17-EFT.WP.Methods.Imaging v1.0 | Chapter 5 Optical Imaging and PSF/OTF/MTF

Chapter 5 Optical Imaging and PSF/OTF/MTF

One-Sentence Goal
Under the linear shift-invariant (LSI) baseline, define and estimate PSF/OTF/MTF, build a system-level modulation transfer chain that accounts for pixel aperture, motion, and defocus, and constrain imaging sharpness and anisotropy with contractible metrics.

I. Scope & Targets

1. Inputs
   • From Chapter 3 (mode binding): SRef.img = { I_lin, masks, mode_axes, tau_map, manifest.imaging.device }.
   • From Chapter 4 (radiometric harmonization): I_corr, L_eff (when energy conservation checks are required).
   • Optical/mechanical metadata: { pixel_pitch, a_px, b_px, f_eff, F_number, focus_pos, T_trans(λ), stab_trace }.
2. Outputs
   • PSF(x,y), OTF(fx,fy), MTF(fx,fy), PTF(fx,fy); optionally field-dependent variants such as PSF(u; x,y).
   • Component factors: MTF_opt, MTF_defocus, MTF_motion, MTF_pixel, MTF_proc, and the system result MTF_sys.
   • Metrics & reports: MTF50, MTF10, anisotropy, alias_ratio, assert_report.optics, and manifest.imaging.optics.
3. Boundaries
   This chapter assumes small-signal linear imaging and weak non-idealities; with strongly nonlinear ISP, report an “effective MTF” only.

II. Terms & Variables

1. Coordinates & frequency domain
   • x,y (image plane in meters), fx,fy (cycles per meter), ρ = sqrt(fx^2 + fy^2).
   • Sampling & Nyquist: f_sx = 1 / pitch_x, f_sy = 1 / pitch_y, f_Nx = f_sx / 2, f_Ny = f_sy / 2.
2. Core functions
   • PSF(x,y) (point spread function), ( ∫∫ PSF dx dy ) = 1.
   • OTF(fx,fy) = F{ PSF(x,y) }, MTF = | OTF |, PTF = arg( OTF ).
   • Pixel-aperture MTF: MTF_pixel(fx,fy) = sinc( π * a_px * fx ) * sinc( π * b_px * fy ).
   • Motion MTF (uniform linear motion, length d_m, motion direction forming angle θ with fx,fy):
     MTF_motion(fx,fy) = sinc( π * d_m * ( fx cosθ + fy sinθ ) ).
3. Objective & aberrations
   • Pupil function P(u,v) = A(u,v) * exp( j * 2π * W(u,v) / λ ), with normalized pupil coordinates u,v.
   • W (wavefront aberration), λ (wavelength), F_number = f_eff / D_aperture.
4. Quality & uncertainty
   u(x), coverage U = k * u_c, q_score, drift, SNR_edge.

III. Axioms P205- (Optics Baseline)*

• P205-1 (Normalization & dimensionless): OTF(0,0) = 1, 0 ≤ MTF(ρ) ≤ 1, PTF(0,0) = 0.
• P205-2 (Explicit spectral weighting): For multi-band or broadband cases, define effective PSF/MTF with a declared W(λ).
• P205-3 (Pixel prior): Pixels are modeled as an equivalent rectangular aperture a_px × b_px, and MTF_pixel must be included in the system chain.
• P205-4 (Linear-domain baseline): All estimates are performed on I_lin or I_corr; do not fit physical OTF directly in gamma/NR-processed nonlinear domains.
• P205-5 (Field consistency): Provide MTF(u) in both chief and off-axis fields and report anisotropy.
• P205-6 (Sampling conservation): Frequency analysis is confined to [ -f_Nx, f_Nx ] × [ -f_Ny, f_Ny ]; out-of-band content is recorded via aliasing measures.
• P205-7 (Auditable contracts): PSF/MTF estimation procedures, templates, and window parameters are written to the manifest and signed.

IV. Minimal Equations S205-*

• S205-1 (Imaging convolution model)
  I_lin(x,y) = ( h * s )(x,y) + n(x,y), where h ≡ PSF and s is the scene radiance projection.
• S205-2 (2-D Fourier link between PSF and OTF)
  OTF(fx,fy) = ( ∫∫ PSF(x,y) * exp( -j 2π ( fx x + fy y ) ) dx dy ) / ( ∫∫ PSF dx dy ).
• S205-3 (Optical OTF via pupil autocorrelation, incoherent imaging)
  OTF_opt(ξ,η) = ( 1 / A_pupil ) * ( ∫∫ P(u,v) * P*(u-ξ, v-η) du dv ), with A_pupil = ( ∫∫ |P(u,v)|^2 du dv ).
• S205-4 (System MTF composition law)
  MTF_sys(fx,fy) = MTF_opt(fx,fy) * MTF_defocus(fx,fy) * MTF_motion(fx,fy) * MTF_pixel(fx,fy) * MTF_proc(fx,fy).
• S205-5 (Defocus MTF, paraxial approximation)
  For a circular pupil and nominal defocus σ_d: MTF_defocus(ρ) = | OTF_defocus(ρ; σ_d ) |, where OTF_defocus derives from the pupil autocorrelation inclusive of W20 (computed numerically).
• S205-6 (Multi-spectral effective PSF/MTF)
  PSF_eff = ( ∫_Λ W(λ) * PSF_λ dλ ) / ( ∫_Λ W(λ) dλ ),
  MTF_eff = | F{ PSF_eff } |, with W(λ) ∝ H_sys(λ) or application-specific.
• S205-7 (Edge method SFR)
  LSF(x) = d/dx ( ESF(x) ), MTF_1D(f) = | F{ LSF(x) } | / | F{ LSF } |_{f=0}.
• S205-8 (Aliasing metric)
  alias_ratio = ( ∫_{Ω_alias} PDS(fx,fy) dfx dfy ) / ( ∫_{Ω_pass} PDS(fx,fy) dfx dfy ), with PDS the power density spectrum.
• S205-9 (Units & dimension checks)
  check_dim( OTF ) = 1, check_dim( MTF ) = 1, check_dim( PSF ) = [L^-2] with integral normalization.

