3D Compression, SM’02 1Jarek Rossignac, GVU Center, Georgia Tech1: T-meshes Triangle meshes Jarek...

3D Compression, SM’02 1Jarek Rossignac, GVU Center, Georgia Tech 1: T-meshes

Triangle meshes

Jarek RossignacGVU Center and College of Computing

Georgia Tech, Atlanta

http://www.gvu.gatech.edu/~jarek


Popular domain

• Surfaces decomposed into simple manifolds (with boundary)

• Represent each manifold surface as a triangle mesh– T-meshes are supported by optimized rendering systems

– Easily derived from polygons and parametric surfaces

Pseudo-manifold Faces (flat or curved)


Focus on explicit representations

• Samples: Location and attributes (color, mass)• Connectivity: Triangle/vertex incidence• Fit: Rule for bending triangles (subdivision surfaces, NURBS)

Samples (vertices):

x y z cx y z cvertex 1

vertex 2

vertex 3 x y z c

Triangle/vertex incidence:

3 2 4

6 5 8

Triangle 4 7 5 6Triangle 5

1 2 3Triangle 1

4 5 2

Triangle 2

Triangle 3

Triangle 6 8 5 1

t3

v4

v2v5

V(3B+k) bits

V(6log2V) bitsT = 2V


Samples, connectivity, & attributes

• Samples (“vertices”)– Location (x,y,z)

• Connectivity (“triangles”)– Define how surface interpolates samples

– Specifies surface as a set of triangles

– Associates each triangle with 3 samples (called corners)

– Define how to interpolate corner attributes over triangle

• Attributes (parameters for color and texture calculations)– One per corner of each triangle

• Could be the same for all 3 corners (flat triangle)

• Could be the same for two adjacent corners (smooth edge)

• Could be the same for all coincident corners (smooth surface)

– Linear interpolation of shape and attributes over triangle


• Vertex: – Location of a sample

• Triangles: – Decompose approximating surface

• Edge: – Bounds one or more triangles– Joins two vertices

• Corner: – Abstract association of a triangle with a vertex– May have its own attributes (not shared by corners with same vertex)

• Used to capture surface discontinuities

• Border (half-edge, dart):– Association of a triangles with a bounding edge.– Defines an orientation of the border

• A triangle has 3 borders and 3 corners

Terminology

triangle

border

vertexcorner

edge


• For each triangle:– For each one of its 3 corners, store:

• Location

• Attributes (may be the same for neighboring corners)

• Each vertex location is repeated (6 times on average)– geometry = 36 B/T (float coordinates: 9x4 B/T)

– Plus 3 attribute-sets per triangle (6 per vertex)

Representation as independent triangles

x y z x y z x y zx y z x y z x y zx y z x y z x y z

vertex 1 vertex 2 vertex 3

Triangle 2

Triangle 1

Triangle 3

Very verbose! Not good for traversal.


Connectivity = Incidence + adjacency

• Triangle/vertex incidence (“corner”)– Associates each triangle with 3 vertices

• Defines Corners

• Triangle/triangle adjacency– Associates triangle with neighboring triangles

• Neighboring triangles share a common edge

– Is completely defined by incidence!

– Convenient to accelerate traversal of triangulated surface• Walk from one triangle to an adjacent one (visit them all once)

• Used to build triangle strips

• Used to estimate surface normals at vertices

• Used to compress triangulated surfaces

• Triangle/edge incidence (border)– Associates triangles with their bounding edges


Triangle orientation

• Orient the plane supporting a triangle– Pick one of 2 possible orientations for the normal

• List corners in counter clockwise order– Cyclic order (equivalence under cyclic permutation)

• Orientation compatibility for adjacent triangles– Common corners follow each other in reverse order

• Can try to propagate consistent orientation– Pick orientation for first triangle

– Propagate to neighbors (edge-connected), needs not use geometry

– What if more than two triangles share same edge?

