11.1 remeshing - hao li

CSCI 621: Digital Geometry Processing

Hao Lihttp://cs621.hao-li.com

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Spring 2017

11.1 Remeshing

http://cs621.hao-li.com

Outline

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• What is remeshing?

• Why remeshing?

• How to do remeshing?

Outline

3


• Why remeshing?


Definition

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Compute another mesh• Satisfy some quality requirements • Approximate well the input mesh

Given a 3D mesh• Already a manifold mesh

Outline

5


• Why remeshing?


Motivation

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Unsatisfactory “raw” mesh• By scanning or implicit representations

Motivation

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Improve mesh quality for further use


Motivation

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Improve mesh quality for further use• Modeling: easy processing • Simulation: numerical robustness • ……


Quality requirements• Local structure • Global structure

Local structure

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Element type• Triangles vs. quadrangles

all-triangle mesh all-quad mesh quad-dominant mesh

Local structure

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Local structure

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Element shape• Isotropic vs. anisotropic


Local structure

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Element distribution• Uniform vs. adaptive

Local structure

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Element alignment• Preserve sharp features and curvature lines

Element distribution• Uniform vs. adaptive

Global structure

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Valence of a regular vertex

Interior vertex Boundary vertexTriangle mesh 6 4

Quadrangle mesh 4 3

Global structure

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Valence of a regular vertex

Different types of mesh structure• Irregular • Semi-regular: multi-resolution analysis / modeling • Highly regular: numerical simulation • Regular: only possible for special models

Interior vertex Boundary vertexTriangle mesh 6 4

Quadrangle mesh 4 3

Outline

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• Why remeshing?


Outline

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• Why remeshing?

• How to do remeshing?- Isotropic remeshing - Anisotropic remeshing

Outline

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• Why remeshing?

• How to do remeshing? - Isotropic remeshing- Anisotropic remeshing

Isotropic remeshing

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Variational remeshing • Energy minimization • Parameterization-based → expensive • Works for coarse input mesh

Greedy remeshing

Incremental remeshing • Simple to implement and robust • Not need parameterization • Efficient for high-resolution input

Isotropic remeshing

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Variational remeshing • Energy minimization • Parameterization-based → expensive • Works for coarse input mesh

Greedy remeshing

Incremental remeshing• Simple to implement and robust • Not need parameterization • Efficient for high-resolution input

Local remeshing operators

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Edge

Split

Vertex

Shift

Edge

Collapse

Edge

Flip

Incremental remeshing

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Specify target edge length L

Lmax = 4/3 * L; Lmin = 4/5 * L;

Iterate:1. Split edges longer than Lmax

2. Collapse edges shorter than Lmin

3. Flip edges to get closer to optimal valence4. Vertex shift by tangential relaxation5. Project vertices onto reference mesh

Edge split

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|Lmax � L| =��12Lmax � L

��

⇥ Lmax =43L

Lmax

12Lmax

12Lmax

Edge

Split

Split edges longer than Lmax

Edge collapse

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|Lmin � L| =��32Lmin � L

��

⇥ Lmin =45L

32Lmin

32Lmin

Lmin Lmin Lmin

Edge

Collapse

Collapse edges shorter than Lmin

Edge flip

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

Edge flip

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Edge Flip

+1 +1

-1

-1

4�

i=1

(valence(vi)� opt valence(vi))2

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

Improve valences• Minimize valence excess

Vertex shift

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Vertex Shift

Local “spring” relaxation• Uniform Laplacian smoothing • Barycenter of one-ring neighborhood

ci =1

valence(vi)

�

j�N(vi)

pj

Vertex shift

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pi

ci


ci =1

valence(vi)

�

j�N(vi)

pj

Vertex shift

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

pi ⇥ pi + ��I� ninT

i

⇥(ci � pi)

ci =1

valence(vi)

�

j�N(vi)

pj

project

tangent

ni

pi

ci

Vertex projection

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Onto original reference mesh• Find closet triangle • Use BSP to accelerate → O(logn) • Barycentric interpolation to

compute position & normal project

Incremental remeshing

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Specify target edge length L

Iterate:1. Split edges longer than Lmax

2. Collapse edges shorter than Lmin

3. Flip edges to get closer to optimal valence4. Vertex shift by tangential relaxation5. Project vertices onto reference mesh

Remeshing result

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Adaptive remeshing

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Adaptive remeshing

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• Compute maximum principle curvature on reference mesh

• Determine local target edge length from max-curvature

• Adjust edge split / collapse criteria accordingly

Feature preservation

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Define feature edges / vertices• Large dihedral angles • Material boundaries

