fullseye

sinogram_center_of_rotation — TOMOGRAPHY geometry op

使い方

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.

参考(サンプルデータ・文献)

実行できる例(この op を実際に呼ぶ検証済みサンプル)

型が繋がる次の op(measurement を入力に取れる)

—

同カテゴリ(geometry)

sinogram_center_shift

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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