Polygonal Mesh – Data Structure and Processing Chiew-Lan Tai

What is a Mesh?

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

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

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)

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.

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

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

Euler Formula Tetrahedron Cube Octahedron V = 4 E = 6 F = 4 4 - 6 + 4 = 2 Cube V = 8 E = 12 F = 6 8 -12 + 6 = 2 Octahedron V = 6 E = 12 F = 8 6 -12 + 8 = 2 V = 8 E = 12 + 1 = 13 F = 6 + 1 = 7 8 - 13 + 7 = 2 V = 8 E = 12 F = 6 8 - 12 + 6 = 2

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 – 32 + 16 = 2 (1 - 1) - 0

Data Structure

Neighborhood Relations

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

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

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)

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

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

Half-Edge Data Structure

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); }

Progressive mesh

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)

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, ...)

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

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

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

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

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

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)

Application – Mesh compression 12964 faces 1000 faces

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

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

Property – Vertex Correspondence Mf 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 ’

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

Mesh Simplification

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

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)

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

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

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

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 …

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

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

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

Operator: Vertex-pair Collapse va vnew vb Enables closing of holes and tunnels – changes topology

Operator: Triangle Collapse va 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.

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

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

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

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

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

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

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

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

QEM Simplification

Mesh Fairing (Smoothing)

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

Mesh Fairing

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

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

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

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

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

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)

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

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

Comparisons

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