# Flow, Surgery, and Boundary in Digital Space

## Atsuchi Imiya

Slides can be found by following the link.

There are three ways for the computation of the curvature from discrete objects, combinatorial curvature, which defines curvature codes on the boundary of digital isotactic polygon and polyhedron, discrete curvature, which computes differential invariant from the discrete analog of differential geometry on the polyhedral surface, numerical differential geometry, directly computes differential features from sampled curves and surfaces. These three methods have advantages and disadvantages. This talk mainly focuses on the first definition, with showing the relation to the second method. We derive the following three fundamental properties based on the curvatures.

1. Curvature flow based boundary evaluation by combinatorial curvature code and its application for topology-preserving skeletonisation.

2. Computation of the Betti number and the definition of Euler formula in the digital space.

3. The Mumford-Shah model for boundary extraction in digital space.