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Advanced Computer Graphics

Lecture 2: Modeling (1): Polygon Meshes

Bernhard JungTU-BAF, Summer 2007

Overview

Polygon Meshes SubdivisionPolygon Mesh Optimization

high-level: Level-of-detail, Progressive meshes low-level: Polygon strips & fans

Further readingA. Watt. 3D Computer Graphics. 3rd edition Addison-Wesley. 2000. Chapter 2.T. Akenine-Möller & E. Haines. Real-Time Rendering. AK Peters. 2002. Chapter 11,12 H. Hoppe. Progressive meshes. Proceedings ACM SIGGRAPH 1996, 99-108.

Computer Graphics Icon:Utah teapot

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Surfaces

Polygon MeshesClassical and ubiquitous in CG

Creating polygonal objects is straightforward Visually efficient algorithms exist to produce shaded images. A machine representation, converted from other user representations such as parametric/implicit surfaces, CSG/volume

Difficulties of polygonal mesh geometric accuracyshape manipulation

Surfaces

Polygon Meshes

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Surfaces - Polygon Meshes –

PolygonsA polygon is a many-sided planar figure composed of vertices and edges.

Vertices are represented by points (x,y).

Edges are represented as line segments which connect two points, (x1,y1) and (x2,y2).

P = { (xi , yi ) } i=1,n

E3

(x3,y3)

E2

(x2,y2)E1(x1,y1)

Surfaces - Polygon Meshes –Convex and Concave Polygons

Convex Polygon - For any two points P1, P2 inside the polygon, all points on the line segment which connects P1 and P2 are inside the polygon.

All points P = uP1 + (1-u)P2, u in [0,1] are inside the polygon provided that P1 and P2 are inside the polygon.

Concave Polygon - A polygon which is not convex.

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Surfaces - Polygon Meshes –Simple and non simple Polygons

Simple Polygons – Polygons whose edges do not cross.

Non simple Polygons – Polygons whose edges cross.Two different OpenGL implementations may render non simple polygons differently. OpenGL does not check if polygons are simple.

Trianglepolygon guaranteed to be convex and simple

SurfacesPolygon Meshes

Mesh Representations

Independent FacesVertex and Face TableAdjacency Lists(Winged Edge)

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Surfaces - Polygon Meshes – Mesh Representation

Independent FacesEach face lists vertex coordinates

Redundant verticesNo topology information

Surfaces - Polygon Meshes – Mesh Representation

Vertex and Face TablesEach face lists vertex references

Shared vertices, more compact representationStill no topology information

VRML: IndexedFaceSet

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Surfaces - Polygon Meshes – Mesh Representation

Adjacency ListsStore all vertex, edge, and face adjacencies

Efficient topology traversalExtra storage

Advanced Mesh Rep: Winged Edge

Stores only some adjacency relationships; can derive others on demandLittle extra storage (fixed records)

Surfaces

SubdivisionProperties

AccurateLocal supportAffine invariantArbitrary topologyGuaranteed continuityEfficient display

used e.g. in animated movies

Pixar: Geri’s Game

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Surfaces

Subdivision

How do you make a smooth curve?

Keep cutting the corners of the polyline until you end up with something that has no corners anymore, i.e., is smooth!

Surfaces

SubdivisionCoarse mesh & subdivision rule

Define smooth surface as limit of sequence of refinements

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Surfaces

Subdivisione.g. “loop” subdivision rule

refine each triangle into 4 triangles by splitting each edge and connecting new vertices

Surfaces

SubdivisionDifferent refinement rules

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Surfaces

Subdivision

All subdivision schemes have 2 steps:

