imaging op• Data kinds: image2d → pairs
• Call: import fullseye as fs; fs.ledger.ensquared_energy(psf, pixel_pitch_um=1.0, samples=64, max_half_width_um=None, centre='centroid') (to call the implementation directly, import optics; optics.ensquared_energy(psf, pixel_pitch_um=1.0, samples=64, max_half_width_um=None, centre='centroid'); from the registry, opsoptics.get("ensquared_energy"))
Ensquared energy of a point-spread function: the fraction of the energy inside a square of half-width w.
The detector-pixel version of encircled energy: how much of the spot falls inside a square pixel (side 2 w) centred on it. The square is axis-aligned with the array and centred like encircled_energy (same *centre* options). The edge pixels are weighted by the exact overlap, so the curve is exact for a PSF that is constant across each pixel.
Returns a (samples, 2) float64 pairs array: column 0 the half-width in micrometres (the pixel side is twice that), column 1 the fraction of the array's energy inside, non-decreasing in [0, 1].
Ground truth it reproduces: a square pupil of side D has the separable PSF sinc^2(D x / (lambda z)) sinc^2(D y / (lambda z)), whose ensquared energy is S(a)^2 with S(a) = (2/pi) (Si(2 pi a) - sin^2(pi a) / (pi a)) and a = w / (lambda N) (checked in tests/test_optdesign_psf_metrics.py).
Raises ValueError: as encircled_energy (with *max_half_width_um*).
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.
• optical_design_metrics — py -3.11 examples/optical_design_metrics.py
pairs as input)mtf50 · energy_to_photons · spectral_product · spectral_integral · spectral_stats · kubelka_munk_reflectance · hyperspectral_capture · film_thickness_fft
imaging)psf_to_mtf · mtf_diffraction · encircled_energy · energy_radii · mtf_tangential_sagittal · wavefront_stats · edge_spread · sfr_from_edge
*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.