informatik yu.ohtake and a.g.belyaev dual/primal mesh optimization for polygonized implicit surfaces...
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INFORMATIKINFORMATIK
Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Dual/Primal Mesh Optimization Dual/Primal Mesh Optimization for Polygonized Implicit Surfacesfor Polygonized Implicit Surfaces
Yutaka Ohtake Yutaka Ohtake Alexander G. BelyaeAlexander G. Belyaevv
Max-Planck-Institut fMax-Planck-Institut füür Informatik, Germanyr Informatik, GermanyUniversity of Aizu, Japan.University of Aizu, Japan.
Yutaka Ohtake Yutaka Ohtake Alexander G. BelyaeAlexander G. Belyaevv
Max-Planck-Institut fMax-Planck-Institut füür Informatik, Germanyr Informatik, GermanyUniversity of Aizu, Japan.University of Aizu, Japan.
INFORMATIKINFORMATIK
Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Implicit SurfacesImplicit Surfaces
Zero sets of implicit functions. Zero sets of implicit functions.
CSG operations.CSG operations.
Zero sets of implicit functions. Zero sets of implicit functions.
CSG operations.CSG operations.
0),,( zyxf
02222 zyxr )0||()0||()0||( zlylxl
=-
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Radial Basis FunctionRadial Basis Function
)()()( xcxx i pfi
RBF fitting Visualization of f=0
Carr et al. “Reconstruction and Representation of 3D Objects with Radial Basis Functions”, SIGGRAPH2001
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Visualization of Visualization of Implicit SurfacesImplicit Surfaces
Ray-tracing Polygonization (e.g. Marching cubes method)
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Problem of PolygonizationProblem of Polygonization
Sharp featuresSharp features are broken are brokenSharp featuresSharp features are broken are broken
503 grid 1003 grid 2003 grid
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Post-processing
InputOutput
Rough Polygonization(Correct topology)
Reconstruction of Sharp FeaturesReconstruction of Sharp Features
DEMO),,( :functionImplicit zyxfandand
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Basic Idea of OptimizationBasic Idea of Optimization
Mesh tangent to implicit surface gives Mesh tangent to implicit surface gives better reconstruction of sharp features.better reconstruction of sharp features. dual meshdual mesh
Mesh tangent to implicit surface gives Mesh tangent to implicit surface gives better reconstruction of sharp features.better reconstruction of sharp features. dual meshdual mesh
Our methodMarching cube method
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Related WorksRelated Works
Extension of Marching CubesExtension of Marching Cubes• Kobbelt, Botsh, Schwanecke, and Seidel “Feature Kobbelt, Botsh, Schwanecke, and Seidel “Feature
Sensitive Surface Extraction from Volume Data”, Sensitive Surface Extraction from Volume Data”, SSIGGRAPH 2001, August.IGGRAPH 2001, August.
Post-processing approach.Post-processing approach.• Ohtake, Belyaev, and Pasko “Accurate PolygonizaOhtake, Belyaev, and Pasko “Accurate Polygoniza
tion of Implicit Surfaces”, tion of Implicit Surfaces”, Shape Modeling InternatiShape Modeling Internatinal 2001nal 2001, May., May.
• Ohtake and BelyaevOhtake and Belyaev“Mesh Optimization for Polygonized Isosurfaces”, “Mesh Optimization for Polygonized Isosurfaces”, Eurographics 2001, September.Eurographics 2001, September.
Extension of Marching CubesExtension of Marching Cubes• Kobbelt, Botsh, Schwanecke, and Seidel “Feature Kobbelt, Botsh, Schwanecke, and Seidel “Feature
Sensitive Surface Extraction from Volume Data”, Sensitive Surface Extraction from Volume Data”, SSIGGRAPH 2001, August.IGGRAPH 2001, August.
Post-processing approach.Post-processing approach.• Ohtake, Belyaev, and Pasko “Accurate PolygonizaOhtake, Belyaev, and Pasko “Accurate Polygoniza
tion of Implicit Surfaces”, tion of Implicit Surfaces”, Shape Modeling InternatiShape Modeling Internatinal 2001nal 2001, May., May.