Adjust local operators• Do not touch corner vertices • Do not flip feature edges • Collapse along features • Univariate smoothing • Project to feature curves

Isotropic remeshing

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Variational remeshing• Energy minimization • Parameterization-based → expensive • Works for coarse input mesh

Greedy remeshing

Incremental remeshing • Simple to implement and robust • Not need parameterization • Efficient for high-resolution input

Voronoi Diagram

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Voronoi Diagram

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Divide space into a number of cells

Voronoi Diagram

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Divide space into a number of cellsDual graph: Delaunay triangulation

Centroidal Voronoi Diagram

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For each cellThe generating point = mass of center

CVDnon CVD


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Compute CVD by Lloyd relaxation1. Compute Voronoi diagram of given points pi 2. Move points pi to centroids ci of their Voronoi cells Vi 3. Repeat steps 1 and 2 until satisfactory convergence

pi ⇥ ci =

�Vi

x · �(x) dx�

Vi�(x) dx


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Compute CVD by Lloyd relaxation1. Compute Voronoi diagram of given points pi 2. Move points pi to centroids ci of their Voronoi cells Vi 3. Repeat steps 1 and 2 until satisfactory convergence

pi ⇥ ci =

�Vi

x · �(x) dx�

Vi�(x) dx

CVD maximizes compactness• Minimize the energy:

�

i

⇥

Vi

�(x) ⇤x� pi⇤2 dx ⇥ min

Variational remeshing

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1. Conformal parameterization of input mesh2. Compute local density3. Perform in 2D parameter space

A. Randomly sample according to local density B. Compute CVD by Lloyd relaxation

4. Lift 2D Delaunay triangulation to 3D

Variational remeshing

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Adaptive remeshing

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Outline

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• Why remeshing?

• How to do remeshing? - Isotropic remeshing - Anisotropic remeshing

Anisotropic remeshing

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Artist-designed models• Conform to the anisotropy of a surface


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[Alliez et al. 2003] Anisotropic Polygonal Remeshing.


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input mesh

principal direction fields

sampling meshingoutput mesh


Anisotropy

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Differential geometry• A local orthogonal frame: min/max curvature directions

and normal

3D curvature tensor

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Isotropic

spherical planar

k > 0 k = 0

Anisotropic

2 principal directions

elliptic parabolic hyperbolic

kmax > 0

kmin > 0 kmin = 0

kmax > 0

kmin < 0

kmax > 0

Principal direction fields

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min curvature max curvature overlay

Flattening to 2D

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one 3D tensor per vertex

discrete conformal parameterization

2D tensor field using barycentric coordinates

piecewise linear interpolation of 2D tensors

2D direction fields

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major foliation principal foliationsminor foliation

• Regular case

2D direction fields

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• Singularities

trisector wedge

umbilic (spherical point) 2D tensor proportional to identity

Umbilics

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Umbilics

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wedge

trisector

Lines of curvature

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minor net major net overlay

Lines of curvature

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major netminor net

Overlay

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• Overlay curvature lines in anisotropic regions

• Add umbilical points in isotropic regions

Vertices

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intersect lines of curvatures

Edges

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straighten lines of curvatures

+ Delaunay triangulation near umbilics

Resolve T-junctions

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T-junction

Smoothing

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quad-triangle subdivision


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input mesh

principal direction fields

sampling meshingoutput mesh


Remeshing results

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min curvature

minor net

max curvature

major net overlay

result

Remeshing results

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Tools

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MeshLab• meshlab.sourceforge.net • open source • available for Windows, MacOSX, and Linux

Graphite• http://alice.loria.fr/index.php/software/3-platform/22-

graphite.html • available for Windows • MacOSX or Linux?

Remeshing via Graphite

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“Mesh” → “remesh” → “pliant” →• [Optional] flag border as feature • [Optional] flag sharp edges as feature (dihedral angle) • [Optional] estimate edge size (bounding box divisions) • remesh (target edge length)

Literature

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• Textbook: Chapter 6 • Alliez et al, “Interactive geometry remeshing”, SIGGRAPH 2002 • Alliez et al, “Isotropic surface remeshing”, SMI 2003 • Alliez et al, “Anisotropic polygonal remeshing”, SIGGRAPH 2003 • Vorsatz et al, “Dynamic remeshing and applications”, Solid Modeling 2003 • Botsch & Kobbelt, “A remeshing approach to multiresolution modeling”,

Symp. on Geometry Processing 2004 • Marinov et al, “Direct anisotropic quad-dominant remeshing”, Pacific

Graphics 2004 • Alliez et al, “Recent advances in remeshing of surfaces”, AIM@Shape

state of the art report, 2006


Thanks!

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11.1 remeshing - hao li

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