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Kobbelt, Slides on Surface Representations )NPUT $ATA 2ANGE 3CAN #!$ 4OMOGRAPHY 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS !NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL 0ARAMETERIZATION Surface Representations 3IMPLIFICATION FOR COMPLEXITY REDUCTION Leif Kobbelt RWTH Aachen University 2EMESHING FOR IMPROVING MESH QUALITY &REEFORM AND MULTIRESOLUTION MODELING 1 Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 2 2 Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 3 3 Mathematical Representations • parametric – range of a function – surface patch 2 3 f : R → R , SΩ = f(Ω) • implicit – kernel of a function – level set 3 F : R → R, Sc = {p : F (p)=c} Leif Kobbelt RWTH Aachen University 4 4 2D-Example: Circle • parametric r cos(t) f : t !→ , S = f([0, 2π]) ! r sin(t) " • implicit 2 2 2 F (x, y)=x + y − r S = {(x, y):F (x, y)=0} Leif Kobbelt RWTH Aachen University 5 5 2D-Example: Island • parametric r cos(???t) f : t !→ , S = f([0, 2π]) ! r sin(???t) " • implicit 2 2 2 F (x, y)=x +???y − r S = {(x, y):F (x, y)=0} Leif Kobbelt RWTH Aachen University 6 6 Approximation Quality • piecewise parametric r cos(???t) f : t !→ , S = f([0, 2π]) ! r sin(???t) " • piecewise implicit 2 2 2 F (x, y)=x +???y − r S = {(x, y):F (x, y)=0} Leif Kobbelt RWTH Aachen University 7 7 Approximation Quality • piecewise parametric r cos(???t) f : t !→ , S = f([0, 2π]) ! r sin(???t) " • piecewise implicit 2 2 2 F (x, y)=x +???y − r S = {(x, y):F (x, y)=0} Leif Kobbelt RWTH Aachen University 8 8 Requirements / Properties • continuity – interpolation / approximation f(ui,vi) ≈ pi • topological consistency – manifold-ness • smoothness – C0, C1, C2, ... Ck • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 9 9 Requirements / Properties • continuity – interpolation / approximation f(ui,vi) ≈ pi • topological consistency – manifold-ness • smoothness – C0, C1, C2, ... Ck • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 10 10 Requirements / Properties • continuity – interpolation / approximation f(ui,vi) ≈ pi • topological consistency – manifold-ness • smoothness – C0, C1, C2, ... Ck • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 11 11 Requirements / Properties • continuity – interpolation / approximation f(ui,vi) ≈ pi • topological consistency – manifold-ness • smoothness – C0, C1, C2, ... Ck • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 12 12 Requirements / Properties • continuity – interpolation / approximation f(ui,vi) ≈ pi • topological consistency – manifold-ness • smoothness – C0, C1, C2, ... Ck • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 13 13 Topological Consistency Leif Kobbelt RWTH Aachen University 14 14 Topological Consistency Leif Kobbelt RWTH Aachen University 14 14 Topological Consistency Mesh Repair ... Leif Kobbelt RWTH Aachen University 14 14 Closed 2-Manifolds • parametric – disk-shaped neighborhoods – f ( D ε [ u, v ]) = D δ [ f ( u, v )] + injectivity • implicit – surface of a “physical” solid – F (x, y, z)=c, !∇F (x, y, z)!#=0 Leif Kobbelt RWTH Aachen University 15 15 Closed 2-Manifolds • parametric – disk-shaped neighborhoods – f ( D ε [ u, v ]) = D δ [ f ( u, v )] + injectivity • implicit – surface of a “physical” solid – F (x, y, z)=c, !∇F (x, y, z)!#=0 Leif Kobbelt RWTH Aachen University 16 16 Closed 2-Manifolds • parametric – disk-shaped neighborhoods – f ( D ε [ u, v ]) = D δ [ f ( u, v )] + injectivity • implicit – surface of a “physical” solid – F (x, y, z)=c, !∇F (x, y, z)!#=0 Leif Kobbelt RWTH Aachen University 17 17 Closed 2-Manifolds • parametric – disk-shaped neighborhoods f(Dε[u, v]) = Dδ[f(u, v)] – • implicit – surface of a “physical” solid – F (x, y, z)=c, !∇F (x, y, z)!#=0 Leif Kobbelt RWTH Aachen University 18 18 Closed 2-Manifolds • parametric – disk-shaped neighborhoods f(Dε[u, v]) = Dδ[f(u, v)] – • implicit – surface of a “physical” solid – F (x, y, z)=c, !∇F (x, y, z)!#=0 Leif Kobbelt RWTH Aachen University 19 19 Smoothness • position continuity : C0 • tangent continuity : C1 • curvature continuity : C2 Leif Kobbelt RWTH Aachen University 20 20 Smoothness • position continuity : C0 • tangent continuity : C1 • curvature continuity : C2 Leif Kobbelt RWTH Aachen University 21 21 Smoothness • position continuity : C0 • tangent continuity : C1 • curvature continuity : C2 Leif Kobbelt RWTH Aachen University 22 22 Fairness • minimum surface area • minimum curvature • minimum curvature variation Leif Kobbelt RWTH Aachen University 23 23 Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 24 24 Polynomials • computable functions p p t ti ! t p( )=! ci = ! ci Φi( ) i=0 i=0 • Taylor expansion p 1 f(h)= f (i)(0) hi + O(hp+1) ! i! i=0 • interpolation error (mean value theorem) p(ti)=f(ti),ti ∈ [0,h] p 1 (p+1) ∗ (p+1) !f(t) − p(t)! = f (t ) (t − ti)=O(h ) (p + 1)! ! i=0 Leif Kobbelt RWTH Aachen University 25 25 Polynomials • computable functions p p t ti ! t p( )=! ci = ! ci Φi( ) i=0 i=0 • Taylor expansion p 1 f(h)= f (i)(0) hi + O(hp+1) ! i! i=0 • interpolation error (mean value theorem) p(ti)=f(ti),ti ∈ [0,h] p 1 (p+1) ∗ (p+1) !f(t) − p(t)! = f (t ) (t − ti)=O(h ) (p + 1)! ! i=0 Leif Kobbelt RWTH Aachen University 26 26 Polynomials • computable functions p p t ti ! t p( )=! ci = ! ci Φi( ) i=0 i=0 • Taylor expansion p 1 f(h)= f (i)(0) hi + O(hp+1) ! i! i=0 • interpolation error (mean value theorem) p(ti)=f(ti),ti ∈ [0,h] p 1 (p+1) ∗ (p+1) !f(t) − p(t)! = f (t ) (t − ti)=O(h ) (p + 1)! ! i=0 Leif Kobbelt RWTH Aachen University 27 27 Implicit Polynomials • interpolation error of the function values !F (x, y, z) − P (x, y, z)! = O(h(p+1)) • approximation error of the contour F (p + !p) − F (p) !p = λ ∇F (p) ≈ #∇F (p)# #!p# Leif Kobbelt RWTH Aachen University 28 28 Implicit Polynomials • interpolation error of the function values !F (x, y, z) − P (x, y, z)! = O(h(p+1)) p • approximation error of the contour p+Δp F (p + !p) − F (p) !p = λ ∇F (p) ≈ #∇F (p)# #!p# Leif Kobbelt RWTH Aachen University 29 29 Implicit Polynomials • interpolation error of the function values !F (x, y, z) − P (x, y, z)! = O(h(p+1)) p • approximation error of the contour p+Δp F (p + "p) − F (p) !p = λ ∇F (p) !"p!≈ !∇F (p)! (gradient bounded from below) Leif Kobbelt RWTH Aachen University 30 30 Implicit Polynomials F(x,y,z) F(x,y,z) x,y,z x,y,z large small gradient gradient Leif Kobbelt RWTH Aachen University 31 31 Polynomial Approximation • approximation error is O(hp+1) • improve approximation quality by – increasing p ... higher order polynomials – decreasing h ... smaller / more segments • issues – smoothness of the target data ( maxt f(p+1)(t) ) – handling higher order patches (e.g. boundary conditions) Leif Kobbelt RWTH Aachen University 32 32 Polynomial Approximation • approximation error is O(hp+1) • improve approximation quality by – increasing p ... higher order polynomials – decreasing h ... smaller / more segments • issues – smoothnessMOTD: of the target data ( maxpt f(p+1)=(t) ) 1 – handling higher order patches (e.g. boundary conditions) Leif Kobbelt RWTH Aachen University 33 33 Piecewise Definition • parametric – Euler formula: V - E + F = 2 (1-g) – regular quad meshes • F ≈ V • E ≈ 2V • average valence = 4 – quasi-regular – semi-regular Leif Kobbelt RWTH Aachen University 34 34 Piecewise Definition • parametric – Euler formula: V - E + F = 2 (1-g) – regular triangle meshes • F ≈ 2V • E ≈ 3V • average valence = 6 – quasi-regular – semi-regular Leif Kobbelt RWTH Aachen University 35 35 Piecewise Definition • quasi regularRemeshing (→ remeshing, Results Pierre) 1 2 4 4 Original ! 2 , " ! 5 , 3 " Leif Kobbelt RWTH Aachen University 36 36 Computer Graphics Group Mario Botsch Piecewise Definition • semi regular Leif Kobbelt RWTH Aachen University 37 37 Piecewise Definition • semi regular Leif Kobbelt RWTH Aachen University 38 38 Piecewise Definition • implicit – regular voxel grids O(h-3) – three color octrees • surface-adaptive refinement O(h-2) • feature-adaptive refinement O(h-1) – irregular hierarchies • binary space partition O(h-1) (BSP) Leif Kobbelt RWTH Aachen University 39 39 3-Color Octree 1048576 cells 12040 cells Leif Kobbelt RWTH Aachen University 40 40 Adaptively Sampled Distance Fields 12040 cells 895 cells Leif Kobbelt RWTH Aachen University 41 41 Binary Space Partitions 895 cells 254 cells Leif Kobbelt RWTH Aachen University 42 42 Message of the Day ... • polygonal meshes are a good compromise – approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement Leif Kobbelt RWTH Aachen University 43 43 Message of the Day ... • polygonal meshes are a good compromise – approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement Leif Kobbelt RWTH Aachen University 44 44 Message of the Day ... • polygonal meshes are a good compromise – approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement Leif Kobbelt RWTH Aachen University 45 45 Message of the Day ... • polygonal meshes are a good compromise – approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement Leif Kobbelt RWTH Aachen University 46 46 Message of the Day ... • polygonal meshes are a good compromise – approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement – efficient rendering Leif Kobbelt RWTH Aachen University 47 47 Message of the Day ..
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