Visibility on continuous shapes is expensive but conceptually clean: two points see each other if the line segment between them stays inside the shape. On digital shapes — shapes defined on a grid — the concept fractures. What does “inside” mean when the boundary is a staircase of pixels? The naive approach (connect lattice points, check if the segment stays within pixels) is both slow and inaccurate, because the digital boundary doesn't have a well-defined tangent or normal.
The solution (arXiv:2603.11851) exploits the specific structure of digital sets. Each row of the shape is an integral interval — a contiguous range of integers. The visibility computation between lattice points reduces to checking whether a discrete line (a sequence of lattice points approximating a continuous segment) stays within the union of these intervals. This representation makes the computation both exact and fast: no floating-point geometry, no epsilon tolerances, just integer arithmetic on interval endpoints.
The application: normal estimation. Given the visibility graph — which boundary points can see each other — the local normal at each point is estimated from the geometry of its visible neighborhood. Points near a corner see a restricted set of neighbors; points on a flat edge see a wide set. The visibility structure automatically detects salient features (corners, cusps) because those features create shadows in the visibility graph.
The structural insight: visibility is the right primitive for feature detection on digital shapes. Rather than detecting corners explicitly through curvature estimation (which requires smoothing, which destroys the corners), visibility detects them implicitly through occlusion. The corner isn't characterized by its local geometry — it's characterized by what it hides.