reconstruct op• Data kinds: sinogram → image2d
• Call: import fullseye as fs; fs.ledger.sart_reconstruct(sinogram, angles_deg=None, size=None, n_iter=10, relaxation=0.3, initial=None, nonnegative=True) (to call the implementation directly, import tomography; tomography.sart_reconstruct(sinogram, angles_deg=None, size=None, n_iter=10, relaxation=0.3, initial=None, nonnegative=True); from the registry, opstomography.get("sart_reconstruct"))
SART — simultaneous algebraic reconstruction, one angle at a time.
An iterative solver for A x = p where A is the projector: for each view in turn, project the current estimate, take the residual, and back-project it with the row and column sums of A as normalisers::
x <- x + lambda * BP_theta( (p_theta - FP_theta(x)) / rowsum_theta )
/ colsum_theta
*rowsum* is the length of each ray through the grid and *colsum* is how many rays touched each pixel, so the update is dimensionally a density and does not depend on the grid size. One "iteration" is one pass over all views.
Why it exists next to filtered_backprojection: FBP inverts an integral transform and therefore *needs* the transform to have been sampled; SART solves a linear system and merely does worse when the system is underdetermined. Measured, it is better at every view count tested (the table
in filtered_backprojection), by 1.43x at 180 views and 2.9x at 8.
The cost is honest and it is the reason this is not the default: 10 sweeps over 180 views is 1800 forward *and* 1800 back-projections against FBP's 180 back-projections, measured at 37.7 s against 0.12 s for a 256-px reconstruction — a factor of 312. At 8 views it is 2.14 s against 0.01 s, the same ratio applied to a much smaller number.
nonnegative=True clips the estimate at zero after every sweep. Attenuation coefficients cannot be negative, so this is a genuine constraint and not a cosmetic clip, and it carries a large part of the advantage above — measured on the analytic Shepp-Logan sinogram, normalised RMS with the constraint against without:
views with without
180 0.0175 0.0300
45 0.0353 0.0626
8 0.1257 0.1428
so at 180 views the constraint alone is worth 1.7x, and it is the *only* reason SART leads FBP there at all (FBP scores 0.0250, between the two).
:param sinogram: (n_angles, n_detectors), rows = angles. :param angles_deg: view angles; None -> uniform [0, 180). :param size: output side; None -> the inscribed square. :param n_iter: sweeps over the full angle set, 1 .. 500. :param relaxation: step size lambda, (0, 2). Over 1 the iteration can
oscillate; over 2 it provably diverges, and is refused.
:param initial: starting estimate, (size, size); None -> zeros. :param nonnegative: clip to >= 0 after each sweep. :returns: (size, size) float64 image. :raises ValueError: as filtered_backprojection, plus a relaxation
outside (0, 2) and an *initial* whose shape is not (size, size).
• 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
image2d as input)reconstruct)backproject_sinogram · filtered_backprojection
*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.