• Ohtake and BelyaevOhtake and Belyaev“Mesh Optimization for Polygonized Isosurfaces”, “Mesh Optimization for Polygonized Isosurfaces”, Eurographics 2001, September.Eurographics 2001, September.
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Previous work (1)Previous work (1)
Kobbelt, Botsh, Schwanecke, and Seidel proKobbelt, Botsh, Schwanecke, and Seidel proposed posed • A new distance field representation A new distance field representation
for detecting accurate vertex positions. for detecting accurate vertex positions.• Vertex insertion rule Vertex insertion rule
for reconstructing sharp features. for reconstructing sharp features. (and edge flipping) (and edge flipping)
Kobbelt, Botsh, Schwanecke, and Seidel proKobbelt, Botsh, Schwanecke, and Seidel proposed posed • A new distance field representation A new distance field representation
for detecting accurate vertex positions. for detecting accurate vertex positions.• Vertex insertion rule Vertex insertion rule
for reconstructing sharp features. for reconstructing sharp features. (and edge flipping) (and edge flipping)newlynewly
insertedinserted
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Related work (2)Related work (2)
Our previous workOur previous work• Mesh evolution forMesh evolution for
– fitting mesh normals to implicit surface normals.fitting mesh normals to implicit surface normals.– keeping mesh vertices close to implicit surface.keeping mesh vertices close to implicit surface.
Our previous workOur previous work• Mesh evolution forMesh evolution for
– fitting mesh normals to implicit surface normals.fitting mesh normals to implicit surface normals.– keeping mesh vertices close to implicit surface.keeping mesh vertices close to implicit surface.
Can not estimate implicit surface normals at high curvature regions
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Advantages of Proposed MethodAdvantages of Proposed Method
Extremely good Extremely good in reconstruction of sharp featuresin reconstruction of sharp features
Adaptive meshingAdaptive meshing
Works better than mesh evolution approachWorks better than mesh evolution approach
Extremely good Extremely good in reconstruction of sharp featuresin reconstruction of sharp features
Adaptive meshingAdaptive meshing
Works better than mesh evolution approachWorks better than mesh evolution approach
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
ContentsContents
1.1. Basic Optimization MethodBasic Optimization Method2.2. Combining with Adaptive RemeshinCombining with Adaptive Remeshin
g and Subdivisiong and Subdivision3.3. DiscussionDiscussion
1.1. Basic Optimization MethodBasic Optimization Method2.2. Combining with Adaptive RemeshinCombining with Adaptive Remeshin
g and Subdivisiong and Subdivision3.3. DiscussionDiscussion
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Basic Optimization AlgorithmBasic Optimization Algorithm
1.1. Triangle centroids are projected Triangle centroids are projected onto the implicit surface. onto the implicit surface.
2.2. Mesh vertices are optimized Mesh vertices are optimized according to tangent planes. according to tangent planes.
1.1. Triangle centroids are projected Triangle centroids are projected onto the implicit surface. onto the implicit surface.
2.2. Mesh vertices are optimized Mesh vertices are optimized according to tangent planes. according to tangent planes.
estimated estimated numericallynumerically
Dual sampling(face points are projected to f=0)
Dual Sampling(fitting to tangent planes)
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Projection ofProjection of face points face points
ffsign )(:Direction
1.1. Find a point at other side of surface.Find a point at other side of surface.2.2. Bisection method along the lines.Bisection method along the lines.1.1. Find a point at other side of surface.Find a point at other side of surface.2.2. Bisection method along the lines.Bisection method along the lines.
f > 0
f < 0
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Fitting to Fitting to Tangent PlanesTangent Planes
2
torianglesincident t
)()( x
xmx
ii PPE
||)(||
)()(
Pf
PfP
mxm(P1)
m(P2)
distance
Same as Garland-Heckbert quadric error metric (SIG’97)
Minimize the sum of squared distance.Minimize the sum of squared distance.Minimize the sum of squared distance.Minimize the sum of squared distance.
