artifact op• Data kinds: sinogram → sinogram
• Call: import fullseye as fs; fs.ledger.beam_hardening_apply(sinogram, high_energy_fraction=0.5, attenuation_ratio=0.4) (to call the implementation directly, import tomography; tomography.beam_hardening_apply(sinogram, high_energy_fraction=0.5, attenuation_ratio=0.4); from the registry, opstomography.get("beam_hardening_apply"))
Turn a monochromatic sinogram into a polychromatic one — cupping.
A real X-ray tube emits a spectrum, and low-energy photons are absorbed more, so the beam that survives a thick path is *harder* (higher mean energy) and therefore attenuated less per unit length than the beam that survives a thin one. The line integral stops being linear in path length, and the reconstruction of a uniform object comes back with a depressed centre: the cupping artefact.
The two-spectrum model used here is the smallest one that is physics and not a
curve::
I/I0 = (1-w) exp(-p) + w exp(-k p)
p_meas = -ln(I/I0)
with *w* the fraction of the beam at the high energy and *k < 1* its relative attenuation. It is exact at p = 0, concave everywhere, and monotone — so it is invertible, which is what beam_hardening_correct inverts.
Measured on a uniform disc (radius 60 px in 256 px, density 1/60 so the peak line integral is 2.0) at w = 0.5, k = 0.4: the FBP reconstruction's centre-to-rim ratio drops to 0.9312, against 0.9981 before hardening,
and beam_hardening_correct returns it to 0.9981 — the clean value in all four digits. (The clean ratio is 0.9981 rather than exactly 1 because of the detector sampling discussed in filtered_backprojection. The cupping is the 6.7-point drop, not the 0.2-point one.)
:param sinogram: (n_angles, n_detectors) monochromatic line integrals,
which must be >= 0.
:param high_energy_fraction: *w*, in [0, 1). 0 is a monochromatic beam and
the operator is then the identity.
:param attenuation_ratio: *k*, in (0, 1). 1 is again monochromatic. :returns: (n_angles, n_detectors) float64 hardened sinogram. :raises ValueError: on a negative sinogram (a negative line integral is not a
measurement this model can harden — the logarithm of the transmitted
intensity has already gone wrong upstream), or parameters outside range.
• 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.
• ct_reconstruction — py -3.11 examples/ct_reconstruction.py
• poc_battery_ct_degradation — py -3.11 examples/poc_battery_ct_degradation.py
sinogram as input)backproject_sinogram · filtered_backprojection · sart_reconstruct · beam_hardening_correct · ring_artifact_apply · ring_artifact_remove · metal_trace_interpolate · sinogram_center_of_rotation
artifact)beam_hardening_correct · ring_artifact_apply · ring_artifact_remove · metal_trace_interpolate
*Provenance: tomography.py — TOMOGRAPHY 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.