V. Imaging Transfer Estimation Process M50-*

• M50-1 Linear domain & masking ready: Start from I_lin or I_corr (Ch. 3/4), apply masks.bad/hot/sat.
• M50-2 Target template: Choose one or more: edge (slanted edge), slit (line), point (point source), deadleaves (texture). Register window, crop, binning.
• M50-3 ESF/LSF/SFR estimation: For edge, perform sub-pixel localization and slant correction; produce MTF_edge(f) and SNR_edge.
• M50-4 2-D OTF reconstruction: Fuse multi-directional estimates to obtain OTF_hat(fx,fy); or derive PSF from point sources and transform.
• M50-5 Pixel & motion separation: Using priors a_px, b_px and stab_trace, decouple MTF_pixel and MTF_motion to obtain MTF_opt * MTF_defocus.
• M50-6 Field-dependent assessment: Repeat Steps 3–5 on a grid {u_i} to obtain MTF(u_i) and anisotropy(u_i).
• M50-7 Aliasing & spectral hygiene: Restrict to the Nyquist window; compute alias_ratio; flag out-of-band components.
• M50-8 Metric extraction & uncertainty: Report MTF50, MTF10, slope@0, and u(MTF50); propagate sampling/noise uncertainties.
• M50-9 Contract checks: Run assert_contract.optics.
• M50-10 Persistence & signature: Write manifest.imaging.optics = { templates, windows, MTF_sys, components, metrics, hashes } and sign.

VI. Contracts & Assertions

• Normalization & energy
  | OTF(0,0) - 1 | ≤ tol_otf_dc; abs( mean(PSF) * A_roi - 1 ) ≤ tol_psf_norm.
• Frequency domain & aliasing
  alias_ratio ≤ tol_alias; MTF_sys(ρ > f_Nyquist) = 0 (by cropping).
• Thresholds for metrics
  MTF50 ≥ mtf50_min (stated either in pixels or angular resolution), MTF10 ≥ mtf10_min, anisotropy ≤ tol_aniso.
• Component consistency
  Composition closure: max| MTF_sys - ∏ MTF_* | ≤ tol_compose; PTF is continuous and var(PTF) ≤ ptf_var_max (within passband).
• Environment & drift
  drift(MTF50; ΔT) ≤ tol_mtf_drift; drift(MTF50; Δt) ≤ tol_age_drift.

VII. Implementation Bindings I50-*

• I50-1 estimate_esf_edge(img, window, angle) -> { ESF, LSF, MTF_1D, SNR_edge }
• I50-2 estimate_psf_point(img, window) -> { PSF, OTF, MTF }
• I50-3 compose_pixel_mtf(a_px, b_px) -> MTF_pixel
• I50-4 estimate_motion_mtf(trace, pitch, ts) -> MTF_motion
• I50-5 decompose_mtf(MTF_sys, MTF_pixel, MTF_motion) -> { MTF_opt_defocus }
• I50-6 field_grid_eval(imgs_or_tiles, grid) -> { MTF(u_i), anisotropy(u_i) }
• I50-7 summarize_mtf_metrics(MTF, f_axes) -> { MTF50, MTF10, slope0, alias_ratio }
• I50-8 validate_optics_contract(metrics, contracts) -> assert_report
• I50-9 emit_manifest_optics(artifacts, params, hashes) -> manifest

VIII. Cross-References

• Linear domain, masking, and mode axes: this volume’s Chapter 3.
• Radiometric harmonization & unit checks: this volume’s Chapter 4; metrological conservation: Methods.Cleaning v1.0, Chapter 4.
• Time axis & motion parameters (trace, ts): Methods.Cleaning v1.0, Chapter 5.
• Environmental correction & reference propagation: Methods.Cleaning v1.0, Chapter 12.
• Data schemas & manifest anchors: EFT.WP.Core.DataSpec v1.0.

IX. Quality Metrics & Risk Control

1. Key indicators
   MTF50, MTF10, anisotropy, alias_ratio, slope@0, u(MTF50), PTF_var, comp_residual.
2. Risk handling
   • If MTF50 or alias_ratio violates limits: reduce publication resolution or enable anti-alias filtering; roll back to the last freeze_release optical parameters.
   • For field-dependent anomalies: mask hotspot regions or trigger refocus/mechanical re-alignment; down-rank q_score and record TraceID.

Summary
Centering on PSF/OTF/MTF, this chapter provides a unified path from data-driven estimation to system-level decomposition and composition—edge/point → PSF/OTF → MTF_components → MTF_sys—and enforces reproducibility and auditability across devices, fields, and environments through contract metrics and manifest signatures.