H2-1. Definition & Why Polarization Helps (DoLP/AoLP in one page)

Goal: define what polarization imaging measures beyond intensity, and how that extra information converts into controllable contrast for industrial inspection (glare, transparent covers, stress patterns, surface defects).

What polarization imaging adds (measurable, not “more clear”)

• S0 (Intensity): the conventional brightness image—still required for geometry, segmentation, and basic defect localization.
• DoLP (Degree of Linear Polarization): a scalar “how polarized” indicator. It often rises in specular/high-glare regions and changes with surface finish, enabling robust masking or contrast re-weighting.
• AoLP (Angle of Linear Polarization): a direction field (orientation). It can highlight stress-induced patterns, fiber/composite direction, or anisotropic surface features when intensity contrast is weak or unstable.

Polarization vs reflection (only the engineering consequence)

• Specular reflections tend to show stronger polarization structure than purely diffuse regions, so DoLP/AoLP can separate “glare energy” from “texture/defect energy.”
• Transparent covers and glossy coatings often break intensity-based inspection (overexposure, bloom, hotspot). DoLP-driven masking can recover edges and defects that vanish in S0.
• Stress patterns and directional materials may be difficult in S0 because contrast depends on lighting angle. AoLP can provide a more repeatable directional signature.

When polarization is worth the cost (decision checklist)

• Glare dominates failure rate: S0 saturates or loses defect visibility; DoLP allows a glare mask or glare-suppressed fusion output.
• Transparent or glossy surfaces: cover glass, films, coatings, or wet surfaces where intensity varies wildly by viewpoint.
• Stress / birefringence patterns: plastic parts, tempered glass stress, adhesives; AoLP/DoLP can stabilize pattern visibility across exposure changes.
• Directional textures: fiber layup, brushed metals, composites—AoLP offers an orientation field for inspection rules.

H2-2. Polarized Pixel Architecture (micro-polarizer array & super-pixel)

Goal: explain the 2×2 polarization super-pixel and why its non-idealities (angle offset, leakage/extinction, crosstalk, offset/gain mismatch) directly translate into DoLP/AoLP errors and artifacts.

Core concept: a 2×2 super-pixel is the measurement unit

• Many polarization sensors use a micro-polarizer mosaic where a 2×2 block measures four analyzer angles: 0° / 45° / 90° / 135°.
• The sensor does not deliver DoLP/AoLP “for free.” It delivers four angle-sampled intensities that must be reordered, interpolated, and converted into polarization features.
• Engineering implication: polarization accuracy is dominated by relative consistency between the four angle channels, not only by absolute image quality.

Minimal math (only what is needed to reason about errors)

Let the four analyzer samples from a super-pixel be I0, I45, I90, I135. A common, engineering-useful approximation is:

• S0 ≈ I0 + I90 (total intensity proxy)
• S1 ≈ I0 − I90 (horizontal vs vertical polarization balance)
• S2 ≈ I45 − I135 (diagonal polarization balance)
• DoLP = √(S1² + S2²) / S0
• AoLP = 0.5 · atan2(S2, S1) (direction field; wrap handling belongs to later chapters)

Why this matters: every non-ideality in the four channels perturbs S1/S2/S0 differently, which shows up as DoLP bias (too high/too low) and AoLP shifts/jumps.

Four dominant non-idealities (error → symptom → fastest discriminator)

• Angle offset (micro-polarizer orientation error)
  Symptom: AoLP is globally rotated or varies spatially (patchy AoLP bias).
  Discriminator: rotating a linear polarizer produces curves with a consistent phase shift (global) or region-dependent phase shifts (spatial).
• Finite extinction / leakage
  Symptom: DoLP ceiling is limited; high-DoLP scenes appear “washed” in polarization contrast.
  Discriminator: under a known strong linear polarization input, measured DoLP fails to approach expected upper bounds even after gain matching.
• Crosstalk (optical/electrical mixing between channels)
  Symptom: DoLP/AoLP textures appear where S0 looks normal, especially near fine repetitive patterns or sharp edges.
  Discriminator: polarization artifacts correlate with spatial frequency/edge orientation and change with focus/MTF or pattern pitch.
• Offset/gain mismatch (black-level / channel gain inconsistency)
  Symptom: dark regions show unstable DoLP, and AoLP becomes noisy or random at low signal levels.
  Discriminator: in a dark/covered condition, non-zero S1/S2 structure persists (indicates offset imbalance before any “real polarization” exists).

