CT Angular-Longitudinal Diagram and SSPz Geometry Reference

The horizontal axis is the distance between a candidate detector-row center and the target plane; the vertical axis is the direct-side relative tube angle β. Solid lines show all N rows on the direct side, and dashed lines show all N rows on the complementary side paired with that angle. Complementary-side positions are plotted against the same reference-angle axis while retaining their longitudinal positions at acquisition; their acquisition times and table positions are not shifted to those of the direct view. Candidate searching retains all N rows in every acquired view and does not exclude candidates according to configured slice thickness T. The left column is a parallel-beam reference with a complementary-angle offset of 180° and a distance ratio of 1. The right column uses the fan-beam complementary angle for the ray through the evaluation point and the source-to-point distance at the acquisition angle. The two conditions coincide at isocenter. When the full-scan comparison is selected, only the solid direct-side trajectories are shown.

These diagrams are not restricted to a fixed central fan channel. The ray toward the selected transverse position is determined at each acquisition angle, so its position within the fan changes during rotation for an off-center evaluation point. The dashed complementary-side trajectories show the acquired views immediately before and after the ideal complementary angle β_c=β+180°+2γ. Here, γ is the signed angle from the fan-center direction to the direction of the evaluation point. Lines connect the positions calculated at every acquired angle as visual guides; they do not represent additional measured data between acquired views. The discrete arrangement of in-plane detector channels is not reproduced.

For the right column, the distance ratio to a fixed evaluation point is defined as q(β) = L(β)/R = √{1 + (r/R)² − 2(r/R)cosβ} Here, R is the source-to-isocenter distance, r is the isocenter-to-point distance, β is the relative tube angle with 0° defined where the source is in the same radial direction as the evaluation point, and L(β) is the source-to-point distance at that angle. The distance ratio q increases from 0° to 180° and decreases from 180° to 360°. In contrast, the difference from the parallel-beam approximation, q − 1, changes sign within one rotation, so the magnitude of its effect decreases and then increases again. A candidate row center is calculated by adding the detector-row position scaled by q(β) to the linear longitudinal displacement from table motion. On the complementary side, the table position and distance ratio are evaluated at the actual complementary-data acquisition angle, not at β on the display axis. Consequently, the solid and dashed trajectories are not related by a simple 180° shift.

Thin lines show the same all-row candidate trajectories as in 2A. Color identifies the detector row. A crossing does not transfer a color to a different row. Overlapping trajectories are shown by color blending. A blended color indicates overlap, not another detector row or a weight. Lines are made fainter for many-row displays. Circles denote selected direct-side candidates and triangles complementary-side candidates. A circle inside a triangle identifies an acquired sample assigned to both sides. Saturation represents total weight w after summing contributions from the two angular branches for the same acquired sample (the same acquired view and detector row). Different acquired samples are not summed merely because they occupy the same position. This shows local FW=0 interpolation at the central position, not all contributors to a thick-slice response. Filter interpolation reselects candidates at each position within FW, using acquired samples outside FW when needed for bracketing. Markers show selected angles; trajectories use every acquired angle.

The horizontal axis is reconstruction-plane position relative to a fixed thin object, zᵣ−zₒ. For each state the object is fixed, the reconstruction plane is scanned, and acquired candidates are reselected at every filter position. Each curve is peak-normalized without additional peak or centroid alignment. The displayed range is derived from the central shape at or above 10%; low-amplitude tails are shown separately.

The filter-interpolated SSPz curves with cone-geometry scaling from 3A are displayed with a logarithmic vertical axis. This separates broad low-amplitude tails, which may persist even when FWHM is nearly constant, from the central profile. Values below 1% are not drawn.

FWHM, FWTM, and standard deviation are calculated from the filter-interpolated SSPz and divided by reference thickness T. T is neither a broadening operation nor a target FWHM. Distinguish effects of changing FW from changes in the ratio caused by T. Inspect tails and asymmetry as well as width metrics.