Segmentation interface curvature balance ($\beta \kappa =(c_j-g{)}^2-(c_i-g{)}^2$)

When grey values are fixed but region labels vary, a smooth interface satisfies $\beta \kappa =(c_j-g{)}^2-(c_i-g{)}^2$. The curvature uses the normal pointing out of region $i$, positive on an outward-oriented circle. It balances the change in data cost against the first variation of interface length. Fixing a full spatial image signal instead fixes its jumps and leaves no such boundary-relocation freedom.

At a freely moving regular junction of three smooth image edges with equal isotropic length weights, the endpoint first variation requires the sum of their outward unit tangent vectors to vanish. Three unit vectors with zero sum have pairwise angles of $120$ degrees. Different interface weights instead give a weighted force balance, and constrained or singular junctions require separate hypotheses.