Chapter 2 Axioms and Minimal Equations (Baseline for Imaging Modeling)
One-Sentence Goal
Provide the modeling preconditions—axioms and a minimal family of equations—for EFT imaging so that subsequent metrology, geometry, color, and time-base processing are computable, auditable, and revertible under a unified convention.
I. Scope & Targets
- Applicability
- Frame-/row-based sensors; CFA/monochrome/multispectral; gated/ToF; HDR multi-exposure; static and mildly dynamic scenes.
- Inputs: D_raw_img = { I_raw(x,y,t), sensor_meta, optics_meta, timing, env }
- Baseline outputs: I_lin, I_corr, I_rgb, I_hdr, manifest.imaging.*.
- Non-goals & boundaries
- Loss functions and training for downstream recognition/retrieval/generation tasks are out of scope for this chapter.
- Systems with strongly nonlinear and widely time-varying PSF must declare chapter-specific deviation terms and residual bounds.
II. Terms & Variables
- Optics & frequency domain: h(x,y), H_otf(nu_x,nu_y), MTF(nu) = | H_otf(nu) |, aperture, f_number, focal.
- Metrology & noise: I_raw, I_lin, gain, offset, SatMax, PRNU, DSNU, σ_shot, σ_read, σ_dark, σ_total.
- Sampling & time: Δx, Δt, bpp, ts, tau_mono, offset/skew/J.
- Color & spectrum: S_sensor(λ), QE(λ), XYZ, RGB_lin, M_rgb2xyz, M_xyz2rgb, CAT, Wp.
- Geometry & projection: K, R,t, dist_params, u, X, eps_reproj.
- Arrival time & path: gamma(ell), n_eff, c_ref, two-form T_arr and the difference delta_form.
- HDR & fusion: t_i, w_i, I_hdr.
- Compliance & traceability: hash_sha256(blob), signature, TraceID, q_score.
III. Axioms P202- (Imaging Modeling Preconditions)*
- P202-1 (LSI, locally stationary): Within a spatial window Ω and temporal window Δt, treat the imaging system as LSI, with ( ∫∫_Ω h dx dy ) = 1 and F{h} = H_otf.
- P202-2 (Radiometric linear domain): After linearization I_lin = gain * ( I_raw - offset ), with unit(I_lin) = "e-"; saturated pixels are flagged by mask_sat and excluded from estimation.
- P202-3 (Additive noise): σ_total^2 = σ_shot^2 + σ_read^2 + σ_dark^2 + σ_fixed^2, and σ_shot^2 ≈ I_lin in the electron-count domain.
- P202-4 (Sampling & aliasing): The effective bandwidth nu_max ≤ 1 / ( 2 * Δx ); out-of-band components are flagged via alias_flag and down-weighted downstream.
- P202-5 (Color separability): Under RefCond, a stable linear mapping exists from sensor space to XYZ/target color space, with explicit white-point adaptation CAT(Wp_src→Wp_dst).
- P202-6 (Geometry & time-base coherence): Internal evaluation on tau_mono; publication on ts with recorded offset/skew/J. Geometry uses a pinhole model plus distortion.
- P202-7 (HDR irradiance consistency): After exposure normalization, scene irradiance across exposures is mergeable; motion differences are handled by de-ghosting terms.
- P202-8 (CFA linear decodability): CFA sampling is a linear operator S_cfa; the baseline demosaic is a linear interpolant D; any nonlinear decode must record explicit deviation terms.
- P202-9 (Flat-/dark-field order): PRNU is multiplicative, DSNU is additive; corrections must precede color transforms and HDR fusion.
- P202-10 (Two-form arrival in parallel): Where T_arr is involved, compute both forms in parallel and record delta_form ≤ tol_Tarr together with path gamma(ell).
- P202-11 (Traceable & revertible): Parameters and versions for every step must be written to the manifest and signed to support freeze and rollback.
*IV. Minimal Equations S202- **
- S202-1 (Imaging convolution): I_lin(x,y,t) = ( h * scene )(x,y,t) + ε, frequency-domain F{I_lin} = H_otf • F{scene} + N.
- S202-2 (Linearization): I_lin = gain * ( I_raw - offset ), mask_sat = [ I_raw ≥ SatMax ].
- S202-3 (Noise synthesis): σ_total^2 = σ_shot^2 + σ_read^2 + σ_dark^2 + σ_fixed^2, SNR = I_lin / σ_total.
- S202-4 (Flat-/dark-field correction): I_corr = ( I_lin - DSNU ) / ( 1 + PRNU ).
- S202-5 (CFA & decoding): I_cfa = S_cfa ∘ I_corr, RGB_lin = D( I_cfa ).
- S202-6 (Color mapping): XYZ = M_rgb2xyz • RGB_lin, RGB_tgt = M_xyz2rgb • CAT(Wp_src→Wp_dst) • XYZ.
- S202-7 (HDR fusion): I_hdr = ( ∑ w_i • I_lin_i / t_i ) / ( ∑ w_i ).
- S202-8 (Geometric projection): u ~ K [ R | t ] X, eps_reproj = mean || u_obs - u_pred ||.
