1. An Incremental Approach to Feature Aligned Quad Dominant Remeshing ACM Symposium on Solid and Physical Modeling 2008 報告者 : 丁琨桓

2. 1 Outline Quad remeshing paper Relax operation

3. 2 Introduction 3D models of objects is now being used in more and more application fields ranging from Computer Graphics and Computer Aided Design. Earlier algorithms for 3D reconstruction have focussed on the generation of highly detailed but unstructured triangle meshes

4. 3 Introduction [Variational Shape Approximation,2004] quad meshes provide a better surface approximation well-shaped quad mesh

5. 4 Introduction This paper present a new algorithm which turns an unstructured triangle mesh into a quad- dominant mesh. Quad mesh with edges aligned to the principal directions.

6. 5 Algorithm

7. 6 Principal Direction Estimation This paper use the methods proposed in [Restricted Delaunay Triangulations and Normal Cycle,2003] and [Robust Principal Curvatures on Multiple Scales,2006] to estimate the principal directions at the vertices of the input mesh.

8. 7 Principal Direction Estimation Restricted Delaunay Triangulations and Normal Cycle,2003 Robust Principal Curvatures on Multiple Scales,2006

9. 8 Initial Mesh Generation The starting configuration is crucial for incremental optimization. This paper use the method proposed in [Robust feature classification and editing, 2007] which produces meshes with the vertex density adapted to the local curvature.

10. 9 Initial Mesh Generation meshes with the vertex density adapted to the local curvature

11. 10 Optimized triangle mesh Original Mesh :M Initial Mesh Generation :M' Sliding the vertices of M' over M such that the edges of M' become aligned to the pricipal direction fields of M. After relocation, project the vertices of M' back to the nearest point of M after every iteration.

12. 11 Optimized triangle mesh Apply a local remeshing procedure to preserve a good mesh quality. Use a variant of the connectivity update procedure described in [A remeshing approach to multiresolution modeling,2004]. remeshingOriginal

13. 12 Optimized triangle mesh Collapse edges that have become shorter than some threshold Flip edges if the maximum inner angle of one of the adjacent triangles is above some threshold

14. 13 Quad Mesh Generation An edge is removed if the quality of the resulting face is superior to the quality of the two faces. The quality of a face is measured by its rectangularity. the quality of the two faces

15. 14 Result Principal DirectionInitial Mesh Generation Optimized triangle mesh Quad Mesh Generation

16. 15 Result Hand model remeshed with the principal direction field

17. 16 Result

18. 17 Conclusion The approach is simple, and produces quad- dominant meshes that show a very good alignment to geometric features. T-vertices provide more flexibility in order to improve the alignment and the shape of the faces.

19. 18 Relax operation 原始模型 分群結果基底模型 細分模型 半規則四角 網格模型 Relax operation

20. 19 Relax operation 增加四角片規則度 將每一個頂點調整移動到其相鄰點的中心點位置 分群區塊邊界上的點不調整 移動到其相鄰點的中心點位置區塊邊界上的點不調整

21. 20 Relax operation 四角化方法點數矩形相似度方型相似度分支度為四 比例 外形誤差 (Hausdroff distance) 無 relaxtion 2406411.371.4299.2%3.833×10 -3 relaxtion2406411.211.4299.2%6.011×10 -3 無 relaxtion relaxtion

22. 21 Relax operation 四角化方法點數矩形相似度方型相似度分支度為四 比例 外形誤差 (Hausdroff distance) 無 relaxtion 829011.831.4996.9%6.252×10 -3 relaxtion829011.491.4896.9%7.187×10 -3 無 relaxtion relaxtion

23. 22 原始模型表面 細分模型表面 投影前點位置 投影頂點法向量 射線 投影後點位置