Surface Representations
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1 )NPUT $ATA 2ANGE 3CAN #!$ 4OMOGRAPHY 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS!NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL 0ARAMETERIZATION 3IMPLIFICATION FOR COMPLEXITY REDUCTION Surface Representations 2EMESHING FOR IMPROVING MESH QUALITY &REEFORM AND MULTIRESOLUTION MODELING 1
2 Outline (mathematical) geometry representations parametric vs. implicit approximation properties types of operations distance queries evaluation modification / deformation data structures 2 2
3 Outline (mathematical) geometry representations parametric vs. implicit approximation properties types of operations distance queries evaluation modification / deformation data structures 3 3
4 Mathematical Representations parametric range of a function surface patch f : R 2 R 3, S Ω = f(ω) implicit kernel of a function level set F : R 3 R, S c = {p : F (p) =c} 4 4
5 2D-Example: Circle parametric ( r cos(t) f : t r sin(t) ), S = f([0, 2π]) implicit F (x, y) =x 2 + y 2 r 2 S = {(x, y) :F (x, y) =0} 5 5
6 2D-Example: Island parametric ( r cos(t)??? f : t r sin(t)??? ), S = f([0, 2π]) implicit F (x, y) =x 2 +??? y 2 r 2 S = {(x, y) :F (x, y) =0} 6 6
7 Approximation Quality piecewise parametric ( ) r cos(t)??? f : t, S = f([0, 2π]) r sin(t)??? piecewise implicit F (x, y) =x 2 +??? y 2 r 2 S = {(x, y) :F (x, y) =0} 7 7
8 Approximation Quality piecewise parametric ( ) r cos(t)??? f : t, S = f([0, 2π]) r sin(t)??? piecewise implicit F (x, y) =x 2 +??? y 2 r 2 S = {(x, y) :F (x, y) =0} 8 8
9 Requirements / Properties continuity interpolation / approximation f(u i,v i ) p i topological consistency manifold-ness smoothness C 0, C 1, C 2,... C k fairness curvature distribution 9 9
10 Requirements / Properties continuity interpolation / approximation f(u i,v i ) p i topological consistency manifold-ness smoothness C 0, C 1, C 2,... C k fairness curvature distribution 10 10
11 Requirements / Properties continuity interpolation / approximation f(u i,v i ) p i topological consistency manifold-ness smoothness C 0, C 1, C 2,... C k fairness curvature distribution 11 11
12 Requirements / Properties continuity interpolation / approximation f(u i,v i ) p i topological consistency manifold-ness smoothness C 0, C 1, C 2,... C k fairness curvature distribution 12 12
13 Requirements / Properties continuity interpolation / approximation f(u i,v i ) p i topological consistency manifold-ness smoothness C 0, C 1, C 2,... C k fairness curvature distribution 13 13
14 Topological Consistency 14 14
15 Topological Consistency 14 14
16 Topological Consistency Mesh Repair
17 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)
18 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)
19 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)
![Closed 2-Manifolds parametric disk-shaped neighborhoods f(d ε [u, v]) = D δ](/docs-images/59/43840650/images/20-1.png "[f(u, v)] implicit surface of a physical solid F (x, y, z) =c, F (x, y, z) 0")
20 Closed 2-Manifolds parametric disk-shaped neighborhoods f(d ε [u, v]) = D δ [f(u, v)] implicit surface of a physical solid F (x, y, z) =c, F (x, y, z)
![Closed 2-Manifolds parametric disk-shaped neighborhoods f(d ε [u, v]) = D δ](/docs-images/59/43840650/images/21-0.png "[f(u, v)] implicit surface of a physical solid F (x, y, z) =c, F (x, y, z) 0")
21 Closed 2-Manifolds parametric disk-shaped neighborhoods f(d ε [u, v]) = D δ [f(u, v)] implicit surface of a physical solid F (x, y, z) =c, F (x, y, z)
22 Smoothness position continuity : C 0 tangent continuity : C 1 curvature continuity : C
23 Smoothness position continuity : C 0 tangent continuity : C 1 curvature continuity : C
24 Smoothness position continuity : C 0 tangent continuity : C 1 curvature continuity : C
25 Fairness minimum surface area minimum curvature minimum curvature variation 23 23
26 Outline (mathematical) geometry representations parametric vs. implicit approximation properties types of operations distance queries evaluation modification / deformation data structures 24 24
27 Polynomials computable functions p(t) = p i=0 c i t i = p i=0 c i Φ i (t) Taylor expansion f(h) = p i=0 1 i! f (i) (0) h i + O(h p+1 ) interpolation error (mean value theorem) f(t) p(t) = p(t i )=f(t i ), t i [0,h] 1 p (p + 1)! f (p+1) (t ) (t t i )=O(h (p+1) ) i=
28 Polynomials computable functions p(t) = Taylor expansion p i=0 c i t i = p i=0 c i Φ i (t) f(h) = p i=0 1 i! f (i) (0) h i + O(h p+1 ) interpolation error (mean value theorem) f(t) p(t) = p(t i )=f(t i ), t i [0,h] 1 p (p + 1)! f (p+1) (t ) (t t i )=O(h (p+1) ) i=
