Multi-Scale Latent Reference Pattern Model for Geometric Drift Estimation in Printing Processes

Multi-scale registration and adaptive feature fusion provide established mechanisms for combining spatial information, but their effectiveness does not establish the benefit of combining drift estimates from independently generated reference patterns. This study proposes a multi-scale latent reference pattern model and its mathematical formulation for geometric drift estimation in printing. Coarse, medium, and fine Poisson-disk patterns provide separate observations for estimating translation, small-angle rotation, and fixed-frequency sinusoidal deformation. Three scale-specific estimators are compared with equal-weight and adaptive multi-scale combinations. The main experiment comprises 250 independent synthetic realizations, each evaluated under 16 combinations of Gaussian noise and blur. All three scales produced valid estimates in all 4000 scenario realizations. The fine scale had the lowest individual estimation errors, while equal weighting achieved the lowest mean errors among the five estimators in the main experiment. Mean deformation RMSE was 0.056848 px for equal weighting and 0.058735 px for adaptive weighting. Additional synthetic experiments examine threshold sensitivity, rigid and thin-plate-spline baselines, pooled estimation, and the relationship between adaptive weights and estimation error. Rigid registration was most precise in the rigid-only control, while pooled estimation achieved the lowest mean exact-field error under sinusoidal deformation. The contribution is a mathematically specified reference-pattern model and a controlled evaluation that identifies scale-dependent precision and limitations of the tested reliability indicator. The results support computational feasibility under the specified synthetic conditions. They do not establish an accuracy advantage for adaptive weighting or measurement performance on physically printed and captured patterns.