Cutoff-Free ECT Perimeter Feature

(1) Integration point: use this as a fixed geometric side channel for a 2D shape classifier or as an auxiliary loss for a segmentation model. For classification, take a binary mask or point cloud M from each sample, compute a scalar perimeter estimate P_hat, normalize it by image width plus height, and concatenate it to the penultimate CNN or PointNet feature before the final MLP. For segmentation, compute P_hat on the predicted mask and add λ|P_hat(prediction)−P_target|, where P_target is computed once from the ground-truth mask. (2) Pseudocode: choose K directions v_k=(cos(2πk/K), sin(2πk/K)) and weights w_k=2π/K; choose anchor a=(0,0), or the image center after consistently centering coordinates. For each direction, assign every foreground pixel, cubical cell, or point its height h=<x,v_k>. Build a lower-star filtration by sorting distinct heights. At each height, update the Euler characteristic using χ=number of connected components minus number of holes for the current binary cubical complex; record c_{k,j} on each interval. Set b_{k,j}=1[t_{k,j}≥<a,v_k>] and accumulate (c_{k,j}−χ(M)b_{k,j})Δt. Average or median estimates over several direction rotations. (3) The theorem supplies the cancellation and equality to perimeter. The implementation estimates χ and critical heights from a pixel cubical complex, point-cloud alpha complex, or a differentiable/topologically relaxed complex. For segmentation, use the exact estimator in the forward pass with a straight-through gradient, or replace hard foreground membership by soft threshold samples and finite differences while retaining the theorem-based value as the evaluation metric. (4) First experiment: train a small ResNet-18 on rotated MNIST silhouettes, MPEG-7 binary shapes, or synthetic polygons and circles, comparing the baseline CNN, raw sampled ECT features, and CNN plus the single P_hat feature. Randomly translate and rotate test shapes while preserving labels. Measure clean and transformed accuracy, calibration, and feature variance across isometric copies; success is improved transformed accuracy and substantially lower P_hat variance across copies without more than a small CPU preprocessing cost. For segmentation, compare U-Net with and without the perimeter loss on noisy synthetic shapes and report boundary F-score, IoU, and whether thin spurious components are reduced at equal parameter count.