Homotopy equivalence of finite digital images
This work addresses a foundational issue in digital topology for researchers in computational geometry and image analysis, though it appears incremental as it builds on existing homotopy equivalence concepts.
The paper tackled the problem of homotopy equivalence in digital images, where classical invariants like Euler characteristic fail, by developing a new numerical invariant and cataloging all connected digital images on small point sets up to homotopy equivalence.
For digital images, there is an established homotopy equivalence relation which parallels that of classical topology. Many classical homotopy equivalence invariants, such as the Euler characteristic and the homology groups, do not remain invariants in the digital setting. This paper develops a numerical digital homotopy invariant and begins to catalog all possible connected digital images on a small number of points, up to homotopy equivalence.