fullseye

filtered_backprojection — TOMOGRAPHY reconstruct op

• データ種: sinogram → image2d
• 呼び出し: import fullseye as fs; fs.ledger.filtered_backprojection(sinogram, angles_deg=None, size=None, filter_name='ramp', cutoff=1.0, span_deg=None, _op='filtered_backprojection') (実装を直接呼ぶなら import tomography; tomography.filtered_backprojection(sinogram, angles_deg=None, size=None, filter_name='ramp', cutoff=1.0, span_deg=None, _op='filtered_backprojection')、台帳から引くなら opstomography.get("filtered_backprojection"))

使い方

Filtered back-projection (FBP) — the standard CT reconstruction.

Filter each projection along the detector axis with the ramp |f| (times an optional apodisation window), then back-project. This is the discretised inverse Radon transform, and with enough samples it is exact: reconstructing a uniform disc of density 1.0 from its analytic sinogram returns an interior mean of 1.0011, and — since 2026-09-06 — the same 1.0011 at 363 and at 727 detector bins, and at 180, 360 and 720 views. The old text here read 0.9954 at 363 bins “converging as the detector is refined”; that was not convergence but the ramp’s missing DC bin, whose size is set by the FFT pad length (:func:_ramlak_spectrum). The absolute value is what pins the ordinary-versus-angular frequency convention in the ramp: the other convention, equally defensible and printed in the same textbooks, would return 2*pi times this, and a CT slice has no absolute grey level for anyone to notice against.

Where it breaks, measured on the Shepp-Logan phantom (256 px, analytic sinogram so the projector contributes no error of its own; normalised RMS error against the truth):

views     FBP (ramp)    SART (10 sweeps)    FBP/SART
  180        0.0250          0.0175           1.43
   90        0.0454          0.0195           2.33
   45        0.1039          0.0353           2.95
   32        0.1362          0.0497           2.74
   16        0.2341          0.0859           2.72
    8        0.3635          0.1257           2.89

There is no crossing point, and the expectation that there would be one was wrong. The received story is that FBP wins when the data is complete and loses only in the sparse regime; measured here, SART with a non-negativity constraint is better at every view count — by 1.43x at 180 views and by about 2.9x once the scan is sparse. What changes with the view count is the price, not the ranking: at 180 views SART costs 312x the wall clock (37.7 s against 0.12 s for a 256-px slice) to buy that 1.43x, which is why filtered back-projection is what production scanners run. At the sparse end the same 2.9x comes nearly free, because both methods scale with the views.

With noise the ranking holds but the margins change, and the apodisation windows stop being decoration (Poisson counts at I0 = 2e4, same phantom):

views    FBP ramp    FBP hann    SART (10 sweeps)
  180      0.0360      0.0371         0.0291
   45      0.1159      0.0766         0.0385
   16      0.2481      0.1921         0.0864
    8      0.3813      0.3093         0.1259

At 180 views the exact ramp beats Hann — the data is complete and the roll-off only blurs. At 45 views and below Hann beats the exact inverse by up to 1.5x, because the frequencies the ramp is busy amplifying were never measured.

Filters, and what they trade: "ramp" is the exact inverse and therefore the sharpest and the noisiest; "shepp-logan", "cosine", "hann" and "hamming" roll the high frequencies off, in that order of aggressiveness. "none" skips the filter entirely and gives :func:backproject_sinogram.

:param sinogram: (n_angles, n_detectors), rows = angles. :param angles_deg: view angles in degrees; None -> uniform [0, 180) with one view per row. :param size: output side; None -> the inscribed square. :param filter_name: one of :data:FILTERS. :param cutoff: fraction of Nyquist to keep, (0, 1]. :param span_deg: angular range for the d(theta) weight; None -> the range the views actually cover, inferred from the angle list (exact for a uniform grid over any span and for any full-coverage irregular set such as golden angle; see :func:_span_weight). :returns: (size, size) float64 image. :raises ValueError: on a non-2-D or non-finite sinogram, an angle count that disagrees with the row count, an unknown filter, a cutoff outside (0, 1], or an output over :data:MAX_IMAGE_ELEMENTS.

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
• poc_ct_fidelity — py -3.11 examples/poc_ct_fidelity.py

型が繋がる次の op(image2d を入力に取れる)

radon_transform

同カテゴリ(reconstruct)

backproject_sinogram · sart_reconstruct

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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.

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