Dual/Primal Mesh Optimization for Polygonized Implicit Surfaces Yutaka Ohtake 　Alexander G. Belyaev Max-Planck-Institut für Informatik, Germany University of Aizu, Japan.

Implicit Surfaces Zero sets of implicit functions. CSG operations. - =

Radial Basis Function Visualization of f=0 RBF fitting Carr et al. “Reconstruction and Representation of 3D Objects with Radial Basis Functions”, SIGGRAPH2001 Visualization of f=0 RBF fitting

Visualization of Implicit Surfaces Polygonization (e.g. Marching cubes method) Ray-tracing

Problem of Polygonization 503 grid 1003 grid 2003 grid Sharp features are broken

Reconstruction of Sharp Features Input Output and Rough Polygonization (Correct topology) Post- processing

Basic Idea of Optimization Mesh tangent to implicit surface gives better reconstruction of sharp features. dual mesh Marching cube method Our method

Related Works Extension of Marching Cubes Post-processing approach. Kobbelt, Botsh, Schwanecke, and Seidel “Feature Sensitive Surface Extraction from Volume Data”, SIGGRAPH 2001, August. Post-processing approach. Ohtake, Belyaev, and Pasko “Accurate Polygonization of Implicit Surfaces”, Shape Modeling Internatinal 2001, May. Ohtake and Belyaev “Mesh Optimization for Polygonized Isosurfaces”, Eurographics 2001, September.

Previous work (1) Kobbelt, Botsh, Schwanecke, and Seidel proposed A new distance field representation for detecting accurate vertex positions. Vertex insertion rule for reconstructing sharp features. (and edge flipping) newly inserted

Related work (2) Our previous work Mesh evolution for fitting mesh normals to implicit surface normals. keeping mesh vertices close to implicit surface. Can not estimate implicit surface normals at high curvature regions

Advantages of Proposed Method Extremely good in reconstruction of sharp features Adaptive meshing Works better than mesh evolution approach

Contents Basic Optimization Method Combining with Adaptive Remeshing and Subdivision Discussion

Basic Optimization Algorithm Triangle centroids are projected onto the implicit surface. Mesh vertices are optimized according to tangent planes. estimated numerically

Dual sampling (face points are projected to f=0) Dual Sampling (fitting to tangent planes)

Projection of face points Find a point at other side of surface. Bisection method along the lines. f < 0 f > 0

Fitting to Tangent Planes Minimize the sum of squared distance. distance m(P2) m(P1) x Same as Garland-Heckbert quadric error metric (SIG’97)

Minimization of the Error Solving system of linear equations. SVD is used (similar to Kobbelt et al. SIG’01). The old primal vertex position is shifted to the origin of coordinates. Small singular values are set to zero.

Thresholding of Small Singular Values

Contents Basic Optimization Method Combining with Adaptive Remeshing and Subdivision Discussion

Improvement of Mesh Sampling Rate Curvature weighted resampling Input Dual/Primal mesh optimization output

Repeated Double Dual Resampling Double dual sampling improves mesh distributions. Averaging by Projection

Curvature Weighted Resampling Sampling should be dense near high curvature regions. Uniform resampling causes a skip here. Small bump Uniform weight Curvature weight

Effectiveness Small bumps are well reconstructed. Uniform resampling + Primal/dual mesh optimization Curvature weighted resampling + Primal/dual mesh optimization

Gathering All Together Curvature weighted resampling Input Adaptive subdivision Dual/Primal mesh optimization else If user is satisfied output

+ Dual/Primal mesh optimization Adaptive subdivision Linear 1-to-4 split rule is applied on highly curved triangles. + Dual/Primal mesh optimization “Cat” model provided by HyperFun project.

Decimation Garland-Heckbert method using Tolerance: 90% reduction

Gathering All Together Curvature weighted resampling Input Adaptive subdivision Dual/Primal mesh optimization else If user is satisfied Mesh Decimation output

The number of triangles

(ε : Threshold of adaptive subdivision) Large adaptive ε 3 subdivision steps Small threshold ε 5 subdivision steps (ε : Threshold of adaptive subdivision)

Contents Basic Optimization Method Combining with Adaptive Remeshing and Subdivision Discussion

Comparison with Mesh Evolution Approach Faster and more accurate than mesh evolution approach. Mesh evolution 20 sec. (stabilized) Primal/Dual mesh optimization 1 sec.

Stanford bunny represented by RBF with 10,000 centers. (FastRBF developed by FarField Technology) Optimization takes several hours (Direct evaluation)

Dual Contouring of Hermite Data SIG’02 Also good for reconstruction of sharp features Tao Ju, Frank Losasso, Scott Schaefer, Joe Warren, “Dual Contouring of Hermite Data”. Dual mesh to marching cubes mesh.

Speed: they(sig’02) > we(sm’02) Their method is not post-processing. Control of sampling rate: we(sm’02) > they(sig’02) Octtree based adaptive sampling. Our Their

Conclusion and Problems A mesh optimization method is developed. Primal/Dual mesh optimization. Not so fast if the implicit function is complex. Adaptive voxelization. Requirement of correct topology in the input mesh. Can not optimize this pattern. Edge flipping

Why It Works Well without Edge Flipping?