robust adaptive meshes for implicit surfaces afonso paiva hélio lopes thomas lewiner matmidia -...
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Robust Adaptive Meshes Robust Adaptive Meshes for Implicit Surfacesfor Implicit Surfaces
Afonso Paiva Hélio Lopes Thomas LewinerMatmidia - Departament of Mathematics – PUC-Rio
Luiz Henrique de FigueiredoVisgraf - IMPA
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MotivationMotivation
Topological Guarantees? – 3D extension of “Robust adaptive approximation of implicit curves” –
Hélio Lopes, João Batista Oliveira and Luiz Henrique de Figueiredo, 2001
3
1
:f
S f c
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ChallengesChallenges
level 8level 7level 6level 5Klein bottle 3D– According to Ian Stewart
22 2 2 2 2 2 2 2 2 2 2 1 2 1 8 16 2 1 0x y z y x y z y z xz x y z y
Guaranteed
Not Guaranteed
• Adaptive Mesh• Topological Robustness• Mesh Quality
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Isosurface ExtrationIsosurface Extration
Marching Cubes– Lorensen and Cline, 1987– Look-up table method– Not adaptive– Sliver triangles
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Isosurface ExtrationIsosurface ExtrationAmbiguities of Marching Cubes :
tri-linear topology = original topology ?
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OverviewOverview
• Numerical tools• Build the octree
– Connected Component Criterion– Topology Criterion– Geometry Criterion
• From octree to dual grid• Mesh generation• Mesh improvements• Future Work
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Numerical ToolsNumerical Tools
Interval Arithmetic (IA)– A set of operations on intervals– Enclosure
Given a box then
, , : , ,
B
F B f B f x y z x y z B
f(B)
F(B)
B
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Numerical ToolsNumerical Tools
Automatic Differentiation (AD)– Speed of numerical differentiation– Precision of symbolic differentiation– Defining an arithmetic for tuples:
– Combining IA & AD: is a tuples of intervals!!
2, , ,
, , , , , ,
sin , , sin( ), cos( ), cos( )
x y
x y x y x x y y
x y x y
u u u u
u u u v v v u v u v u v u v u v
u u u u u u u u
nF B
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f < 0
f > 0
S
Build the OctreeBuild the Octree
F(Ω)
0
B1
0
F(B1)
F
F
Connected Components Criterion
0 n nF B Bif then discard
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0
Build the OctreeBuild the Octree
Topology Criterion
0,0,0 n nF B Bif then subdivide
Bn nF B
n
-n , ,n f x y z
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n
Build the OctreeBuild the Octree
Geometry Criterion
max
nn
n
F BDiam k B
F B
if then subdivide
maxd k
Bn nF B
high curvature
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Adaptive Marching CubesAdaptive Marching Cubes
• Shu et al, 1995
• Cracks & holes
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Dual ContouringDual Contouring
• Ju et al., SIGGRAPH 2002• Subdivision controlled by
QEFs• Well-shaped triangles and
quads• Allows more freedom in
positioning vertices• High vertex valence
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From Octree to DualFrom Octree to Dual
• “Dual marching cubes: primal countouring of dual grids” – S. Schaefer & J. Warren, PG, 2004.
• Generates cells for poligonization using the dual of the octree
• Creates adaptive, crack-free partitioning of space
• Uses Marching Cubes on dual cells to construct triangles
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From Octree to DualFrom Octree to Dual
Recursive procedures– It does not require any explicit neighbour
representation in octree data-structure – Three types of procedures:
FaceProc
EdgeProc
VertProc
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Mesh GenerationMesh Generation
Cell key generation
• The vertices of the triangles are computed using bisection method along the dual edge
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Mesh GenerationMesh Generation
“Efficient implementation of Marching Cubes’ cases with topological guarantees”, T. Lewiner, H. Lopes, A. Vieira and G. Tavares, JGT, 2003.
• Topological MC: 730 cases• Original MC: 256 cases
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Mesh GenerationMesh Generation
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v
Mesh ImprovementsMesh Improvements
• Vertex smoothing– Improves the aspect ratio of the triangles– “A remeshing approach to multiresolution modeling”,
M. Botsch and L. Kobbelt, SGP, 2004.
• Project the vertices back to surface using bisection method
,v v v vv v b b n n v vb
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level 7level 6level 5
Results: robustnessResults: robustness
Torus
level 4
2
22 2 21.5 1.35 0x y z
Guaranteed
Not Guaranteed
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Results: topological guaranteeResults: topological guarantee
Complex models– Two torus
level 8level 7level 6
Guaranteed
Not Guaranteed
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level 10
Results: robust to singularitiesResults: robust to singularities
– Teardrop surface
5 4 2 20.5 0x x y z
level 9level 8level 7level 6level 5
Guaranteed
Not Guaranteed
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ResultsResults
Algebraic Surface Non-Algebraic Surface
3 32 2 2 2 22 1 0.1 0x y z x y 2 2 2 sin(4 ) sin(4 ) sin(4 ) -1 0x y z x y z
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Results: adaptativityResults: adaptativity
The effect of geometry criterion
max 0.5k
4 42 2 2 2 21 1 0y x y x y z
maxk max 0.95k # triang = 25172 # triang = 22408 # triang = 4948
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Results: mesh qualityResults: mesh quality
Mesh processing– Cyclide surface– Aspect ratio histograms
Marching Cubes# triang = 11664
Our method without smooth# triang = 5396
Our method with smooth
# triang = 5396
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Results: no makeup!Results: no makeup!
Our algorithm does not suffer of symmetry artefacts– Chair surface
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ResultsResults
Boolean operation Non-manifoldxy = 0
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Limitations and Future Work
• Tighter bounds for less subdivisions– Replace Interval Arithmetics
by Affine Arithmetics
• Only implicit surfaces– Implicit modeling such as MPU
• Infinite subdivision:– Horned sphere → no solution
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That’s all That’s all folks!!!!folks!!!!
http://www.mat.puc-rio.br/~apneto