1. Polygonal Mesh – Data Structure and Smoothing
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 boundary loops
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

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. Mesh Fairing (Smoothing)

21. 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

22. Mesh Fairing

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

24. 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
...

25. 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

26. General idea X = - K X Explicit updating – find each and update each
In matrix form:
X = - K X
where K = I - W
Explicit updating – find each and update each

27. 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

28. 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)

29. 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

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

31. Comparisons

32. 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