shape_from_polarization op• Data kinds: polsweep → table
• Call: import fullseye as fs; fs.ledger.sfp_normals(sweep, refractive_index=1.5, model='specular', angles_deg=(0.0, 45.0, 90.0, 135.0), violation_tol=0.05, out_of_range='invalid') (to call the implementation directly, import sfpolar; sfpolar.sfp_normals(sweep, refractive_index=1.5, model='specular', angles_deg=(0.0, 45.0, 90.0, 135.0), violation_tol=0.05, out_of_range='invalid'); from the registry, opsoptics.get("sfp_normals"))
• GPU: none (CPU reference implementation only)
From a polariser sweep to the explicit set of candidate surface normals per pixel — 2 for the diffuse model, 4 for the specular model (2 zeniths x 2 azimuths).
*sweep* is an (N, H, W) stack through linear analysers at *angles_deg* (the polsweep sort; optics.polarization_demosaic gives it from a raw mosaic). Per pixel the sinusoid I(t) = a0 + a1 cos 2t + a2 sin 2t is fitted (the same least-squares fit as specularity.polarization_dolp_map), giving DoLP = hypot(a1, a2)/a0 and AoLP = 0.5 atan2(a2, a1); then dolp_to_zenith and aolp_to_azimuth give the zenith and azimuth candidates.
A fitted minimum below zero (modulation above the mean, DoLP > 1) is impossible for light. Near Brewster's angle the true minimum is ~0, so sensor noise pushes it slightly negative; *violation_tol* is the largest excess DoLP - 1 that is read as noise (DoLP clipped to 1). A larger excess marks the pixel invalid — the frames do not match *angles_deg*, or the pixel saw something other than a polariser sweep. Such a pixel is reported in violation, never silently clipped. (specularity.polarization_dolp_map refuses the whole image instead; a shape map with one hot pixel is still a shape map, so here the decision is per pixel and sfp_disambiguate refuses invalid pixels inside its mask by default.)
*out_of_range*: a DoLP that no zenith produces (diffuse model, above (n^2-1)/(n^2+1)) makes the pixel "invalid" (default) or is "clamp"-ed to the reachable limit and kept, flagged in clamped.
Returns a dict (a table, marked kind == "sfp_candidates"):
• candidates (K, H, W, 3) unit normals; specular order is [low zenith & azimuth 0, low & azimuth 1, high & azimuth 0, high & azimuth 1], diffuse [azimuth 0, azimuth 1]. Zero vectors at invalid pixels (see valid) — never NaN, never a made-up unit normal.
• zenith_deg (1|2, H, W), azimuth_deg (2, H, W), dolp, aolp_deg, intensity (S0) (H, W).
• n_distinct (H, W) — how many genuinely different normals the pixel admits: 4 (specular), 2 at Brewster's angle or for the diffuse model, 0 where nothing fits or the pixel is dark.
• valid (H, W) bool — lit, physically possible and invertible; violation / clamped (H, W) bool — why a pixel is suspect.
• dark (H, W) bool — the fitted mean a0 = S0/2 is at most specularity.POLAR_DARK_K (3) × the noise of the modulation amplitude σ_A = σ̂·√(v₁+v₂) (v = the diagonal of the fit's (MᵀM)⁻¹; σ̂ = one noise level for the image, from the median per-pixel residual of the N > 3 frames, χ²-median corrected), i.e. the DoLP's own noise would be 1/3 or more: no degree of polarisation is defined there, so the pixel is not lit (valid False, n_distinct 0). ★Before 2026-11-09 only a0 > 0 was required, and clipped sensor noise in a dark region (σ 0.001 of a scene peaking at 1) gave DoLP ≈ 1 and Brewster-angle normals on 26–28 % of the dark pixels, all valid. (A first fix used a floor relative to the brightest 1 % of the image, which ignored the noise: it killed a dark diffuse object next to specular highlights and fell into the noise when the bright part was under 1 % of the image.) A frame that does not belong to the sweep raises the residual to the level of the signal, and then every pixel is dark — the sweep defines no DoLP anywhere (fail-closed); the violation flags are still computed independently of dark. With three angles there is no residual and only a0 <= 0 is dark — noise-only dark pixels are not screened.
