remeshing i

Remeshing Ig

RemeshingRemeshing

Given a 3D mesh find a “better” discreteGiven a 3D mesh, find a better discrete representation of the underlying surface

2

What is a good mesh?What is a good mesh?

• Equal edge lengths• Equal edge lengths

• Equilateral triangles

• Valence close to 6

3





U if d ti li• Uniform vs. adaptive sampling

4





U if d ti li• Uniform vs. adaptive sampling

• Feature preservationp

5





Uniform vs adaptive sampling• Uniform vs. adaptive sampling

• Feature preservation

• Alignment to curvature lines

I t i i t i• Isotropic vs. anisotropic

6

What is a good mesh?What is a good mesh?• Equal edge lengthsEqual edge lengths



Uniform vs adaptive sampling• Uniform vs. adaptive sampling

• Feature preservation

• Alignment to curvature lines

• Isotropic vs. anisotropic

• Triangles vs. quadrangles

7

Triangles vs. quadrangles

ApplicationsApplications

– Design– Reverse engineering– Simulation– Simulation– Visualization

8

Local StructureLocal Structure

• Element type• Element type– Triangle– Quadrangle– Polygonyg

9


• Element shape (isotropy vs anisotropy)• Element shape (isotropy vs anisotropy)

10

Local StructureLocal StructureElement distribution (sizing, grading)

Uniform AdaptiveInput 11

Local StructureLocal Structure• Element orientation

12


• Element orientation• Element orientation

13

Isotropic RemeshingIsotropic RemeshingWell-shaped elementsp

– for processing & simulation (numerical stability & efficiency)efficiency)

14

Two Fundamental ApproachesTwo Fundamental Approaches

• Parameterization based• Parameterization-based– map to 2D domain / 2D problem– computationally more expensive– works even for coarse resolution remeshingg

• Surface-oriented– operate directly on the surface– treat surface as a set of points / polygons in space– efficient for high resolution remeshing

15

Parameterization BasedParameterization Based

16


parameter domain

17


parameterize

18


re-parameterize

19


apply

20

Surface OrientedSurface Oriented

21


sample

22


filter

23


back-project

24

Variational MethodsVariational Methods

• Minimize a global energy• Minimize a global energy– Both discrete and continuous degrees of freedom

• Connectivity, geometry

• What to optimizeWhat to optimize– Edge lengths, angles, triangle shapes,…

• Isotropic point sampling well shaped trianglesIsotropic point sampling well shaped triangles– Simple and effective

25

Parameterization-Based Remeshing[Alli t l ‘03][Alliez et al. ‘03]

• Compute 2D parameterization• Compute 2D parameterization– Conformal: only area distortion

S l 2D d i• Sample 2D domain– Density based on area distortion

• Triangulate

• Project back to 3D• Project back to 3D26

Isotropic 2D SamplingIsotropic 2D Sampling

• Density based random sampling• Density based random sampling– Does not guarantee uniform distance between

samples

27

Sampling EnergySampling Energy

• Given sites x and regions R minimize• Given sites xi and regions Ri, minimize

2( ) || ||k kE x x R R x x dx= −∑ ∫

S d t i t

1 11..

( , .., , , ..., ) || ||i

k k ii k x R

E x x R R x x dx= ∈∑ ∫

• Spreads out points

xi

R

xi

28



2( ) || ||k kE x x R R x x dx= −∑ ∫

• If sites are fixed energy is minimized by the

1 11..

( , .., , , ..., ) || ||i

k k ii k x R


• If sites are fixed, energy is minimized by the Voronoi TesselationVoronoi cell Ri =

All points closer to xi than to other xj

29



2( ) || ||k kE x x R R x x dx= −∑ ∫• If sites are fixed, energy is minimized by the Voronoi

1 11..

( , .., , , ..., ) || ||i

k k ii k x R


Tesselation

• Among all Voronoi tesselations, energy is minimized when sitesgyare centroids of cells =Centroidal Voronoi Tesselation

30

Varying Density2( ) ( ) || || d∑ ∫

Varying Density2

1 11..

