
2. Path: eta_path, fov_overlap, res_path.
3. Arrival: delta_form, T_arr confidence intervals.
4. Risk playbooks
   • Trigger phase drift: self-calibrate pulse trains; tighten tol_phase; drop nonconforming frames.
   • Rolling mismatch / fast motion: row-time correction and ts(i,j)-aware motion compensation; otherwise downgrade to a global-blur model.
   • Refractive-index drift: update n_eff and recompute T_arr; if delta_form exceeds threshold, mark degraded publication.
   • FOV mismatch: crop to common FOV via mask_path or reconfigure optics.

Summary
This chapter unifies time gating G(t), path gating A_path(•), and two-form arrival-time consistency. Contracts on eta_G/eta_path, delta_form, and the sync triplet offset/skew/J guarantee that multi-modality/multi-device data align under a common physical semantics—remaining computable and auditable end to end.