polarization op• Data kinds: signal → matrix
• Call: import fullseye as fs; fs.ledger.mueller_from_intensities(intensities, psg, psa, rank_tol=1e-09) (to call the implementation directly, import optics; optics.mueller_from_intensities(intensities, psg, psa, rank_tol=1e-09); from the registry, opsoptics.get("mueller_from_intensities"))
Recover a Mueller matrix from intensities measured through known generator / analyser states (polarimetric least squares).
Model: the detector behind a polarisation-state analyser sees
`I_i = (A_i @ M @ G_i)[0, 0] = a_i^T M g_i where g_i = G_i[:, 0]` is
the Stokes vector the generator emits from unpolarised light and
`a_i = A_i[0, :]` is the analyser's first row. That is **linear in the 16
entries of M**: `I_i = w_i . vec(M) with w_i = outer(a_i, g_i).ravel()`.
Stacking the N measurements gives `W (N x 16)`, solved by least squares.
*intensities* is `(N,) (one detector) or (N, ...)` (an image per
state: the same system solved per pixel, so the result is `(4, 4, ...)`).
*psg* and *psa* are sequences of N Mueller matrices (4x4) — build them with
:func:mueller_element and matrix products.
Fail-closed on an under-determined design: if `W` has rank < 16 the
design cannot see every entry (the classic case — linear polarisers only,
no retarder — leaves the S3 row and column unobservable, rank 9) and the
function raises with the rank instead of returning a pseudo-inverse answer
that would look plausible and be wrong in the unobserved entries.
Ground truth in the tests: a known `M` (retarder then polariser),
36 generator/analyser pairs with quarter-wave plates, intensities computed
by the forward model → `M` recovered to 1e-10; the same with polarisers
only → rank 9, refused.
Raises `ValueError`: shapes disagree, fewer than 16 measurements,
non-finite input, or rank < 16.
Provenance: the observation matrix `outer(a, g)` is the standard
formulation (Chipman, "Polarimetry", *Handbook of Optics* ch. 15); the
`pinv variant is what Polanalyser's calcMueller` does. Re-implemented
with an explicit rank check.
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.**
• 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.
• 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.
• polarization_camera_pipeline — py -3.11 examples/polarization_camera_pipeline.py
matrix as input)abcd_trace · mueller_apply · mueller_checks
polarization)jones_element · jones_apply · stokes_from_jones · mueller_element · mueller_apply · stokes_analyze · polarization_demosaic · polarization_demosaic_color
*Provenance: optics.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.