informatik viii computer graphics & multimedia martin marinov and leif kobbelt direct...
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Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Direct Quad-Dominated Anisotropic Remeshing
Martin Marinov and Leif Kobbelt
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
CAD meshes
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Meshes from scanned data
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing
Goal: Produce CAD quality mesh elements
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
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Connectivity Regularization [Kobbelt et al.'99],
[Surazhsky&Gotsman'03]
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Isotropic Remeshing [Alliez et al.'03], [Surazhsky et al.'03], [Botsch&Kobbelt'04]
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Normal Noise Reduction [Botsch&Kobbelt'01]
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Variational Shape Approximation [Cohen-Steiner et al.'04]
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Remeshing algorithms
Anisotropic Remeshing [Alliez et al.'03]
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]
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
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]
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Overview• Previous work• Anisotropic remeshing overview• Curvature tensor field estimation• Curvature lines integration• Meshing • Results
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Curvature tensor field estimation
Minimum principal curvature directions Maximum principal curvature directions
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Curvature lines sampling
Minimum curvature lines Maximum curvature lines
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Meshing
Minimum and maximum curvature lines intersection Anisotropic remesh
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
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
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Local tensor flattening3 2
Flattening
Barycentric interpolation
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]
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Tensor field singularities
Where the estimated curvature directions are reliable?
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Confidence estimation2
0 1
2
f i minj 0..2
cos j
High confidence
Low confidence
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Tensor field filtering
vi1K f j Nb vi
f j
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
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
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Local parameterization2
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
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Proximity queries
Sampling density
P-cell
Lines' samples are associated with the original mesh faces
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
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
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Line snapping
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
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
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
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
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Contributions and results• Direction propagation into isotropic regions
– Confidence estimation– Confidence-based filtering– Line-snapping
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Contributions and results• Local parameterization approach
– Arbitrary genus meshes handling– Reduces computational burden and
simplifies implementation
Informatik VIIIComputer Graphics & MultimediaMartin Marinov and Leif Kobbelt
Contributions and results• Proximity queries algorithm
– Dramatically improves the scalability and the performance
– Simple data structures
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
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