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Minimization of the ErrorMinimization of the Error
Solving system of linear equations. Solving system of linear equations.
SVD is used SVD is used (similar to Kobbelt et al. SIG’01). (similar to Kobbelt et al. SIG’01).• The old primal vertex position The old primal vertex position
is shifted to the origin of coordinates. is shifted to the origin of coordinates.• Small singular values are set to zero.Small singular values are set to zero.
Solving system of linear equations. Solving system of linear equations.
SVD is used SVD is used (similar to Kobbelt et al. SIG’01). (similar to Kobbelt et al. SIG’01).• The old primal vertex position The old primal vertex position
is shifted to the origin of coordinates. is shifted to the origin of coordinates.• Small singular values are set to zero.Small singular values are set to zero.
0)(
x
x
d
dEbxA
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Thresholding of Thresholding of Small Singular ValuesSmall Singular Values
|)||,||,max(|10 aluesingular v Small 323
110 510410310210
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
ContentsContents
1.1. Basic Optimization MethodBasic Optimization Method2.2. Combining with Adaptive RemeshinCombining with Adaptive Remeshin
g and Subdivisiong and Subdivision3.3. DiscussionDiscussion
1.1. Basic Optimization MethodBasic Optimization Method2.2. Combining with Adaptive RemeshinCombining with Adaptive Remeshin
g and Subdivisiong and Subdivision3.3. DiscussionDiscussion
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Improvement of Improvement of Mesh Sampling RateMesh Sampling Rate
Curvature weighted Curvature weighted resamplingresampling
Dual/Primal Dual/Primal mesh optimizationmesh optimization
InputInput
outputoutput
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Repeated Double Repeated Double Dual Resampling Dual Resampling
Double dual sampling Double dual sampling improves mesh distributions.improves mesh distributions.Double dual sampling Double dual sampling improves mesh distributions.improves mesh distributions.
i
ii
w
Pw
xProjectionProjection Averaging by Averaging by
xiP
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Curvature Weighted Curvature Weighted Resampling Resampling
Sampling should be dense near high Sampling should be dense near high curvature regions.curvature regions.Sampling should be dense near high Sampling should be dense near high curvature regions.curvature regions.
Small bump
Uniform resampling causes a skip here.
i
ii
w
Pw
x
curvature:iwUniform weightUniform weight Curvature weightCurvature weight
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
EffectivenessEffectiveness
Small bumps are well reconstructed.Small bumps are well reconstructed.Small bumps are well reconstructed.Small bumps are well reconstructed.Curvature weightedCurvature weighted
resamplingresampling++
Primal/dual mesh Primal/dual mesh optimizationoptimization
Uniform Uniform resamplingresampling
++Primal/dual mesh Primal/dual mesh
optimizationoptimization
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Gathering All TogetherGathering All Together
Curvature weighted Curvature weighted resamplingresampling
Dual/Primal Dual/Primal mesh optimizationmesh optimization
Adaptive Adaptive subdivisionsubdivision
InputInput
outputoutput
elseelseIf user is If user is satisfiedsatisfied
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Adaptive subdivisionAdaptive subdivision
Linear 1-to-4 split rule is applied Linear 1-to-4 split rule is applied on highly curved triangles. on highly curved triangles.Linear 1-to-4 split rule is applied Linear 1-to-4 split rule is applied on highly curved triangles. on highly curved triangles.
““Cat” model provided Cat” model provided by by HyperFunHyperFun project. project.
