1. Polygonal Mesh – Data Structure and Processing
Chiew-Lan Tai

2. What is a Mesh?

3. What is a Mesh?
A Mesh is a pair (P,K), where P is a set of point positions
and K is an abstract simplicial complex which contains all topological information.
K is a set of subsets of :
Vertices
Edges
Faces

4. What is a Mesh? Each edge must belong to at least one face, i.e.
Each vertex must belong to at least one edge, i.e.
An edge is a boundary edge if it only belongs to one face

5. What is a Mesh? A mesh is a manifold if A mesh is a polyhedron if
Every edge is adjacent to one (boundary) or two faces
For every vertex, its adjacent polygons form a disk (internal vertex) or a half-disk (boundary vertex)
Manifold
Non-manifold
A mesh is a polyhedron if
It is a manifold mesh and it is closed (no boundary)
Every vertex belongs to a cyclically ordered set of faces (local shape is a disk)

6. Orientation of Faces
Each face can be assigned an orientation by defining the ordering of its vertices
Orientation can be clockwise or counter-clockwise.
The orientation determines the normal direction of face. Usually counterclockwise order is the “front” side.

7. non-orientable surfaces
Orientation of Faces
A mesh is well oriented (orientable) if all faces can be oriented consistently (all CCW or all CW) such that each edge has two opposite orientations for its two adjacent faces
Not every mesh can be well oriented. e.g. Klein bottle, Möbius strip
non-orientable surfaces

8. Euler Formula
The relation between the number of vertices, edges, and faces.
where
V : number of vertices
E : number of edges
F : number of faces

9. Euler Formula Tetrahedron Cube Octahedron V = 4 E = 6 F = 4
= 2
Cube
V = 8
E = 12
F = 6
= 2
Octahedron
V = 6
E = 12
F = 8
= 2
V = 8
E = = 13
F = = 7
= 2
V = 8
E = 12
F = 6
= 2

10. Euler Formula More general rule where V : number of vertices
E : number of edges
F : number of faces
C : number of connected components
G : number of genus (holes, handles)
B : number of boundaries
V = 16
E = 32
F = 16
C = 1
G = 1
B = 0
16 – = 2 (1 - 1) - 0

11. Data Structure

12. Neighborhood Relations

13. Neighborhood Relations
For a vertex
All neighboring vertices
All neighboring edges
All neighboring faces
Knowing some types of relation, we can discover other (but not necessary all) topological information
e.g. if in addition to VV, VE and VF, we know neighboring vertices of a face, we can discover all neighboring edges of the face

14. Choice of Data Structure
Criteria for choosing a particular data structure
Size of mesh (# of vertices and faces)
Speed and memory of computer
Types of meshes (triangles only, arbitrary polygons)
Redundancy of data
Operations to be preformed (see next slide)
Tradeoff between updating and query
More redundancy of data, faster query but slower updating

15. Choice of Data Structure
Face-based data structure
Problem: different topological structure for triangles and quadrangles
Edge-based data structure
Winged-edge data structure
Problem: traveling the neighborhood requires one case distinction
Half-edge data structure
Aka doubly connected edge list (DCEL)

16. Half-Edge Data Structure
Each edge is divided into two half-edges
Each half-edge has 5 references:
The face on left side (assume counter-clockwise order)
Previous and next half-edge in counterclockwise order
The “twin” edge
The starting vertex
Each face has a pointer to one of its edges
Each vertex has a
pointer to a half
edge that has this vertex as the start vertex

17. Half-edge data structure: example
origin
twin
incident face
next
prev
e3,1 v2
e3,2 f1
e1,1 v3 f2
e5,1
e4,1 v4
e4,2 f3
e7,1
e6,1

18. Half-Edge Data Structure

19. Half-Edge Data Structure
Here is another view of half-edge data structure.
Next pointers provide links around each face in counterclockwise (prev pointers can be used to go around in clockwise)
Example: finding all vertices adjacent to vertex v.
/* Assume closed mesh and using counterclockwise order */
HalfEdge he = v.he; HalfEdge curr = he; output (curr.twin.start); while (curr.twin.next != he) { curr = curr.twin.next; output (curr.twin.start); }

20. Progressive mesh

21. Progressive Mesh New representation of triangular meshes
References:
Hoppe, Progressive mesh, Siggraph 96
Hoppe, View-dependent Refinement of Progressive Meshes, Siggraph 97
New representation of triangular meshes
Simplify meshes through sequence of edge collapse transformations
Record the sequence of inverse transformations (vertex splits)

22. Traditional Mesh Representation
mesh M V F
Vertex 1 x1 y1 z1
Vertex 2 x2 y2 z2 …
Face 1 2 3
Face 3 2 4
Face 4 2 7 …
(appearance attributes: normals, colors, textures, ...)