1. Splitting or refinement step topological rule which introduces new vertices and modifies connectivity

2. Averaging or smoothing step geometric rule which computes the positions for new or all vertices by weighted averages

face split

vertexsplit

Surfaces

Subdivision

Quadrilateral subdivision

Computer Graphics Icon:Ivan Sutherland's Volkswagen,

Utah, late 1970ies

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Surfaces

SubdivisionSubdivision in 3DS Max, tessellate modifier

original

1 iteration

2 iterations

4 iterations

Surfaces

Subdivision

adaptive approximation: surface is subdivided only where topology of the surface requires more detail

selection of critical mesh areas (vertices) usually by hand

loop subdivision example

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Surfaces

Polygonal Models vs SubdivisionFrom 3DS Max Help:

You create a subdivision surface by applying a modifier to an object. Two kinds of subdivision surfaces are supported:

The HSDS modifier provides hierarchical subdivision surfaces [i.e. at multiple levels of detail]The MeshSmooth modifier provides smoothing.

Both these modifiers work best as finishing tools for models.…The HSDS modifier implements Hierarchical SubDivision Surfaces.

It is intended primarily as a finishing tool rather than as a modeling tool. For best results, perform most of your modeling using low-polygon methods, and then use HSDS to add detail and adaptively refine the model.

Surfaces

Parametric SurfacesGeneralization of parametric curvesbi-polynomial, e.g. bi-cubicparametric surfacespiecewise parametric surfacese.g. Bézier surface, NURBS-surface, …

C1-continuity requires aligning boundary curves and derivatives (difficult; a reason to prefer subdivision surfaces)

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Surfaces

SummaryPolygon meshes

games

Subdivision surfacesmovies

Parametric SurfacesCAD/CAM

increased accuracy

slower rendering

Polygon mesh optimization

Main disadvantage of polygon meshes:often high polygon count to synthesize an object for a high quality rendition

Quote from 1976

today: objects composed of > 1000000 polygons

It makes no sense to use 500 polygons in describing an object if it covers only 20 raster units on the display … For example, when we view the human body from a very large distance, we might need to present only specks for the eyes, or perhaps a block for the head, totally eliminating the eyes from consideration

(James H. Clark, 1976)

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Polygon mesh optimizationMesh Simplification

Reduce polygons to a level that is adequate for the quality required

Application: Level of detail approximation [Clark76, Funkhouser93]VRML: LOD node

problem: may "pop"

distancefrom viewer?

distancefrom viewer?

close far

10,000 2,000 1,000 500 250

Polygon mesh optimizationProgressive meshes [Hoppe, 96]

Idea: apply sequence of edge collapses

to build hierarchy of meshes:

edge collapseedge collapse

vvll vvrr

vvtt

vvss

vvssvvll vvrr’’

13,54613,546 500500 152152 150150

MM00MM11MM175175ecolecol00ecolecoliiecolecolnn--11

M=MM=Mnn^̂

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Polygon mesh optimizationProgressive meshes [Hoppe, 96]

Which edges to select?simple heuristic: edges connecting close polygons with similar orientationor more complex heuristics

edge collapse is invertible!inverse operation: vertex splitcan store required vertex information in coarser meshlossless

Applicationscontinuous level of detailprogressive transmissionselective refinement, e.g. viewer dependentmesh compression

edge collapseedge collapse

vvll vvrr

vvtt

vvss

vvssvvll vvrr’’

vertex splitvertex split

Polygon mesh optimizationcontinuous level of detail

http://gts.sourceforge.net/

A multi-resolution model obtained by simplification of the original model (in the top left corner). The number of triangles are respectively 96966, 13334, 3334 and 668.

Animation of a continuous Level Of Detail model of the horse. The initial number of vertices is 32 and is increased to 10,000.

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Triangle strips

Without strips: 8 triangles * 3 vertices = 24 verticesWith strips: use 1 vertex per triangle instead of 3 !

What we send to graphics hardware:Startup cost: v0, v1 then v2 (T0), v3 (T1), v4 (T2), v5 (T3), v6 (T4), v7 (T5), v8 (T6), v9 (T7).

