1 Global Structure Optimization of Quadrilateral Meshes Published in Eurographics 2011 報告者 : 丁琨桓

2 Introduction this paper proposing an algorithm that takes an automatically generated quad mesh as input and converts it into a quad mesh with improved global structure Optimization : NURBS fitting ab

3 Introduction NURBS

4 Related Work Quad mesh decimation [DSSC08] Quadrilateral mesh simplification Poly-Chord 、 Quadrilateral Collapse originalsimplified

5 Related Work Quad-remeshing [BZK09] Mixed-Integer Quadrangulation

6 Algorithm Input mesh Base complex Search q-helices GP-operator Smoothing

7 Definition A vertex is called regular if it has valence 4 otherwise it is a singular vertex. An important property of the dual of a quadrilateral mesh is that all vertices are regular (valence 4).

8 Base Complex The base complex is the union of all parametric lines which start and end at singular vertices.

9 Algorithm the input mesh with a fine base-complex

10 Grid-Preserving Operators dual half-edge shift leftcollapseshift right

11 Grid-Preserving Operators The most important property of a GP-operator is that it does not introduce new singularities or non-quadrilateral elements.

12 Grid-Preserving Operators Quad loop the triangle and the pentagon will cancel out at the last dual half-edge to the state at the first dual half-edge.

13 Quad loop Creating three vertices for each dual half edge, the different states shift left (red), collapse (yellow) and shift right (green).

14 Quad loop Singular Breadth-first search all possible paths generated by the state- machine with increasing length until the shortest cycle is found. Exponential time

15 q-helices topological helices in quadrilateral meshes, which call q-helices

16 q-helices pitch 1 pitch =1 pitch >1

17 Algorithm Two q-helices (blue and yellow curve) are identified Base complexq-helices

18 Algorithm belonging to the blue helix is extended to a GP-operator q-helicesQuad loop

19 Algorithm mesh after applying all GP-operations tangential smoothing improves the per element quality apply GP-operationssmoothing

20 Result Base complex Optimization

21 Result Base complexOptimization

22 Result

23 Result # Hel : the number of helices # F : the number of quadrilaterals of the mesh # BC : he number of quadrilaterals of the base-complex

24 Result a bc Grid-Preserving Operators 使用時機