Interleaving-Calibrated Pixel Topology

Integrate this into a radial-persistence feature pipeline used during segmentation training, shape classification, or consistency regularization. For every contour extracted from an image, record the physical pixel spacing h and compute vertex radii in physical coordinates rather than raw pixel indices. Build the boundary filtration with vertex value r_i=||x_i-c||_2 and edge value max(r_i,r_j). Compute extended persistence intervals and assign each interval a confidence weight w=min(1,lifetime/(2*h)); use this weight in the topology loss, feature pooling, or teacher-student consistency term. Pseudocode: coords=voxel_spacing*pixel_coords; h=max_edge_length_bound; bars=radial_extended_persistence(coords,c); for (b,d) in bars: weight=min(1,abs(d-b)/(2*h)); L_top += weight*match_cost(b,d,target_bars). For multi-resolution augmentation, recompute h after resizing and either normalize radii by physical units or update the threshold automatically. The mathematical quantity computed from the paper is the uniform function-approximation and interleaving tolerance; the engineer estimates the actual local h from the contour mesh and may replace the conservative global h by the maximum edge length. First test a U-Net on synthetic circles, annuli, and branching shapes rendered at several resolutions, comparing unweighted persistence loss, weighted loss, and no topology loss. Measure topology error under rasterization and random boundary jitter, plus interval survival across resolutions. Success is fewer false topological corrections on sub-pixel features and stable predictions when resolution changes, without sacrificing Dice or increasing persistence computation.