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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Direct Quad-Dominated Anisotropic Remeshing
Martin Marinov and Leif Kobbelt
![Page 2: Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt](https://reader035.vdocuments.net/reader035/viewer/2022062423/56649e175503460f94b02a33/html5/thumbnails/2.jpg)
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
CAD meshes
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Meshes from scanned data
![Page 4: Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt](https://reader035.vdocuments.net/reader035/viewer/2022062423/56649e175503460f94b02a33/html5/thumbnails/4.jpg)
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing
Goal: Produce CAD quality mesh elements
![Page 5: Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt](https://reader035.vdocuments.net/reader035/viewer/2022062423/56649e175503460f94b02a33/html5/thumbnails/5.jpg)
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Overview
• Previous work– Remeshing algorithms– Anisotropic alignment
• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing • Contributions and results
![Page 6: Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt](https://reader035.vdocuments.net/reader035/viewer/2022062423/56649e175503460f94b02a33/html5/thumbnails/6.jpg)
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Connectivity Regularization [Kobbelt et al.'99],
[Surazhsky&Gotsman'03]
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Isotropic Remeshing [Alliez et al.'03], [Surazhsky et al.'03], [Botsch&Kobbelt'04]
![Page 8: Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt](https://reader035.vdocuments.net/reader035/viewer/2022062423/56649e175503460f94b02a33/html5/thumbnails/8.jpg)
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Normal Noise Reduction [Botsch&Kobbelt'01]
![Page 9: Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt](https://reader035.vdocuments.net/reader035/viewer/2022062423/56649e175503460f94b02a33/html5/thumbnails/9.jpg)
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Variational Shape Approximation [Cohen-Steiner et al.'04]
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Anisotropic Remeshing [Alliez et al.'03]
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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]
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Anisotropic elements in CADDesigners and engineers compose models using
combinations of anisotropic and isotropic objects
Quad elements reflect the symmetry of the shape
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Lines of principal curvature
• Visual perception of the shape: Line-art drawing
[Bradly et al.'85] [Elber'98] [Hertzmann&Zorin'00] [Rössl&Kobbelt'01]
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Overview• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing • Results
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Curvature tensor field estimation
Minimum principal curvature directions Maximum principal curvature directions
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Curvature lines sampling
Minimum curvature lines Maximum curvature lines
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Meshing
Minimum and maximum curvature lines intersection Anisotropic remesh
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Overview
• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation
– Local tensor flattening– Confidence estimation– Filtering
• Curvature lines integration• Meshing • Results
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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
vBe
T e e e et
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Local tensor flattening3 2
Flattening
Barycentric interpolation
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Umbilic points
Trisector Wedge
min max
Spherical or flat point on the tensor field [Delmarcelle&Hesselink'92'92], [Tricoche'02]
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Tensor field singularities
Where the estimated curvature directions are reliable?
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Confidence estimation2
0 1
2
f i minj 0..2
cos j
High confidence
Low confidence
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Tensor field filtering
vi1K f j Nb vi
f j
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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
![Page 26: Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt](https://reader035.vdocuments.net/reader035/viewer/2022062423/56649e175503460f94b02a33/html5/thumbnails/26.jpg)
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Local parameterization2
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Proximity queries
Sampling density
P-cell
Lines' samples are associated with the original mesh faces
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Isotropic regions
• Flat, spherical and transition regions
• Low confidence: Curvature directions not reliable
• Curvature line-based sampling senseless
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Line snapping
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Overview
• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing
– Vertices, edges and faces construction
• Results
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Vertices construction
Lines intersections constitute the new mesh vertices
Use sample groups corresponding to the original mesh faces for
effective computation
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Overview
• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing • Contributions and results
![Page 37: Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt](https://reader035.vdocuments.net/reader035/viewer/2022062423/56649e175503460f94b02a33/html5/thumbnails/37.jpg)
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Contributions and results• Direction propagation into isotropic regions
– Confidence estimation– Confidence-based filtering– Line-snapping
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Contributions and results• Local parameterization approach
– Arbitrary genus meshes handling– Reduces computational burden and
simplifies implementation
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Contributions and results• Proximity queries algorithm
– Dramatically improves the scalability and the performance
– Simple data structures
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
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