• Orientable set of triangles– Can all triangles be oriented to be consistent with their neighbors?

• Only if the mesh is orientable


• Integer Ids for vertices (0, 1, 2… V-1) & triangles (0, 1, 2…T-1)

• Triangle orientation: cyclic order in which corners are listed

• Other connectivity info may be derived– Borders: defined by 3 pairs of corners for each triangle

– Edges: Set of borders with same two vertices

– Adjacency: Triangles incident upon the same edge

• Incidence graph representation:– List of corners (id for vertex & attribute)

– Corners of each triangle are consecutive

• Samples defined separately:– List of vertex locations (x,y,z)

– List of attributes (color,normal, texture…)

• Not practical for traversing mesh

vertex 1 x y z

vertex 2 x y z

vertex 3 x y z

vertex 4 x y z

Incidence table

1

2

34

Triangle 0 1 a

Triangle 0 2 b

Triangle 0 3 c

Triangle 1 2 c

Triangle 1 1 d

Triangle 1 4 e

Triangle 2 1 a

v a

attribute a red

attribute b blue


T-meshes: manifold connectivity graph

• T-mesh = triangle set with a manifold connectivity graph– The corners of each triangle refer to different vertices

– Each edge has exactly two incident triangles

– Manifold vertices: The star of each vertex is connected• Star = union of edges and triangles incident upon the vertex

• Triangles form a surface that may be globally oriented– All triangle orientations are consistent (No Klein bottle)

• All triangles form a connected set– All pairs of triangles are connected

• Two adjacent triangles are connected

• Connectivity is a transitive relation


Connectivity/geometry discrepancy

• Connectivity of T-mesh may conflict with actual geometry– Vertices with different names may be coincident

– Edges with different names may be coincident

– Triangles, edges, and vertices may intersect

• T-mesh with consistent geometry– Triangles, edges, vertices are pairwise disjoint

• We consider edges and triangles to be open – I.e., not containing their boundary

• Manifold graphs may be used with invalid geometry– Coincident edges and vertices: Non-manifold singularities

– Self-intersecting surfaces

Non-manifold shapeManifold graph


Handles in T-meshes

• Handles correspond to through-holes– A sphere has zero handles, a torus has one

• The number of handles is well defined in a T-mesh– A handle cannot be identified as a particular set of triangles

• An edge-loop is a cycle of oriented edges– Each starts where the previous one ended, no repetition of vertices

• A T-mesh has k handles if and only if you can remove at most 2k edge-loops without disconnecting the mesh

• The genus of a T-mesh is the number of handles it has

connected


Simple T-meshes (STM) and T-patches

• We first look at simple meshes (no handles)– Homeomorphic to a sphere

– Incidence graph is a planar triangle graph

• We will also use the notion of a T-patch– Connected portion of an STM

– Bounded by a single edge-loop


Simple mesh

• A simple mesh is homeomorphic to a triangulated sphere– Orientable

– Manifold

– No boundary (no holes)

– No handles (no throu-holes)

• Properties– Each edge has exactly 2 incident triangles

– Each vertex has a single cycle of incident triangles

– May be drawn as a planar graph


Dual graphs and spanning trees

• Dual graph:– Nodes represent triangles

– Links represent edges• Join centers of adjacent triangles

• Vertex spanning tree (VST)– Edge-set connecting all vertices

– No cycles

– Cuts mesh into simply connected polygon with no interior vertices

• Triangle-spanning tree (TST)– Graph of remaining vertices

– No loops

– Connects all triangles

From Bosen


Euler formula for Simple Meshes

• Mesh has V vertices, E edges, and T triangles• E = (V-1)+(T-1)

– VST has V nodes and thus V-1 links– TST has T nodes and thus T-1 links

• E = 3T/2– There are 3 borders (edge-uses) per triangle– There are twice more edge-uses then edges

• T=2V-4– Because (V-1)+(T-1) = 3T/2– And hence V-2 = 3T/2-T = T/2

• There are twice more triangles than vertices• The number C of corners (vertex-uses) is about 6V

