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

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt CAD meshes

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Meshes from scanned data

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing Goal: Produce CAD quality mesh elements

Informatik VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Connectivity Regularization [Kobbelt et al.'99], [Surazhsky&Gotsman'03]

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Isotropic Remeshing [Alliez et al.'03], [Surazhsky et al.'03], [Botsch&Kobbelt'04]

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Normal Noise Reduction [Botsch&Kobbelt'01]

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Variational Shape Approximation [Cohen-Steiner et al.'04]

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Anisotropic Remeshing [Alliez et al.'03]

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Anisotropic alignment Approximation properties: – Optimal L 2 [Nadler'86] and L p [Simpson'91] approximation (except for hyperbolic regions) – Optimal normal field approximation (L 2,1 metric)[Cohen-Steiner et al.'04]

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Anisotropic elements in CAD Designers and engineers compose models using combinations of anisotropic and isotropic objects Quad elements reflect the symmetry of the shape

Informatik VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Overview Previous work Anisotropic remeshing overview Curvature tensor field estimation Curvature lines integration Meshing Results

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Curvature tensor field estimation Minimum principal curvature directionsMaximum principal curvature directions

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Curvature lines sampling Minimum curvature linesMaximum curvature lines

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Meshing Minimum and maximum curvature lines intersectionAnisotropic remesh

Informatik VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Curvature tensor field estimation Integrate edge tensors over a surface area B [Cohen-Steiner&Morvan'03]

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Local tensor flattening Flattening Barycentric interpolation

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Umbilic points Trisector Wedge Spherical or flat point on the tensor field '92 [Delmarcelle&Hesselink'92], [Tricoche'02]

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Tensor field singularities Where the estimated curvature directions are reliable?

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Confidence estimation High confidence Low confidence

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Tensor field filtering

Informatik VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Curvature line integration Current sample Integrate the next sample parameterization Next sample Current sample parameterization

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Local parameterization

Informatik VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Proximity queries Sampling density P-cell Lines' samples are associated with the original mesh faces

Informatik VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Isotropic regions Flat, spherical and transition regions Low confidence: Curvature directions not reliable Curvature line-based sampling senseless

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Line snapping

Informatik VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Edges and faces construction Meshing 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 VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Overview Previous work Anisotropic remeshing overview Curvature tensor field estimation Curvature lines integration Meshing Contributions and results

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Contributions and results Direction propagation into isotropic regions – Confidence estimation – Confidence-based filtering – Line-snapping

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Contributions and results Local parameterization approach – Arbitrary genus meshes handling – Reduces computational burden and simplifies implementation

Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Contributions and results Proximity queries algorithm – Dramatically improves the scalability and the performance – Simple data structures

Informatik VIII Computer Graphics & Multimedia Martin 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 VIII Computer Graphics & Multimedia Martin 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