An Incremental Approach to Feature Aligned Quad Dominant Remeshing

ACM Symposium on Solid and Physical Modeling 2008

報告者 :丁琨桓

2

Outline

Quad remeshing paper Relax operation

3

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

Introduction [Variational Shape Approximation,2004] quad meshes provide a better surface approximation

well-shaped quad mesh

5

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

Algorithm

7

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

Principal Direction Estimation

Restricted Delaunay Triangulations and Normal Cycle,2003

Robust Principal Curvatures on Multiple Scales,2006

9

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

Initial Mesh Generation

meshes with the vertex density adapted to the local curvature

11

Optimized triangle mesh

Original Mesh :M

Initial Mesh Generation :M' Sliding the vertices of M' over M such that the ed

ges 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

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

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

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

Result

Principal Direction Initial Mesh Generation

Optimized triangle mesh

Quad Mesh Generation

16

Result

Hand model remeshed with the principal direction field

17

Result

18

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

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

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

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 24064 11.37 1.42 99.2% 3.833×10-3

relaxtion 24064 11.21 1.42 99.2% 6.011×10-3

無 relaxtion relaxtion

22

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 8290 11.83 1.49 96.9% 6.252×10-3

relaxtion 8290 11.49 1.48 96.9% 7.187×10-3

無 relaxtion relaxtion

23

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