geometry op• Data kinds: sinogram → measurement
• Call: import tomography; tomography.sinogram_center_of_rotation(sinogram, angles_deg=None, min_condition=0.02) (or opstomography.get("sinogram_center_of_rotation"))
Where the axis of rotation actually is, in detector bins from the centre.
The centre-of-mass identity, which is exact and needs no reconstruction: the
first moment of a projection is the projection of the object's centre of
mass, so::
s_cm(theta) = x0 cos(theta) + y0 sin(theta) + c
with `(x0, y0) the centre of mass in the slice and c` the offset of the
rotation axis from the detector centre. Fitting the three unknowns by least
squares over all views gives *c* directly. Measured on the Shepp-Logan
phantom with 180 views, recovering a deliberately introduced shift, together
with the cost of not correcting it (normalised RMS error of the FBP
reconstruction against the truth):
true shift estimated error uncorrected after this fix
0.00 px +0.0029 px 0.0029 0.0250 0.0249
0.50 px +0.5029 px 0.0029 0.0537 0.0358
1.00 px +1.0029 px 0.0029 0.1016 0.0249
2.00 px +2.0029 px 0.0029 0.1630 0.0249
Three things in that table are worth reading twice. **Half a pixel already
doubles the error** (0.0250 -> 0.0537) and does not look like a mistake — it
looks like a slightly soft reconstruction, which is why this is a measurement
and not an inspection. The estimator's own bias is a constant 0.0029 px
across every shift, so it is a property of the phantom and the detector
sampling, not of the size of the error being measured. And the half-pixel row
is the only one the fix does not fully repair (0.0358 against 0.0249),
because correcting a *fractional* shift means resampling, and the linear
interpolation costs more than the integer shifts do — see
:func:sinogram_center_shift.
Two things this needs, both refused rather than assumed. The object must be
entirely inside the field of view — the identity is about the whole mass,
and a truncated object has a different mass at every angle. And the views must
span enough angle for `[cos, sin, 1]` to be independent: over a narrow
wedge, `cos(theta)` and the constant are nearly the same vector and the fit
puts the object's own offset into *c*. The condition number is checked and a
degenerate design is refused.
:param sinogram: `(n_angles, n_detectors)`, rows = angles.
:param angles_deg: view angles; `None -> uniform [0, 180)`.
:param min_condition: smallest acceptable reciprocal condition number of the
`[cos, sin, 1]` design matrix. The default of 0.02 is calibrated,
not chosen — the reciprocal condition number and the error it lets
through, on a sinogram with a true 1.00-px shift:
span rcond estimate error
180 deg 2.15e-01 +0.9939 0.0061 px
120 deg 8.92e-02 +0.9900 0.0100 px
90 deg 4.85e-02 +0.9485 0.0515 px
60 deg 2.09e-02 +0.8993 0.1007 px <- the default admits this
45 deg 1.17e-02 +0.7041 0.2959 px <- and refuses this
20 deg 2.26e-03 +1.7113 0.7113 px
10 deg 5.54e-04 -6.9765 7.9765 px
The 10-degree row is why the check exists at all: the answer is finite,
the sign is wrong, and the magnitude is eight pixels on a one-pixel
error. Nothing about it looks like a failure.
:returns: `float` — the axis offset in detector bins, positive towards
higher bin indices.
:raises ValueError: on a sinogram whose total mass is zero, on fewer than 3
views, or on an angular span too narrow to separate the offset from the
object's own position.
• 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
measurement as input)—
geometry)*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.