beam_hardening_apply — TOMOGRAPHY artifact op

• データ種: sinogram → sinogram

• 呼び出し: import tomography; tomography.beam_hardening_apply(sinogram, high_energy_fraction=0.5, attenuation_ratio=0.4) (または opstomography.get("beam_hardening_apply"))

使い方

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 :func: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 :func: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 :func: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.

参考(サンプルデータ・文献)

• サンプルデータ カタログ(DL URL / ライセンス) — 2-D は skimage.data(BSD/public)+ 合成、3-D は実データ源(Stanford/PDS 等)の DL URL。

• 演算子の来歴・参考文献 — この op 族の元になった研究/手法の出典。

• アルゴリズムの正典(著者・年)と用途は上記ファミリ使い方ガイドに記載。

実行できる例(この op を実際に呼ぶ検証済みサンプル)

• ct_reconstruction — py -3.11 examples/ct_reconstruction.py

型が繋がる次の op(sinogram を入力に取れる)

backproject_sinogram · filtered_backprojection · sart_reconstruct · beam_hardening_correct · ring_artifact_apply · ring_artifact_remove · metal_trace_interpolate · sinogram_center_of_rotation

同カテゴリ(artifact)

beam_hardening_correct · ring_artifact_apply · ring_artifact_remove · metal_trace_interpolate

*Provenance: tomography.py — TOMOGRAPHY operator registry. この per-op ノートは tools/opdocs.py md が自動生成(手編集しない)。*

© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.