Connectivity Editing for Quadrilateral Meshes

1. Connectivity Editing for Quadrilateral Meshes SIGGRAPH ASIA 2011 報告者:丁琨桓

2. Irregular vertex • Vertex with valence (# adjacent edges) != 4 • irregular vertices are vertices with adjacent edges not equal to four Valence 2 Valence 3 Valence 5 Valence 6

3. Irregular vertex • quadrilateral meshes without any irregular vertices? At least 8 v3 are needed to form a (closed) genus 0 mesh 8 v3 80 v3, 12 v5, 30 v6 genus 1 mesh :Free of irregular vertices

4. Total valence deficit theorem • Total valence deficit is a constant determined only by the mesh’s genus g l(v) is the valence of v.4 - l(v) is the vertex’ valence deficit.(v3: +1, v4: 0, v5: -1,…)χ is the mesh’s Euler characteristic = 2 – 2g

5. Total valence deficit theorem Total valence deficit for genus 0 mesh is 8: 8 v3: 8*1=8 80 v3, 12 v5, 30 v6: 80*1+12*(-1)+30*(-2)=8 22 v3, 14 v5: 22*1+14*(-1)=8

6. Total valence deficit theorem 0 for genus 1 mesh: -32 for genus 5 mesh: Free of irregular vertices 4X = 4(2 – 2g) = 4(2 – 2) = 0 32 v5 4X = 4(2 – 2g) = 4(2 – 10) = -32

7. Vertex editing • Create, delete, or just move a single irregularvertex is impossible because it will violate the total valence deficit equation • Since there is little freedom in editing one single irregular vertex, authors are instead interested in editing irregular vertices in pairs.

8. Vertex editing • To simplify the discussion, authors only consider irregular vertices with a valence of 3 or 5. • The focus of this paper is to introduce and analyze the following three operations: 3-5, 3-3, and 5-5 pair movement operations.

9. Basic Operations • Basic operations include quad collapse, edge split, and edge flip. • The set of basic operations has the following desirable properties: (i) the support for these operations is local (ii) implementations are relatively easy with a low computational cost (iii) it is straightforward to combine multiple basic operations, incurring no limitations and special cases

10. Quad Collapse • merging a pair of diagonally opposing vertices (v1,v2) sharing the same face.

11. Edge Split • bloating a pair of connected edges (e1,e2) into one face and the central vertex v between them is split into two.

12. Edge Flip • rotating an edge in either the counter-clockwise or clockwise direction.

13. Adjacent 3-5 Pair Movement Quad Collapse Quad Collapse Edge Flip Edge Flip Edge Split Edge Split Each step is realized by one basic operation

14. v3 Movement and 3-5 Pair Generation(Removal) Quad Collapse Edge Flip

15. v5 Movement and 3-5 Pair Generation(Removal) Edge Split Edge Flip

16. Type change (valence increasing/decreasing) Convert v6+ vertex to multiple v5 by edge split Convert v2 vertex to multiple v3 by quad collapse

17. Pipeline of a 3-5 pair movement • Move the v5 to a user-specified location by an edge split.A byproduct 3-5 pairB will appear. • Transport B toward the v3 to collapse B’s v5 end with the v3. They will cancel each other. • B’s v3 end becomes the new v3. B

18. Multiple ways to move the byproduct 3-5 pair Taking an alternative path that avoids passing through the sharp feature (grey).

19. Multiple ways to move the byproduct 3-5 pair Taking an alternative path that avoids passing through the sharp feature (grey).

20. Pipeline for 3-3 pair movement Same pipeline as 3-5 pair’s, but their relative distance willchange in four possible ways: (+1,-1), (-1,+1), (+1,+1), (-1,-1). spinning away closer

21. Pipeline for 3-3 pair movement Same pipeline as 3-5 pair’s, but their relative distance willchange in four possible ways: (+1,-1), (-1,+1), (+1,+1), (-1,-1). spinning away closer

22. 5-5 pairmovement Same as 3-3 pair movement. (+1,-1), (-1,+1), (+1,+1), (-1,-1). spinning away closer

23. 5-5 pair movement Same as 3-3 pair movement. (+1,-1), (-1,+1), (+1,+1), (-1,-1). spinning away closer

24. Merge two v5 vertices into a v6 Yes, when the sum of relative distance is even. No, when the sum of relative distance is odd.

25. Topological Analysis:merge two v5 vertices into a v6 Even configurations:degenerate to a v6. Odd configurations:become an adjacent pair.

26. Merge two v3 vertices into a v2 Yes when the sum of relative distance is even. No when the sum of relative distance is odd.

27. Topological Analysis:merge two v3 vertices into a v2 Yes when the sum of relative distance is even. No when the sum of relative distance is odd. Only even (red) configurations can degenerate to a v2.

28. Alignirregular vertices Any 3-3 or 5-5 pair can be aligned by pair-wise movements (1,3) (-1,+1)(0,4) (1,3) (-1,-1)(0,2)

29. cancel a 3-5 pair It is not possible cancel a pair of vertices 3-5 pair without other irregular vertices, but can cancel a 3-5 pair by involving a third irregular vertex.

30. Geometry impacted • Laplacian Smoothing. • Projecting vertices back to the original mesh. Before smoothing After smoothing

31. Applications • Irregular Vertex Cancellation and Alignment • Connectivity Improvement Original mesh Edited mesh

32. Result reduce the numbers of irregular vertices from 168 (92 v3, 62 v5, and 14 v6) to 24 (12 v3 and 12 v5) by multiple 3-5 pair cancellation operations, and then align the irregular vertices to improve the flow of the panel structure.

33. Result Progressive irregular vertex cancellation with bunny model 956 irr-vertices 400 irr-vertices 48 irr-vertices 8 irr-vertices

34. Conclusion • This paper propose editing operations for quad meshes to explicitly control the location, orientation and number of irregular vertices. • The questions that plan to address are how to select candidate operations, how to evaluate the cost of a candidate operation, and how to preserve the shape of the input model. It’s an an NP-complete problem. • Future Work: Automated irregular vertices cancellation

35. 實驗結果 • 在分群時將平坦度相近的三角片歸為一群 • 增加投影時的準確度 • E( Ti, Ri ) + α*L2,1( Ti , Pi ) Lp-norm 法向量差異度 α= 0.2 α= 0.0

36. 實驗結果 • 矩形相似度: 四角片長寬比 • 方形相似度: 四角片的四個角和90度的差異度 六面法(89群) 無六面法(89群)

37. 基底模型修正 • 使基底模型上盡可能只有四角片 • 移除長度過短的邊，移除基底模型上小於(平均邊長/ 3)的邊

38. 實驗結果 • 矩形相似度: 四角片長寬比 • 方形相似度: 四角片的四個角和90度的差異度 無修正 基底模型修正