Next chapters (not included in this batch) will use these foundations to explain polarization demosaicing, polarization-specific artifacts (de-moiré / angle wrap), calibration maps, and synchronization evidence.

H2-3. Polarization Demosaicing (from mosaic to per-pixel Stokes)

Goal: convert the polarized-pixel mosaic into stable per-pixel polarization features (S0/S1/S2, DoLP/AoLP), while preventing edge artifacts, beat-pattern moiré, and low-signal angle instability.

Two reconstruction modes (choose by risk & resolution)

• Super-pixel output (lower resolution, higher robustness): compute one polarization solution per 2×2 super-pixel (minimal interpolation). Best when the priority is stable inspection signals and predictable AoLP continuity.
• Full-resolution output (higher detail, higher artifact risk): reorder into four angle-channels, then interpolate (prefer edge-aware) before Stokes. Best when small defect localization matters—but must be paired with artifact checks.

Reference pipeline (safe ordering)

• 1) Mosaic capture: raw polarized pixel mosaic is the measurement; DoLP/AoLP does not exist yet.
• 2) Channel split + reorder: form the four sub-images I0, I45, I90, I135 from the mosaic grid.
• 3) Channel alignment (registration): compensate the intrinsic sub-pixel sampling offsets between channels to avoid “texture → polarization” leakage.
• 4) Interpolation (edge-aware): fill missing samples per channel while preserving edges; this step is where most AoLP failures are created or avoided.
• 5) Stokes compute: derive S0, S1, S2 (using the chapter-2 definitions) from the aligned channels.
• 6) DoLP/AoLP: compute polarization degree and angle; apply AoLP representation rules (wrap/unwrap hooks are validated in H2-4).

Failure modes that matter (symptom → mechanism → fastest discriminator)

• Edge AoLP jumps
  Mechanism: edge interpolation biases one channel, perturbing the S1/S2 ratio and flipping AoLP near boundaries.
  Discriminator: on a slanted-edge target, AoLP should remain locally continuous after unwrap; repeated “tile flips” indicates interpolation/registration issues.
• Beat-pattern false texture (near pixel-period frequency)
  Mechanism: scene spatial frequency interacts with the polarization sampling lattice; differencing produces polarization-only stripes.
  Discriminator: DoLP shows regular stripes that are absent (or much weaker) in S0 for the same ROI.
• Low-S0 angle instability
  Mechanism: when S0 is small, DoLP/AoLP become highly sensitive to noise and offsets; AoLP becomes speckled or random.
  Discriminator: masking low-S0 or low-DoLP regions stabilizes AoLP immediately (proves numerical instability, not “real polarization”).

H2-4. Polarization-Specific Artifacts (moiré, angle wrap, crosstalk)

Goal: explain why polarization outputs show stripes or angle discontinuities that do not exist in intensity, and provide fast evidence-based isolation: moiré/beat, AoLP wrap, or cross-channel mixing.

Fast isolation rule (2-image test)

• Generate S0 (Intensity) and DoLP for the same ROI.
• If DoLP shows regular stripes or grid patterns that are absent in S0, suspect a polarization-specific artifact chain (demosaic / alias / crosstalk) before blaming the part.

Artifact A — de-moiré / beat-pattern stripes

• What it looks like: periodic stripes in DoLP/AoLP (often aligned to the sensor lattice or a scene grating) while S0 looks relatively normal.
• Root cause: the polarization sampling lattice interacts with scene texture frequency; channel differencing amplifies the beat frequency.
• Fast discriminator: change distance/focus slightly or rotate the textured target—if stripe frequency/phase shifts sharply, it is lattice/texture beating.
• First fix: prefer super-pixel output for robustness, or apply light edge-preserving smoothing in the polarization domain (DoLP/AoLP) after validating registration.

