imaging op• Data kinds: table → table
• Call: import fullseye as fs; fs.ledger.wavefront_stats(coeffs, radial=128, angular=192) (to call the implementation directly, import optics; optics.wavefront_stats(coeffs, radial=128, angular=192); from the registry, opsoptics.get("wavefront_stats"))
Wavefront error statistics from a Zernike expansion: RMS, PV and Strehl.
*coeffs* is exactly the dict match3d.fit_zernike returns — {(n, m): coefficient}, coefficients in waves — and this op re-uses match3d's own basis builder, so the two cannot drift apart in normalisation or in the (n, m) convention. Fit with fit_zernike, characterise here.
The wavefront is reconstructed on a polar grid (*radial* x *angular*) over the unit pupil and reduced with the area element rho d(rho) d(theta) — an unweighted mean over a uniform-in-rho grid over-counts the centre and is a real, silent, ~10% error.
Returns a dict: rms_waves (piston removed — piston is not an aberration) · pv_waves peak-to-valley over the pupil · strehl the Marechal estimate exp(-(2*pi*rms)^2) · marechal_valid whether rms_waves <= MARECHAL_RMS_LIMIT (0.1), because past that the estimate is optimistic and reporting the number without the caveat is the dishonest option · terms and n_max of the expansion.
Ground truth it reproduces (measured at the defaults): pure defocus {(2, 0): 0.1} — for which Z = 2*rho^2 - 1 has an exact pupil RMS of 1/sqrt(3) — gives rms_waves = 0.0577422 against the exact 0.0577350, a relative error of 1.2e-4 from the discrete quadrature (3.1e-5 at radial=256), and pv_waves = 0.2 exactly; the Strehl is 0.8766676 against the exact 0.8766962. Pure astigmatism {(2, 2): 0.1} (exact RMS 1/sqrt(6)) gives 0.0408280 against 0.0408248. Piston alone ({(0, 0): c}) gives rms 0 and Strehl 1 for any c, and RMS scales exactly linearly in the coefficients (doubling them doubles the RMS to machine precision).
Raises ValueError: *coeffs* is not a dict, is empty, holds more than MAX_ZERNIKE_TERMS terms, has a key that is not an (n, m) int pair or is not a valid Zernike index (n >= 0, |m| <= n, n-|m| even), or a non-finite coefficient; a radial order above MAX_ZERNIKE_ORDER (40 — the shared basis builder's factorial recurrence breaks its own |Z| <= 1 bound at n = 46, measured, and the same bound is re-checked at runtime); *radial* / *angular* outside [8, MAX_GRID].
The radial quadrature is discrete, so its error grows with the order being
integrated: measured, the relative RMS error tracks (n_max/radial)^2 within a factor 2 — 1.2e-4 at n_max=2, 1.7e-3 at n_max=6 and 4.3e-2 at n_max=20, all at the default radial=128. Below radial >= 16*n_max a RuntimeWarning says so rather than letting a 12%-wrong Strehl look authoritative. Raising *radial* fixes it at O(1/radial^2), but note the basis is built for *all* orders up to n_max, so the working set grows as n_max^2 * radial * angular and is capped by MAX_ZERNIKE_BASIS.
Marechal is a small-aberration approximation and the RMS is over the *fitted* expansion, so it says nothing about wavefront structure finer than n_max — and fit_zernike itself discloses ~10% inter-mode crosstalk at its default sampling. Both limits compound; treat the Strehl as an indicator, not a measurement.
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.
• measurement_uncertainty — Measurement uncertainty and calibration — what it takes to claim you are measuring
• mv_cameras — Industrial camera makers (sensor mapping, line scan / TDI)
• virtual_machine_vision — Virtual machine vision — enumerating the parameters and an object model
• 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.
• optics_imaging — py -3.11 examples/optics_imaging.py
• poc_zernike_aberrations — py -3.11 examples/poc_zernike_aberrations.py
table as input)abcd_matrix · sfr_from_edge · paraxial_trace · seidel_coefficients · spot_stats · tolerance_analysis · wavefront_from_opd · spot_diagram
imaging)psf_to_mtf · mtf_diffraction · edge_spread · sfr_from_edge · mtf50 · veiling_glare_index
*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.