23. Progressive Mesh Representation
150
152
500
13,546 M0
base mesh M1
M175 Mn
Original mesh

24. Simplification: Edge Collapse
Idea: apply a sequence of edge collapses:
ecol(vs ,vt , vs ) ’ vt vl vr vl vr vs ’
(optimization) vs

25. Simplification: Edge Collapse
150
152
500
13,546 M0 M1
M175 Mn
ecol0
… ecoli …
ecoln-1
Mn=M ^ M0 Mi

26. Reconstruction: Vertex Split
Invertible & lossless!
attributes
vspl(vs ,vl ,vr , vs ,vt ,…) ’ ’ vt ’ vl vl vr vr vs vs ’

27. Reconstruction: Vertex Split
150
152
500
13,546 M0 M1
M175 Mn
vspl0
… vspli …
vspln-1
Mn=M ^ M0 Mi

28. Progressive Mesh: Benefits^ PM V F M0
lossless
vspl
single-resolution
efficient
continuous-resolution
space-efficient
progressive
Optimization process:
off-line process
various metrics (could use simpler heuristics)

29. Application – Mesh compression
12964 faces
1000 faces

30. Application – Progressive Transmission
Transmit the records progressively:
time
vspln-1 M ^
(~ progressive JPEG)
vspli-1 Mi M0
vspl0
vspl1
Receiver displays: M0

31. Application – Selective refinement
vspl0
vspl1
vspli-1
vspln-1
(e.g. view frustum)

32. Property – Vertex CorrespondenceMf
Mf-1 v1 v2 v3 v4 v5 v6 v7
Mf-2 v1 v2 v3 v4 v5 v6 Mc Mn v1 v1 M0 v2 v2
ecol v3
ecol v3 v4
ecol v5 v6 v7 v8
ecol(vs ,vt , v’s ) vl vr vt vs ’

33. Application – Smooth Transitions
Correspondence is a surjection (onto function): Mf Mc v1 v1 v2 v2
Mf«c v3 v3 v4 V V F v5 v6 v7
® can form a smooth visual transition: geomorph v8

34. Mesh Simplification

35. References
Garland and Heckbert, Surface Simplification Using Quadric Error Metrics, Siggraph 97

36. Applications
Create progressively coarser versions of objects (levels of detail LOD); render less detailed version for small, distant, unimportant parts of scene
Inverse decimation for progressive transmission
Multiresolution decomposition (for compression, editing) – decompose an original mesh M0 into a low frequency component (simplified mesh Ms) and a high frequency component (M0 – Ms)

37. Introduction LOD frameworks: Simplification algorithms can be
discrete LOD
continuous LOD
view-dependent (anisotropic) LOD
Simplification algorithms can be
Fidelity-based – for generating accurate image
Budget-based – simplify until a targeted number of polygons; for time-critical rendering

38. Basic Approaches Bottom-up approaches (this lecture)
Start with the original fine mesh
Remove elements (vertices, edges, triangles, tetrahedra) and replace with fewer elements
Top-down approaches (wavelet, subdivision-based)
Start with very coarse approximation of original mesh
New points are inserted to generate better meshes

39. Bottom-up approaches
Most methods operate from a priority queue of element
Vertices, edges, triangles, tetrahedra are ordered in a priority queue and process one-by-one as they are de-queued
The cost function, which assigns the “priority”, determines the order of processing

40. Local simplification operators
Edge collapse (full edge and half edge collapse) – Hoppe96, Xia96, Bajaj99
Vertex-pair collapse – Shroeder97, Garland97,Popovic97
Triangle collapse – Hamann94, Gieng98
Vertex removal – Shroeder92
etc …

41. Operator: Edge Collapse
vnew va vb
Edge collapse
Vertex split
Full-edge collapse: vnew = optimized position
Half-edge collapse: vnew = va or vb

42. Operator: Edge Collapse
Beware triangle folding!
Lead to visual artifacts, e.g., illumination and texture discontinuities.
Can be detected by measuring the change in the normals of the corresponding triangles before and after an edge collapse

43. Operator: Edge Collapse
Beware topological inconsistence!
vnew va vb
Edge collapse
A manifold mesh may become non-manifold due to edge collapse

44. Operator: Vertex-pair Collapseva
vnew vb
Enables closing of holes and tunnels – changes topology

45. Operator: Triangle Collapseva vc
vnew vb
A triangle collapse is equivalent to two edge collapses. Triangle collapse hierarchies are shallower, but also less adaptable since this is a less fine-grained operation.

