Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated...

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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt

Direct Quad-Dominated Anisotropic Remeshing

Martin Marinov and Leif Kobbelt

CAD meshes

Meshes from scanned data

Remeshing

Goal: Produce CAD quality mesh elements

Overview

• Previous work– Remeshing algorithms– Anisotropic alignment

• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing • Contributions and results

Remeshing algorithms

Connectivity Regularization [Kobbelt et al.'99],

[Surazhsky&Gotsman'03]

Isotropic Remeshing [Alliez et al.'03], [Surazhsky et al.'03], [Botsch&Kobbelt'04]

Normal Noise Reduction [Botsch&Kobbelt'01]

Variational Shape Approximation [Cohen-Steiner et al.'04]

Anisotropic Remeshing [Alliez et al.'03]

Anisotropic alignment

• Approximation properties:– Optimal L2 [Nadler'86] and Lp [Simpson'91]

approximation (except for hyperbolic regions)– Optimal normal field approximation (L2,1 metric)

[Cohen-Steiner et al.'04]

Anisotropic elements in CADDesigners and engineers compose models using

combinations of anisotropic and isotropic objects

Quad elements reflect the symmetry of the shape

Lines of principal curvature

• Visual perception of the shape: Line-art drawing

[Bradly et al.'85] [Elber'98] [Hertzmann&Zorin'00] [Rössl&Kobbelt'01]

Overview• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing • Results

Curvature tensor field estimation

Minimum principal curvature directions Maximum principal curvature directions

Curvature lines sampling

Minimum curvature lines Maximum curvature lines

Meshing

Minimum and maximum curvature lines intersection Anisotropic remesh

Overview

• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation

– Local tensor flattening– Confidence estimation– Filtering

• Curvature lines integration• Meshing • Results

Curvature tensor field estimation

Integrate edge tensors over a surface area B [Cohen-Steiner&Morvan'03]

T v1B e B

e B e e et

T e e e et

Local tensor flattening3 2

Flattening

Barycentric interpolation

Umbilic points

Trisector Wedge

min max

Spherical or flat point on the tensor field [Delmarcelle&Hesselink'92'92], [Tricoche'02]

Tensor field singularities

Where the estimated curvature directions are reliable?

Confidence estimation2

f i minj 0..2

High confidence

Low confidence

Tensor field filtering

vi1K f j Nb vi

Overview

• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration

– Local parameterization approach– Seeding and density estimation– Proximity queries– Line snapping

• Meshing • Results

Curvature line integration

si f i , ui , vi ,w i

Current sample

i 1 RK4i ,hi ,d i

Integrate the next sample parameterizationNext sample

i i , i2

Current sample parameterization

Local parameterization2

Seeding and density estimation

– Placing seeds in the most confident regions– Minimum curvature lines samples seed

maximum curvature lines and vice versa– Lines density depends on the local curvature

estimation and the user-specified approx. tolerance

Proximity queries

Sampling density

P-cell

Lines' samples are associated with the original mesh faces

P-cell evolution• P-cell contents

maintenance:– Include adjacent

faces which are inside the local sampling density r

– Remove already included faces which are now outside r

Isotropic regions

• Flat, spherical and transition regions

• Low confidence: Curvature directions not reliable

• Curvature line-based sampling senseless

Line snapping

Overview

• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing

– Vertices, edges and faces construction

• Results

Vertices construction

Lines intersections constitute the new mesh vertices

Use sample groups corresponding to the original mesh faces for

effective computation

Edges and faces constructionMeshing directly in 3D:

– Edges are defined between subsequent line's intersections

– Use vertex normal information obtained on the original surface

– Halfedge structure construction

– Convex partitioning of concave faces

Overview

• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing • Contributions and results

Contributions and results• Direction propagation into isotropic regions

– Confidence estimation– Confidence-based filtering– Line-snapping

Contributions and results• Local parameterization approach

– Arbitrary genus meshes handling– Reduces computational burden and

simplifies implementation

Contributions and results• Proximity queries algorithm

– Dramatically improves the scalability and the performance

– Simple data structures

Comparison to Alliez et al.'03

• Alliez et al.'03– Seeding at umbilic

points – Point based sampling

in isotropic regions and CDT meshing

– Global parameterization

– CDT proximity queries

• Marinov&Kobbelt'04– Seeding in anisotropic

regions– Geodesic line

sampling in isotropic regions, quad-dominated meshing

– Local parameterization– Fast proximity queries

based on the original mesh connectivity

Future work

• Improving the distribution of the integrated curvature lines– Global uniformity constraints– Multiresolution approach

• Improved approximation– Using asymptotic instead of principal directions

in hyperbolic regions

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated...

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