Anchored periodic phase-manifold registration for respiratory 4D-CT

Background Respiratory 4D-CT registration requires correspondence across multiple phases and a coherent description of cyclic motion. We developed anchored periodic phase-manifold registration (PMR) with shared temporal coefficients and subject-specific optimisation. Methods PMR combines four anchored harmonic pairs, a phase-dependent affine branch and residual refinement. We evaluated ten DIR-Lab and six CREATIS cases at 2-mm working resolution. Seed 0 supplied the primary results. Three seeds were used for the complete model, NoAffine, NoResidual and controlled protocol contrasts; two-harmonic and coarse-only variants used seed 0. Landmark errors and physical-coordinate Jacobians were evaluated from saved fields. Paired differences were averaged within each case before exploratory case-bootstrap analysis. Results Primary mean target registration error (TRE) was 1.586 mm on DIR-Lab and 1.128 mm on CREATIS. DIR-Lab comparator means were 2.555 mm for elastix, 3.032 mm for the VoxelMorph-class implementation and 2.801 mm for the LapIRN-class implementation. Primary structural-ablation means were 1.701 mm for NoAffine, 1.991 mm for NoResidual, 1.717 mm for 2Harmonics and 2.071 mm for CoarseOnly. Across three seeds, NoAffine and NoResidual changed case-averaged TRE by +0.091 mm and +0.484 mm relative to full PMR. Lung-supported similarity reduced mean TRE by 31.8% relative to whole-image support. Adding a ninth temporal function changed mean TRE by +0.033 mm (exploratory 95% interval [−0.016, 0.106] mm). Endpoint identity held at numerical precision, while saved fields contained local folds and concentrated errors at large displacement. Conclusions PMR provides a compact periodic respiratory-motion representation with millimetre-scale mean landmark accuracy on two CT datasets. Residual refinement improved landmark correspondence while increasing folding. The component and protocol analyses identify specific targets for improving large-motion accuracy and spatial regularity.