46. Comparisons
Collapse operators: simplest to implement; well-suited for implementing geomorphing between successive levels of detail
Half-edge collapse: advs: less triangles are modified than full-edge collapse; vertices are a subset of original mesh => simplifies bookkeeping

47. Priority Queue Methods
Using the cost function (priority), prioritize all the elements in the mesh
Collapse the element with the lowest cost (priority)
Readjust the priorities of all the elements in the queue affected by the collapse

48. Error Metrics
Usually incorporate some form of object-space geometric error measure
Object-space error measure may be converted to a screen-space distance during runtime
May also incorporate measure of attribute errors (color, normal and texture coordinate)
Measure incremental error or total error

49. Quadric Error Metric (QEM) (Garland and Heckbert, Siggraph 97)
Maybe the “best” method
Fast error computation, compact storage, good visual quality
Measures sum of squared vertex-plane distances

50. Quadric Error Metric (QEM)P1 P2 P3 s1 v s3 s2
P1 = (a b c d) representing a plane ax + by + cz + d = 0, and v = [vx vy vz 1]
s1 : distance from vertex v to plane P1
squared vertex-plane distances:
s2 = (p·v)2 = (vTp)(pTv)
Total square distances:
The quadratic form Qp is a 4x4 symmetric matrix, represented using 10 unique floating point numbers. Summing all the matrices gives Qv

51. Quadric Error Metric (QEM)
Introduces QEM to prioritize collapses
a 4x4 matrix Q is associated with each vertex
Error of a vertex v is vTQv
Initially, compute matrices Q for all vertices
Compute the collapse cost of each edge by summing the QEM of its two vertices
The error (cost) of collapsing the edge (va , vb ) v is vT(Qa+Qb)v
Repeat
Collapse the edge with least cost
Update collapse costs of v and its neighborhood
Until user requirement is achieved

52. Quadric Error Metric (QEM)
Ellipsoids shown are level surfaces of the error quadrics, illustrate the size and orientation of error quadrics

53. Quadric Error Metric (QEM)
69,451 triangles
1,000 triangles
100 triangles

54. QEM Simplification

55. Mesh Fairing (Smoothing)

56. References
Taubin, A signal processing approach to fair surface design, Siggraph 95.
Kobbelt et al., Interactive multi-resolution modeling on arbitrary meshes, Siggraph 98
Desbrun et al. Implicit fairing of irregular meshes using diffusion and curvature flow, Siggraph 99

57. Mesh Fairing

58. Fairing operators for meshes
Umbrella [Kobbelt98]
Improved umbrella [Desbrun99]
Taubin l|m [Taubin95]
Curvature flow [Desbrun99]
etc …

59. Definition: 1-ring Neighbors of a Vertexp1 pn p p2
1-ring neighbors of p,
N1(p)={p1,p2,…, pn}
Valence of p is n
...

60. General Idea Predict a vertex position from its neighbor
for a vertex vi and its neighbor vj, let the weight be wij such that
For vertex vi, predict
Iterate through all vertices and update the positions
where
Is a specific normalized curvature operator,  is a damping factor

61. General idea Two ways to do smoothing
Explicit updating – find each and update each
Compute all , solve for all xi as a linear system:
Let K = I – W, then in matrix form, X = - K X

62. Umbrella Operator
Pros: simple, fast; work well for meshes with small variation in edge length and face angles
Cons:
for irregular connectivity meshes lead to artifacts
weights only based on the connectivity, not the geometry (e.g., edge length, face area, angle between edges)
vertex drifting problem, smoothing affects parametrization, not just geometry

63. Improved Umbrella Operator
Scale-dependent umbrella operator
Still has a non-zero tangential component that causes vertex drifting
Discussion: no longer linear (linearized by assuming the edge lengths do not change during one smoothing round)

64. Curvature Flow Operator
A noise removal procedure that does not depend on parametrization
Vertices move along the surface normal with a speed equal to the mean curvature
Geometry is smoothed without affecting the sampling rate aj xi xj bj

65. Curvature Flow Operator
Vertices can only move along their normal
no vertex drifting in parameter space

66. Comparisons

67. Extensions and Applications
Volume preservation
Fairing meshes with constraints
Stopband filters and enhancement
Multiresolution editing of arbitrary meshes
Non-uniform subdivision
Mesh edge detection
Fairing of non-manifold models