29 Polynomials computable functions p(t) = p i=0 c i t i = p i=0 c i Φ i (t) Taylor expansion f(h) = p i=0 1 i! f (i) (0) h i + O(h p+1 ) interpolation error (mean value theorem) p(t i )=f(t i ), t i [0,h] f(t) p(t) = 1 (p + 1)! f (p+1) (t ) p (t t i )=O(h (p+1) ) i=
30 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 p = λ F (p) F (p + p) F (p) p F (p) 28 28
31 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 p+δp p p = λ F (p) F (p + p) F (p) p F (p) 29 29
32 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 p+δp p p = λ F (p) p F (p + p) F (p) F (p) (gradient bounded from below) 30 30
33 Implicit Polynomials F(x,y,z) F(x,y,z) x,y,z x,y,z large gradient small gradient 31 31
34 Polynomial Approximation approximation error is O(h p+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) 32 32
35 Polynomial Approximation approximation error is O(h p+1 ) improve approximation quality by increasing p... higher order polynomials decreasing h... smaller / more segments issues MOTD: p = 1 smoothness of the target data ( maxt f (p+1) (t) ) handling higher order patches (e.g. boundary conditions) 33 33
36 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 34 34
37 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 35 35
38 Piecewise Definition quasi regular ( remeshing, Pierre) 36 36
39 Piecewise Definition semi regular 37 37
40 Piecewise Definition semi regular 38 38
41 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) 39 39
42 3-Color Octree cells cells 40 40
43 Adaptively Sampled Distance Fields cells 895 cells 41 41
44 Binary Space Partitions 895 cells 254 cells 42 42
45 Message of the Day... polygonal meshes are a good compromise approximation O(h 2 )... error * #faces = const. arbitrary topology flexibility for piecewise smooth surfaces flexibility for adaptive refinement 43 43
46 Message of the Day... polygonal meshes are a good compromise approximation O(h 2 )... error * #faces = const. arbitrary topology flexibility for piecewise smooth surfaces flexibility for adaptive refinement 44 44
47 Message of the Day... polygonal meshes are a good compromise approximation O(h 2 )... error * #faces = const. arbitrary topology flexibility for piecewise smooth surfaces flexibility for adaptive refinement 45 45
48 Message of the Day... polygonal meshes are a good compromise approximation O(h 2 )... error * #faces = const. arbitrary topology flexibility for piecewise smooth surfaces flexibility for adaptive refinement 46 46
49 Message of the Day... polygonal meshes are a good compromise approximation O(h 2 )... error * #faces = const. arbitrary topology flexibility for piecewise smooth surfaces flexibility for adaptive refinement efficient rendering 47 47
50 Message of the Day... polygonal meshes are a good compromise approximation O(h 2 )... error * #faces = const. arbitrary topology flexibility for piecewise smooth surfaces flexibility for adaptive refinement efficient rendering implicit representation can support efficient access to vertices, faces,
51 Outline (mathematical) geometry representations parametric vs. implicit approximation properties types of operations evaluation distance queries modification / deformation data structures 49 49
52 Evaluation smooth parametric surfaces positions normals curvatures f(u, v) n(u, v) =f u (u, v) f v (u, v) ( ) c(u, v) =C f uu (u, v), f uv (u, v), f vv (u, v) 50 50
53 Evaluation smooth parametric surfaces positions normals curvatures f(u, v) n(u, v) =f u (u, v) f v (u, v) ( ) c(u, v) =C f uu (u, v), f uv (u, v), f vv (u, v) generalization to triangle meshes positions (barycentric coordinates) (α, β) α P 1 + β P 2 + (1 α β) P 3 0 α, 0 β, α + β
54 Evaluation smooth parametric surfaces positions normals curvatures f(u, v) n(u, v) =f u (u, v) f v (u, v) ( ) c(u, v) =C f uu (u, v), f uv (u, v), f vv (u, v) generalization to triangle meshes positions (barycentric coordinates) (α, β, γ) α P 1 + β P 2 + γ P 3 α + β + γ =
55 Evaluation smooth parametric surfaces positions normals curvatures f(u, v) n(u, v) =f u (u, v) f v (u, v) ( ) c(u, v) =C f uu (u, v), f uv (u, v), f vv (u, v) generalization to triangle meshes positions (barycentric coordinates) α u + β v + γ w α P 1 + β P 2 + γ P 3 α + β + γ =
56 Evaluation smooth parametric surfaces positions normals curvatures f(u, v) n(u, v) =f u (u, v) f v (u, v) ( ) c(u, v) =C f uu (u, v), f uv (u, v), f vv (u, v) generalization to triangle meshes positions (barycentric coordinates) Parametrization Bruno α u + β v + γ w α P 1 + β P 2 + γ P 3 α + β + γ =