( , .., , , ..., ) ( ) || ||i

k k ii k x R

E x x R R x x x dx= ∈

= −∑ ∫ ρ

R

xj

Initial Sample ScatterInitial Sample Scatter

Voronoi tesselation32

Optimized Sample PlacementOptimized Sample Placement

LloydLloyd

centroidal Voronoi tesselation

Uniform RemeshingUniform Remeshing

back-project

50kV50kV

Lloyd AlgorithmLloyd Algorithm

• Alternate:– Voronoi partitioning

Move sites to respective centroids– Move sites to respective centroids

35

Centroidal Voronoi DiagramsCentroidal Voronoi Diagrams

36

demo


37

demo


38

demo


39

demo

Centroidal Voronoi TesselationCentroidal Voronoi Tesselation

• Lloyd converges slowly• Lloyd converges slowly– Stop when points “stop” moving

• Recent fast algorithm: direct optimization of the• Recent fast algorithm: direct optimization of the energy using quasi-Newton“On centroidal voronoi tessellation energy smoothness and fastOn centroidal voronoi tessellation—energy smoothness and fast computation” [Liu et al., TOG ’09]

40

Uniform vs. AdaptiveUniform vs. Adaptive

41

Uniform SamplingUniform Sampling

42

Adaptive SamplingAdaptive Sampling

43

LimitationsLimitations

• Closed meshes• Closed meshes– Need a good cut– Free boundary parameteriztion– Stitch seams afterwards

• Protruding legs– Sampling– Numerical problemsNumerical problems

44

Smart Cut, Free BoundarySmart Cut, Free Boundary

45

Free vs. Fixed BoundaryFree vs. Fixed Boundary

46

Visible SeamsVisible Seams

47

Naive Cut, Numerical ProblemsNaive Cut, Numerical Problems

48

Direct Surface Remeshing[Botsch et al. ’04]

• Avoid global parameterization• Avoid global parameterization– Numerically very sensitive– Topological restrictions

• Avoid local parameterizations– Expensive computationsExpensive computations

• Use local operators & back-projectionsUse local operators & back-projections– Resampling of 100k triangles in < 5s

49

Local Remeshing OperatorsLocal Remeshing Operators

Edge VertexEdge EdgeEdgeSplit

VertexShift

EdgeCollapse

EdgeFlip

50

Isotropic RemeshingIsotropic Remeshing

S if t t d l th LSpecify target edge length L

Compute edge length range [Lmin, Lmax]p g g g [ ]

Iterate:1. Split edges longer than Lmax

2. Collapse edges shorter than Lmin2. Collapse edges shorter than Lmin

3. Flip edges to get closer to valence 64 V t hift b t ti l l ti4. Vertex shift by tangential relaxation5. Project vertices onto reference mesh

51

Edge Collapse / SplitEdge Collapse / Split

52

Edge FlipEdge Flip

• Improve valences• Improve valences– Avg. valence is 6 (Euler)– Reduce variation

• Optimal valence is• Optimal valence is– 6 for interior vertices– 4 for boundary vertices

53

Edge FlipEdge Flip

• Improve valences• Improve valences– Avg. valence is 6 (Euler)– Reduce variation

• Optimal valence is Edge• Optimal valence is– 6 for interior vertices

EdgeFlip

1– 4 for boundary vertices

• Minimize valence excess +1 +1

-1

Minimize valence excess-1

54

Vertex ShiftVertex Shift

• Local “spring” relaxation• Local spring relaxation– Uniform Laplacian smoothing– Bary-center of one-ring neighbors

VertexVertexShift

55



56



tangent

• Keep vertex (approx.) on surface– Restrict movement to tangent plane

projectRestrict movement to tangent plane

57

Vertex ProjectionVertex Projection

• Project vertices onto• Project vertices onto original reference mesh

• Assign position & interpolated normal

project

interpolated normal

58

Remeshing ResultsRemeshing Results

Original

59

Remeshing ResultsRemeshing Results

Edge length DeviationEdge lengthdeviation (%)

Deviationfrom 60º

60

Feature Preservation?Feature Preservation?

61

Feature PreservationFeature Preservation

• Define features• Define features– Sharp edges– Material boundaries

• Adjust local operators– Don’t move corners– Collapse only along features

Don’t flip feature edges– Don t flip feature edges– Univariate smoothing

62

– Project to feature curves

Adaptive RemeshingAdaptive Remeshing

• Precompute max curvature• Precompute max. curvature on reference mesh

• Target edge length locally determined by curvaturedetermined by curvature

• Adjust split / collapse criteria

63

Next Time:Quad Dominant Meshing

64

remeshing i

Documents