Artifact B — AoLP wrap (angle discontinuity)

• What it looks like: abrupt AoLP “jumps” (e.g., +89° to −89°) forming hard seams or color breaks.
• Root cause: AoLP is an angle; the chosen representation range (±90° or 0–180°) creates an inherent discontinuity.
• Fast discriminator: apply unwrap (or convert to a continuous representation) and re-check continuity. If the seam disappears, it was representation—not interpolation.
• First fix: define a single AoLP convention for the system and mask low-DoLP/low-S0 regions where AoLP is physically meaningless and numerically unstable.

Artifact C — crosstalk (channel mixing → DoLP bias)

• What it looks like: DoLP appears too high/too low in highlights or dark regions; polarization “texture” changes with edge direction or pattern orientation.
• Root cause: optical/electrical mixing between analyzer channels distorts S1/S2; leakage/extinction and offset/gain mismatch can compound the effect.
• Fast discriminator: compare a bright specular ROI and a dark ROI across exposure/gain steps—abnormal DoLP behavior suggests mixing + channel mismatch.
• First fix: restore four-channel offset/gain consistency (Chapter 2 checks) before deeper model calibration.

H2-5. ISP Hooks for Polarization (de-moiré, contrast, glare suppression)

Goal: use polarization-only signals (DoLP/AoLP/Stokes) as ISP hooks for controllable gains: suppress polar moiré, reduce specular glare, and enhance polarization contrast without over-smoothing intensity.

Hook 1 — Polar de-moiré (polar-domain vs intensity-domain)

• Why polar-domain helps: beat-pattern striping often appears strongly in DoLP/AoLP while being weak in S0. Filtering in the polarization domain can suppress “polar-only” artifacts without flattening real intensity detail.
• When intensity-domain is unavoidable: if the same periodic pattern is already strong in S0, intensity-domain pre-cleaning may be required before polarization features become stable.
• Fast split rule: if DoLP shows regular stripes that are absent (or much weaker) in S0, prioritize polar-domain de-moiré first.

Hook 2 — Glare suppression (DoLP-guided mask / reweight)

• Core idea: specular glare can be separated as a controllable component by using a DoLP-guided mask (with safety gates). A common pattern is mask = (DoLP high) AND (S0 high) AND (not saturated).
• What the mask enables: reweight glare regions (reduce intensity contribution) or fuse with neighborhood/alternate exposures while preserving non-glare texture.
• Common failure to avoid: saturation breaks the channel relations and can invalidate DoLP; always include a “not saturated” gate.

Hook 3 — Polar contrast & AoLP stability

• DoLP contrast: local contrast on DoLP can reveal features that are weak in S0 (coatings, micro-scratches, stress marks), but should be limited to avoid amplifying noise in low-S0 areas.
• AoLP stabilizer: AoLP is an angle field; apply edge-preserving stabilization with wrap-aware handling and suppress AoLP output where DoLP/S0 indicates low confidence.

H2-6. Calibration for Polarization (angle map, extinction, flat-field)

Goal: make polarization features quantifiable and repeatable by calibrating per-pixel angle response, channel gain/offset consistency, extinction/leakage behavior, and flat-field uniformity.

What must be calibrated (and what breaks if it is not)

• Per-pixel angle offset: prevents spatial AoLP “patchiness” on uniform scenes.
• Channel gain/offset: controls DoLP bias in dark regions and stabilizes thresholds for masking/fusion.
• Extinction / leakage indicator: prevents highlight DoLP distortion when analyzer channels are not ideal.
• Flat-field: removes fixed-pattern structure that otherwise looks like false polarization texture.

Practical calibration method (rotating polarizer + uniform light)

• Jig: uniform illumination (integrating sphere / uniform panel) + rotating linear polarizer + rigid camera mount.
• Capture: a sweep of polarizer angles (multiple steps) plus dark frames and flat frames.
• Fit objective: per-pixel response vs angle should be phase-consistent with expected spacing across I0/I45/I90/I135; residuals should be low and spatially unstructured.

Outputs (maps) that enable engineering acceptance

• Angle map: per-pixel angle offset (or equivalent parameterization) used to stabilize AoLP and DoLP estimation.
• Gain/offset maps: per-channel normalization maps to remove relative channel mismatch.
• Extinction/leakage indicator: a per-pixel (or per-region) quality parameter used for mask confidence and bias control.
• Flat-field map: S0 and/or channel flat-field to eliminate fixed spatial structure.
• Quality mask (optional): flags pixels/regions with high residuals for downstream avoidance.