- S202-9 (Sampling bound): nu_nyq = 1 / ( 2 * Δx ), require nu_max ≤ nu_nyq.
- S202-10 (Arrival two forms & gap):
T_arr = ( 1 / c_ref ) * ( ∫_{gamma} n_eff d ell ),
T_arr = ( ∫_{gamma} ( n_eff / c_ref ) d ell ),
delta_form = | • |. - S202-11 (Time-base mapping): ts = tau_mono + offset + skew * ( tau_mono - tau0 ) + jitter, with | jitter | ≤ J.
*V. Baseline Modeling Process M20- **
- M20-20 Metadata readiness: Validate completeness and dimensional coherence of sensor_meta / optics_meta / timing / env.
- M20-21 Linearization & saturation mask: Apply S202-2 to obtain I_lin and mask_sat; persist metrology parameters and units.
- M20-22 Dark/flat correction: Estimate DSNU/PRNU and apply S202-4 to get I_corr; record residual statistics.
- M20-23 Noise modeling: Fit σ_shot / σ_read / σ_dark; produce σ_total and SNR panels.
- M20-24 Sampling & aliasing check: Compute nu_nyq and nu_max; if violated, issue alias_flag and down-weight policy.
- M20-25 Optics & MTF: Estimate h / H_otf / MTF and validate the energy normalization in S202-1.
- M20-26 CFA & decoding: Apply S_cfa and D to obtain RGB_lin; record interpolation kernels and versions.
- M20-27 Color & white point: Execute S202-6; record M_xyz2rgb / CAT / Wp and ΔE metrics.
- M20-28 HDR fusion: Generate I_hdr via S202-7; output ghost_rate and saturation coverage.
- M20-29 Geometry & time base: Estimate K,R,t,dist_params and eps_reproj; execute S202-11 and, when ToF is present, compute S202-10 in parallel.
- M20-30 Baseline freeze: Emit manifest.imaging.baseline, run assert_contract, and sign.
VI. Contracts & Assertions
- Optics & sampling
abs( ( ∫∫ h dx dy ) - 1 ) ≤ eps_psf_int; nu_max ≤ nu_nyq or alias_flag = true. - Metrology & noise
check_dim( I_lin - gain*(I_raw-offset) ) = true; regression for σ_shot^2 ≈ I_lin has R^2 ≥ R2_min. - Color & geometry
ΔE ≤ ΔE_max; eps_reproj ≤ eps_reproj_max. - HDR & arrival time
ghost_rate ≤ ghost_max; delta_form ≤ tol_Tarr. - Publication & traceability
manifest_signed = true and hash_sha256(blob) consistent.
*VII. Implementation Bindings I20- **
- I20-20 fit_noise_model(I_lin, dark, flat) -> { σ_shot, σ_read, σ_dark, σ_total }
- I20-21 linearize_response(I_raw, gain, offset, SatMax) -> { I_lin, mask_sat }
- I20-22 apply_flatfield(I_lin, PRNU, DSNU) -> I_corr
- I20-23 check_nyquist(I_corr, Δx) -> { nu_max, nu_nyq, alias_flag }
- I20-24 estimate_psf_mtf(I_corr) -> { h, H_otf, MTF }
- I20-25 demosaic_linear(I_corr, S_cfa, kernel) -> RGB_lin
- I20-26 color_pipeline(RGB_lin, M_rgb2xyz, M_xyz2rgb, CAT, Wp) -> { RGB_tgt, ΔE }
- I20-27 hdr_merge(frames, t_i, w_i) -> { I_hdr, ghost_rate }
- I20-28 calibrate_geometry(frames, pattern) -> { K,R,t,dist_params, eps_reproj }
- I20-29 map_time_and_arrival(tau_mono, timing, path, n_eff, c_ref) -> { ts, T_arr_forms, delta_form }
- I20-30 assert_contract(ds, tests) -> report
VIII. Cross-References
- Schema & manifests: see EFT.WP.Core.DataSpec v1.0; time base & arrival: see EFT.WP.Core.Sea v1.0.
- Streaming & SLI/SLO: see EFT.WP.Core.Threads v1.0.
- Dimensions & uncertainty propagation: see Methods.Cleaning v1.0, Chapter 4 and Appendix E.
- Arrival & path coherence: see Methods.Cleaning v1.0, Chapter 6.
- Probability & normalization: see Methods.Cleaning v1.0, Chapter 13.
IX. Quality Metrics & Risk Control
- Key indicators
mtf50, ΔE, eps_reproj, SNR@P50/P95, ghost_rate, alias_flag, σ_read, PRNU/DSNU drift. - Risk & rollback
On any contract failure or metric drift beyond thresholds, roll back to the latest freeze_release(tag_prev) and emit audit records plus a postmortem.
Summary
This chapter establishes the foundational axioms P202-* and minimal equations S202-* for imaging modeling, along with an executable baseline process M20-20 … 30 and implementation bindings I20-*. Subsequent chapters build on this baseline to refine optics, color, HDR, geometry, and time-base topics and to govern deviations.