57 Evaluation smooth parametric surfaces positions normals curvatures f(u, v) n(u, v) =f u (u, v) f v (u, v) ( ) c(u, v) =C f uu (u, v), f uv (u, v), f vv (u, v) generalization to triangle meshes positions (barycentric coordinates) normals (per face, Phong) N =(P 2 P 1 ) (P 3 P 1 ) 55 55
58 Evaluation smooth parametric surfaces positions normals curvatures f(u, v) n(u, v) =f u (u, v) f v (u, v) ( ) c(u, v) =C f uu (u, v), f uv (u, v), f vv (u, v) generalization to triangle meshes positions (barycentric coordinates) normals (per face, Phong) α u + β v + γ w α N 1 + β N 2 + γ N
59 Evaluation smooth parametric surfaces positions normals curvatures f(u, v) n(u, v) =f u (u, v) f v (u, v) ( ) c(u, v) =C f uu (u, v), f uv (u, v), f vv (u, v) generalization to triangle meshes positions (barycentric coordinates) normals (per face, Phong) curvatures... ( smoothing, Mark) 57 57
60 Distance Queries parametric for smooth surfaces: find orthogonal base point [p f(u, v)] n(u, v) =0 for triangle meshes use kd-tree or BSP to find closest triangle find base point by Newton iteration (use Phong normal field) 58 58
61 Modifications parameteric control vertices free-form deformation n f (u, v) = boundary constraint modeling i=0 m j=0 c ij N n i (u) N m j (v) 59 59
62 Modifications parameteric control vertices free-form deformation n f (u, v) = boundary constraint modeling i=0 m j=0 c ij N n i (u) N m j (v) 60 60
63 Modifications parameteric control vertices free-form deformation boundary constraint modeling Remeshing Pierre 61 61
64 Modifications parameteric control vertices free-form deformation boundary constraint modeling 62 62
65 Modifications parameteric control vertices free-form deformation boundary constraint modeling 63 63
66 Modifications parameteric control vertices free-form deformation boundary constraint modeling 63 63
67 Modifications parameteric control vertices free-form deformation boundary constraint modeling Mesh Editing Mario 64 64
68 Outline (mathematical) geometry representations parametric vs. implicit approximation properties types of operations distance queries evaluation modification / deformation data structures 65 65
69 Mesh Data Structures how to store geometry & connectivity? compact storage file formats efficient algorithms on meshes identify time-critical operations all vertices/edges of a face all incident vertices/edges/faces of a vertex 66 66
70 Face Set (STL) face: 3 positions Triangles x11 y11 z11 x12 y12 z12 x13 y13 z13 x21 y21 z21 x22 y22 z22 x23 y23 z xf1 yf1 zf1 xf2 yf2 zf2 xf3 yf3 zf3 36 B/f = 72 B/v no connectivity! 67 67
71 Shared Vertex (OBJ, OFF) vertex: position face: vertex indices Vertices x1 y1 z1... xv yv zv Triangles v11 v12 v vf1 vf2 vf3 12 B/v + 12 B/f = 36 B/v no neighborhood info 68 68
72 Face-Based Connectivity vertex: position 1 face face: 3 vertices 3 face neighbors 64 B/v no edges! 69 69
73 Edge-Based Connectivity vertex position 1 edge edge 2 vertices 2 faces 4 edges face 120 B/v edge orientation? 1 edge 70 70
74 Halfedge-Based Connectivity vertex position 1 halfedge halfedge 1 vertex 1 face 1, 2, or 3 halfedges face 1 halfedge 96 to 144 B/v no case distinctions during traversal 71 71
75 One-Ring Traversal 1. Start at vertex 72 72
76 One-Ring Traversal 1. Start at vertex 2. Outgoing halfedge 73 73
77 One-Ring Traversal 1. Start at vertex 2. Outgoing halfedge 3. Opposite halfedge 74 74
78 One-Ring Traversal 1. Start at vertex 2. Outgoing halfedge 3. Opposite halfedge 4. Next halfedge 75 75
79 One-Ring Traversal 1. Start at vertex 2. Outgoing halfedge 3. Opposite halfedge 4. Next halfedge 5. Opposite 76 76
80 One-Ring Traversal 1. Start at vertex 2. Outgoing halfedge 3. Opposite halfedge 4. Next halfedge 5. Opposite 6. Next
81 Halfedge-Based Libraries CGAL computational geometry free for non-commercial use OpenMesh " mesh processing free, LGPL licence 78 78
82 Literature Kettner, Using generic programming for designing a data structure for polyhedral surfaces, Symp. on Comp. Geom., 1998 Campagna et al, Directed Edges - A Scalable Representation for Triangle Meshes, Journal of Graphics Tools 4(3), 1998 Botsch et al, OpenMesh - A generic and efficient polygon mesh data structure, OpenSG Symp
83 Outline (mathematical) geometry representations parametric vs. implicit approximation properties types of operations distance queries evaluation modification / deformation data structures 80 80
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