Temperature drift (keep it practical)

• Why it matters: angle offset and DoLP thresholds can shift with temperature, causing masks and fusion outputs to drift.
• Minimum mitigation: calibrate at multiple temperature points and store a small LUT (or fitted coefficients) used at runtime.
• Boundary note: parameter storage governance and versioning belong to the “Calibration & NVM” page.

H2-7. Synchronization & Multi-Frame Consistency (trigger/FSYNC, timestamp alignment)

Goal: polarization outputs (DoLP/AoLP) stay stable only when trigger, exposure, strobe, and frame timestamps are aligned within a controlled skew/jitter window. This chapter focuses on practical alignment strategy (no PTP algorithm deep dive).

Three synchronization scenarios (and why polarization is sensitive)

• Strobe ↔ exposure alignment (µs-class): strobe drift or overlap error changes highlight energy and can create false DoLP/AoLP changes frame-to-frame.
• Multi-camera polarization fusion: same trigger and aligned frame timing are required before any spatial fusion is trusted.
• Moving targets: time misalignment shifts the sampled physical point and appears as “material polarization change” unless timing is controlled.

Practical engineering strategy (no timing-system deep dive)

• Use one capture event: distribute a single trigger/FSYNC to define exposure start consistently.
• Align by fixed offsets: measure and correct deterministic delay (deskew) between trigger, exposure-active, and strobe enable.
• Gate by confidence: suppress DoLP/AoLP evaluation when exposure is saturated or signal is too low; timing errors are easiest to see on stable ROIs.
• Define a skew/jitter window: specify acceptable timing skew and jitter bounds so verification can be repeated and signed off.

H2-8. Optics & Illumination Constraints (birefringence, lens effects, polarized lighting)

Goal: identify system-level “pollution sources” that bias AoLP/DoLP before algorithms are blamed. Lens/window stress (birefringence) and polarized illumination can shift baselines and inject spatial patterns. This chapter focuses on verification and baseline SOP (no driver/controller implementation).

Lens & window birefringence (symptoms → fastest verification)

• Typical symptom: AoLP shows a structured field (patches or gradients) on a uniform target; the pattern can remain even when the scene lacks polarization structure.
• Fastest verification: capture an “empty scene” or uniform diffuse reference board. If DoLP/AoLP are non-zero or spatially patterned beyond a stable baseline, suspect optics/window stress first.
• Rotation check: rotating the window or camera and observing pattern rotation indicates the bias originates in the optical chain.

Polarized illumination & geometry (baseline shifts that look like defects)

• Typical symptom: DoLP baseline stays elevated (or changes with light direction) even without a polarized target.
• Fastest verification: change illumination angle or add diffusion; if DoLP baseline shifts, illumination/geometry is dominating.
• Engineering requirement: define a system baseline under a fixed illumination setup and treat deviations as a diagnosis signal.

H2-9. Validation Test Plan (pass/fail metrics for DoLP/AoLP)

Goal: make polarization imaging measurable and sign-off ready. This chapter defines tools, test flows, pass/fail metrics, and an executable test matrix that ties conditions to criteria and report outputs.

Minimum lab toolkit (repeatable, low ambiguity)

• Rotating linear polarizer: provides a controllable AoLP ground-truth sweep.
• Uniform light source: isolates spatial artifacts and baseline bias (flat-field style checks).
• Standard reflection targets: diffuse + specular regions for glare-benefit validation.
• Angle stage / rotation table: locks geometry so changes are attributable to polarization, not pose drift.

Metrics (what to measure, how to interpret)

Scenario benefit validation (prove controllable gain)

• Glare suppression gain: show Intensity vs DoLP mask vs glare-reduced output under the same setup.
• Defect contrast gain: quantify contrast uplift on scratches/coatings/stress patterns using consistent geometry.

Test Matrix (Conditions × Metrics × Criteria)

H2-10. Field Debug Playbook (symptom → evidence → isolate → fix)

Goal: provide a repeatable, evidence-first troubleshooting SOP for polarization pipelines. Each symptom uses the same structure: First 2 measurements → Discriminator → First fix.

Playbook template (use for every symptom)

Symptom playbooks (high-frequency field issues)