9 vertices 100*9/24= 37.5% or 9/8=1.125 verts/tri

v0

v1

v2

T0

v3

T1

v4

T2

v5

T3

v6

T4

v7

T5

v8

T6

v9

T7

Triangle strips9 vertices instead of 24

100*9/24= 37.5% of data 9/8=1.125 vertices/tri

I.e., we can expect the geometry stage to run almost 3 times faster!

Definition of triangle strip makes orientation change between neighboring triangles in strip

Internally, counterclockwise order is kept consistent by traversing vertices 0-1-2, 1-3-2, 2-3-4, 3-5-4, …

v0

v1

v2

T0

v3

T1

v4

T2

v5

T3

v6

T4

v7

T5

v8

T6

v9

T7

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Triangle stripsOpenGL: glBegin(GL_TRIANGLE_STRIP); … glEnd()

Swaps in triangle stripsWhat can we do for this case?(T3 = v2 v4 v5 )

Implement a swap!Startup cost: v0, v1 then

v2 (T0)v3 (T1)v2,v4 (T2)v5 (T3), v6 (T4)

Degenerate triangle (0-area): v2, v3, v2

v0

v1

v2

v3

v4

v5 v6

T1

T2

T3

T0

T4

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Swaps…

cost of degenerate triangle:1 extra vertexstill cheaper than restarting triangle strip

this example: 8 sent vertices / 5 triangles = 1.6 vertices/triangle

restarting triangle strip costs more:4 vertices (2 triangles) +5 vertices (3 triangles)

(v0, v1, v2, v3, v2, v4, v5, v6)

(v0, v1, v2, v3, swap, v4, v5, v6) implemented by sending v0

v1

v2

v3

v4

v5 v6

T1

T2

T3

T0

T4

Swaps…also, hardware determines degenerate triangles efficiently and skips thesecan use swaps to connect non-connected triangles too

the idea is to avoid API call overheadhardware needs caches for this to be efficient

send these vertices: 0,1,2,3, 3,4, 4,5,6,710 vertices (sending as 2 strips: 8 vertices)if 3 and 4 are cached, then 8 vertices

v0

v1

v2

T0

v3

T1

v4

v5

v6

T2

v7

T3Generates

4 degeneratetriangles

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How to create triangle strips from a 3D model?

manually only doable for small models, and not fun…

NVIDIA Tri Strip Library http://developer.nvidia.com/object/nvtristrip_library.htmlor make your own program

need to know triangle’s adjacencies

1. Choose a starting triangle 2. Build 3 different triangle strips; one for each edge of the triangle3. Extend these triangle strips in the opposite direction4. Choose the longest of these 3 strips5. Repeat Step 1 until all triangles are included in a strip

SGI stripping algorithm

• greedy algorithm; linear time if appropriate data structures are used• choice of starting triangle:

• original algorithm: a triangle with smallest number of neighbors (at least 1) • in practice, arbitrary choice produces almost equally good results

Triangle fans

analysis - same as for strips:without fans: 6 triangles * 3 vertices = 18 verticeswith fans: use 1 vertex per triangle instead of 3 !

in practice: harder to find long fans than strips

smaller benefit than strips

easy to convert arbitrary polygons to triangle fans

can convert every triangle fan to a triangle strip (with many swaps) - but not vice versa

v0

v1 v2

v3

v4

v5

v6

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Vertex arrays / buffers

Idea: most efficient to transform a vertex only once (using strips or fans a vertex will be transformed once for each strip / fan the vertex is contained in)

Specifiy each vertex (position, normal, diffuse color, specular color, texture coordinates, etc.) once & send to bufferThen send (e.g., 16 bit) indices to identify vertices in bufferExample: make a buffer of 100 unique vertices

A triangle is sent as 3 indices into this buffer: 97, 5, 32This method mimimizes coyping from main memory to graphics card

OpenGL vertex arrays, DirectX vertex bufffersTriangle strips and vertex buffers are (often) fastest way to send triangles today

Summary

3D object representation propertiesAccuracyAccuracy vs. data volumeEase of editingOptimization potential

data volume, network transmission, rendering