24
1
)(area)(i
imnim normal
suraceImplicit
+ Dual/Primal + Dual/Primal mesh optimizationmesh optimization
DEMO
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
DecimationDecimation
Garland-Heckbert method usingGarland-Heckbert method using
Tolerance:Tolerance:
Garland-Heckbert method usingGarland-Heckbert method using
Tolerance:Tolerance: 2)()( ii PPE xmx
90%90%reductionreduction
2m)-area)(n(
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Gathering All TogetherGathering All Together
Curvature weighted Curvature weighted resamplingresampling
Dual/Primal Dual/Primal mesh optimizationmesh optimization
Adaptive Adaptive subdivisionsubdivision
Mesh DecimationMesh Decimation
InputInput
outputoutput
elseelseIf user is If user is satisfiedsatisfied
The number of triangles
Large adaptive ε 3 subdivision steps
Small threshold ε 5 subdivision steps
(ε : Threshold of adaptive subdivision)
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
ContentsContents
1.1. Basic Optimization MethodBasic Optimization Method2.2. Combining with Adaptive RemeshinCombining with Adaptive Remeshin
g and Subdivisiong and Subdivision3.3. DiscussionDiscussion
1.1. Basic Optimization MethodBasic Optimization Method2.2. Combining with Adaptive RemeshinCombining with Adaptive Remeshin
g and Subdivisiong and Subdivision3.3. DiscussionDiscussion
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Comparison with Comparison with Mesh Evolution ApproachMesh Evolution Approach
Faster and more accurate Faster and more accurate than mesh evolution approach. than mesh evolution approach.Faster and more accurate Faster and more accurate than mesh evolution approach. than mesh evolution approach.
Primal/Dual Primal/Dual mesh optimizationmesh optimization
1 sec.1 sec.
Mesh evolutionMesh evolution20 sec.20 sec.
(stabilized)(stabilized)
Optimizationtakes several hours(Direct evaluation)
Stanford bunny represented by RBF with 10,000 centers.Stanford bunny represented by RBF with 10,000 centers.((FastRBF developed by FarField Technology)
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Dual Contouring Dual Contouring of Hermite Data SIG’02of Hermite Data SIG’02
Also good Also good for reconstruction of sharp features for reconstruction of sharp features• Tao Ju, Frank Losasso, Scott Schaefer, Joe Warren,Tao Ju, Frank Losasso, Scott Schaefer, Joe Warren,
“Dual Contouring of Hermite Data”.“Dual Contouring of Hermite Data”.• Dual mesh to marching cubes mesh.Dual mesh to marching cubes mesh.
Also good Also good for reconstruction of sharp features for reconstruction of sharp features• Tao Ju, Frank Losasso, Scott Schaefer, Joe Warren,Tao Ju, Frank Losasso, Scott Schaefer, Joe Warren,
“Dual Contouring of Hermite Data”.“Dual Contouring of Hermite Data”.• Dual mesh to marching cubes mesh.Dual mesh to marching cubes mesh.
Speed: they(sig’02) > we(sm’02)Speed: they(sig’02) > we(sm’02)• Their method is not post-processing.Their method is not post-processing.
Control of sampling rate: we(sm’02) > they(sig’02)Control of sampling rate: we(sm’02) > they(sig’02)• Octtree based adaptive sampling.Octtree based adaptive sampling.
Speed: they(sig’02) > we(sm’02)Speed: they(sig’02) > we(sm’02)• Their method is not post-processing.Their method is not post-processing.
Control of sampling rate: we(sm’02) > they(sig’02)Control of sampling rate: we(sm’02) > they(sig’02)• Octtree based adaptive sampling.Octtree based adaptive sampling.
OurOur TheirTheir
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Yu.Ohtake and A.G.BelyaevYu.Ohtake and A.G.Belyaev
Conclusion and ProblemsConclusion and Problems
A mesh optimization method is developed.A mesh optimization method is developed.• Primal/Dual mesh optimization.Primal/Dual mesh optimization.
Not so fast Not so fast if the implicit function is complex. if the implicit function is complex.• Adaptive voxelization.Adaptive voxelization.
Requirement of correct topology Requirement of correct topology in the input mesh. in the input mesh.Can not optimize this pattern.Can not optimize this pattern.
A mesh optimization method is developed.A mesh optimization method is developed.• Primal/Dual mesh optimization.Primal/Dual mesh optimization.
Not so fast Not so fast if the implicit function is complex. if the implicit function is complex.• Adaptive voxelization.Adaptive voxelization.
Requirement of correct topology Requirement of correct topology in the input mesh. in the input mesh.Can not optimize this pattern.Can not optimize this pattern.
Edge Edge flippingflipping