– C=3T=6V-12

• On average, a vertex is used 6 times


Properties of manifold meshes

• T triangles, E edges, V vertices, H handles, S shells• Euler: T-E+V=2S -2H

– Example: a tetrahedron has T=4, E=6, V=4, S=1, H=0…4-6+4=2-0

• Number of handles: H=S-(T-E+V)/2• Shared edges: E=3T/2

– 3 borders per triangle, 2 borders per edge

• Twice more triangles than vertices: T=2V+4(H-S)– T-3T/2+V=2S-2H– Assume H and S are much smaller than V

• Three times more edges than vertices: E=3V-6+6H– 2E/3-E+V=2-2H

vertices

triangles

edgesAdd 1 vertex, 2 triangles, and 3 edges


• Table of corners, for each corner c store:– c.v : integer reference to vertex table

– c.o : integer reference to opposite corner

– c.a : index to table of corner attributes

• Make the corners of each triangle consecutive– List them according to consistent orientation of triangles

– Can retrieve triangle number: c.t = c DIV 3

– Can retrieve next corner around triangle: c.n = 3t + (c+1)MOD 3

Corner table:data structure for T-meshes

c.t

c

c.v

c.o

c.n c.n.n

vertex 1 x y z

vertex 2 x y z

vertex 3 x y z

vertex 4 x y z

attribute a red

attribute b blue

Triangle 0 corner 0 1 7 a

Triangle 0 corner 1 2 8 b

Triangle 0 corner 2 3 5 c


Triangle 1 corner 4 1 6 d

Triangle 1 corner 5 4 2 e

v o a

1

2

340

12

3

4 5


• c.o can be derived from c.v (needs not be transmitted):

• Build table of triplets {min(c.n.v, c.n.n.v), max(c.n.v, c.n.n.v), c}– 230, 131, 122, 143, 244, 125, …

• Sort: – 122, 125 ...131... 143 ...230...244 …

• Pair-up consecutive entries 2k and 2k+1– (122, 125)...131... 143...230...244…

• Their corners are opposite– (122,125)...131...143...230...244…

Computing adjacency from incidence

Triangle 1 corner 0 1 a

Triangle 1 corner 1 2 b

Triangle 1 corner 2 3 c


Triangle 2 corner 4 1 d

Triangle 2 corner 5 4 e

v o a

Triangle 1 corner 0 1 a

Triangle 1 corner 1 2 b



Triangle 2 corner 4 1 d

Triangle 2 corner 5 4 2 e

v o a

1

2

340

12

3

4 5

0

12

3

4 5

1

2

34


Accessing left and right neighbors

• Can identify opposite corners of right and left neighbors– c.r = c.n.o

– c.l = c.n.n.o

c

c.v

c.o

c.l = c.n.n.oc.r = c.n.o

c.n c.p=c.n.n


Using adjacency table for T-mesh traversal

• Visit T-mesh (triangle-spanning tree)– Mark triangles as you visit

– Start with any corner c and call Visit(c)

– Visit(c) • mark c.t;

• IF NOT marked(c.r.t) THEN visit(c.r);

• IF NOT marked(c.l.t) THEN visit(c.l);

• Label vertices– Label vertices with consecutive integers

– Label(c.n.v); Label(c.n.n.v); Visit(c);

– Visit(c) • IF NOT labeled(c.v) THEN Label(c.v);

• mark c.t;

• IF NOT marked(c.r.t) THEN visit(c.r);

• IF NOT marked(c.l.t) THEN visit(c.l);


Summary

• Samples+incidence graph define triangles and corners (c.v)

• Attributes attached to corners (may be same for neighbors)

• T-mesh: oriented manifold incidence (consistent geometry?)

• Adjacency (c.o) supports T-mesh traversal (derived from c.v)

• Simple meshes, T=2V-4, H=0, S=1

Date post:	17-Dec-2015
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