• dolp_noise (H, W) — the DoLP's noise level σ_A/a0 (0 on unlit pixels — they have no DoLP, see valid / dark), and noisy = lit pixels with dolp_noise > violation_tol — a flag only (valid is not changed by it): their DoLP, hence their zenith, is mostly noise. With three angles dolp_noise is None and noisy all False. ★One σ̂ for the image assumes stationary noise; shot noise grows with the signal, so treat the flag as a screen.
• model, refractive_index, brewster_deg.
Nothing here chooses among the candidates; a single "best" normal map returned at this stage would be a guess dressed as a measurement.
Raises ValueError: not an (N, H, W) stack of >= 3 finite non-negative frames; angles that do not match or do not determine the fit; *violation_tol* outside [0, 1]; index not > 1; unknown *model* / *out_of_range*.
Every optics op validates its input before computing (nothing slips through silently):
• Units are baked into the argument name — _mm / _um / _deg / _mrad. Confusing mm with µm does not crash; it yields a plausible-looking wrong answer, so the name prevents it. Nothing here guesses the unit from the magnitude.
• Strings raise ValueError — float('50') succeeds, so an unparsed configuration value would slip through as a length (measured: thin_lens('50', '200') returned a plausible 66.667 mm). bool is refused too, as the implicit promotion True == 1.
• complex / masked arrays raise ValueError (real-valued slots only; silently dropping the imaginary part or peeling off the mask is refused). NaN/Inf raises ValueError on every input. Exception, declared: the u_map of bottle_fit_from_pattern takes pixels that could not be measured as NaN (infinity is still refused).
• Division by zero and its relatives are refused by name: focal length 0, radius of curvature 0, refractive index <= 0, a fully opaque aperture (all zeros, so the normalisation is 0/0), a PSF whose sum is <= 0, a Stokes vector with S0 = 0, and an object sitting at the front focal point (the image is at infinity).
• Only two ops return a non-finite value, and both state it as a contract: depth_of_field returns far_mm = inf beyond the hyperfocal distance (that is what the hyperfocal distance means), and gaussian_beam returns wavefront_radius_mm = inf at the waist (the radius of curvature of a plane wavefront). Both also return a finite companion (far_is_infinite / curvature_per_mm). Any other silent NaN/Inf is detected internally and raises ValueError — "float64 overflowed" and "the answer is infinite" are different claims, so the first is never returned wearing the face of the second. Exception, declared: bottle_background_map returns NaN in u_mm for pixels whose ray is unusable (total internal reflection and the like), together with a valid mask and the counts.
• Size caps: generated grids are capped by optics.MAX_GRID (4096); supplied fields/PSFs/apertures by optics.MAX_FIELD_ELEMENTS (2^24); ABCD element chains by optics.MAX_SYSTEM_ELEMENTS (1024); Zernike by MAX_ZERNIKE_TERMS (512) / MAX_ZERNIKE_ORDER (40) / MAX_ZERNIKE_BASIS (2^25). This closes, fail-closed, the paths where a small argument triggers a huge internal allocation (measured: n_max=40 × 4096² needs 108 GB).
• Physically impossible states are refused too: a Stokes vector with degree of polarisation > 1, negative transmittance, negative intensity, and invalid Zernike indices such as n-|m| odd.
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_shape_from_polarization — py -3.11 examples/poc_shape_from_polarization.py
table as input)abcd_matrix · wavefront_stats · sfr_from_edge · paraxial_trace · seidel_coefficients · spot_stats · tolerance_analysis · wavefront_from_opd
shape_from_polarization)sfp_dolp_model · dolp_to_zenith · aolp_to_azimuth · sfp_render · sfp_disambiguate · sfp_estimate_index · sfp_height
*Provenance: sfpolar.py — OPTICS operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.