1. Cutting a surface into a Disk Jie Gao Nov. 27, 2002

2. Papers Optimally cutting a surface into a disk, by Jeff Erickson and Sariel Har-Peled, in SoCG’02. Geometry images, by Xianfeng Gu, Steven J. Gortler and Hugues Hoppe, in Siggraph’02.

3. Given a 3d mesh M, find a set of edges (called cut graph) whose removal transforms the surface into a topological disk.

4. Outline Theory work: –Minimize the length of the cut graph –Algorithms for exact/approximate solutions. In practice: –Geometry Images. –Heuristics

5. Theory work Optimally cutting a surface into a disk, by Jeff Erickson and Sariel Har-Peled, in SoCG’02. Minimize the total weight of the cut graph. – e.g., the total length of the cut.

6. Definitions M : compact 2-manifold with boundary. Genus g: maximum number of disjoint non- separating cycles of M. k: number of boundary components. 1-skeleton M 1 of M is the graph consisting of all the vertices and edges. A cut graph G is a subgraph of M 1 so that M \G (polyhedral schema) is homeomorphic to a disk. Goal: find a polyhedral schema with minimum perimeter.

7. Computing min cut graph of M with fixed g “or” k is NP- hard: reduction from rectilinear Steiner tree problem. Each point puncture Cut graph Steiner tree To get high genus: attach tori or cross-caps to punctures.

8. Cut path: from one branch point (degree>2) or boundary point to another, without branching point in the middle. Lemma: Any cut path in the minimum cut graph G * can be decomposed into 2 equal- length shortest path in M 1. Proof: If there is a shorter path, then we can “cut and re-glue” and get a shorter cut graph.

11. Min cut graph Reduced cut graph –Remove vertices of degree-1 and their edges. –Replace maximal path through degree-2 vertices by a single edge –Degree-3, 4g+2k-2 vertices, 6g+3k-3 edges (Euler’s formula). Min cut graph of M with fixed g “and” k can be computed in time O(n O(g+k) ). –Find all-pairs shortest paths –Enumerating all possible solutions.

12. Approximate Min Cut Graph 1.Convert M to punctured manifold M’ without boundary. Contract every boundary to a puncture point. Claim: |G( M )|=|G( M’ )|, M’ includes all punctures. 2.Cut along short essential cycles (doesn’t bound a disk with <2 punctures) until we get a set of punctured spheres. 3.Connect the punctures by cutting along a MST. 4.Re-glue some previously cut edges back.

13. Approximate Min Cut Graph Computing shortest essential cycle is expensive – O(n 2 logn). –Use 2-approximation – O(nlogn). O(log 2 g)-approximate min cut graph. Running time: O(g 2 nlogn).

14. In practice… Geometry images, by Xianfeng Gu, Steven J. Gortler and Hugues Hoppe, in Siggraph’02. We want not only the cut, but also a geometry image D – a unit square. Find a cut ρ and parametrization Φ. –Φ: piecewise linear map from D to M\ρ.

16. Why a disk? Texture mapping. Hardware rendering Compression and decompression –Wavelet-based coder

17. Cut & parametrize Find an initial cut & parametrization. Improve the cut with the info from the parametrization. Iterate until no improvement.

18. Cut & parametrize Find an initial cut & parametrization. Any cut, e.g., the previous one. Heuristics: find a cut and locally shorten a cut path. Improve the cut with the info from the parametrization. Iterate until no improvement.

19. Cut & parametrize Find an initial cut & parametrization. boundary refinement Improve the cut with the info from the parametrization. Iterate until no improvement. optimize a few interior pts

20. Cut & parametrize Find an initial cut & parametrization. Improve the cut with the info from the parametrization. Search for regions with large geometric stretch. Pick a extremal vertex v. Add to ρ the shortest path from v to the current boundary. Iterate until no improvement.

21. Example

22. More example…