CSL 859: CSL 859: Advanced Advanced Computer Computer GraphicsGraphicsDept of Computer Sc. & Engg.Dept of Computer Sc. & Engg.

IIT DelhiIIT Delhi

MeshMesh List of trianglesList of triangles

Each is a triplet of VerticesEach is a triplet of Vertices Each is an array of attributesEach is an array of attributes

Euler’s relation:Euler’s relation: V + F -2 = EV + F -2 = E

Adjacency listAdjacency list List of verticesList of vertices List of “pointers”List of “pointers” A vertex referred ~6 times (closed model)A vertex referred ~6 times (closed model)

Vertex & TopologyVertex & Topology

A

G

F

B

CD

E

..A..

..B..

..C..

..D..

..E..

..F..

Vertex AttributesVertex Attributes

TopologyTopology0 1 51 3 51 2 35 3 46 5 40 5 6What order What order

should we should we choose?choose?Post vertex shader cachePost vertex shader cache

Triangle StripTriangle Strip

A

G

F

B

CD

E

..A..

..B..

..C..

..D..

..E..

..F..

Vertex AttributesVertex Attributes

TopologyTopology

ABFDEAFGEBCD

Can we turn any Can we turn any given model into a given model into a single strip?single strip?

Would that eliminate Would that eliminate the need for storing the need for storing topology?topology?

Post Shader CachePost Shader Cache Perfect Triangle Strip of n triangles =>Perfect Triangle Strip of n triangles =>

1 new vertex per triangle1 new vertex per triangle n-2 cache missesn-2 cache misses

Can we do better?Can we do better? Euler’s law: V + F = E + 2Euler’s law: V + F = E + 2 V ~ 0.5F (E = 1.5F for closed model)V ~ 0.5F (E = 1.5F for closed model)

Yes (Ideal: 0.5verts/triangle)Yes (Ideal: 0.5verts/triangle) [Chhugani & Kumar, I3D 2007][Chhugani & Kumar, I3D 2007]

Vertex ArraysVertex Arrays Array of structuresArray of structures

Better triangle localityBetter triangle locality

Structure of arraysStructure of arrays Better shader localityBetter shader locality

V0x y z u v a b c

x0 x1 x2 x3

V1x y z u v a b c

y0 y1 y2 y3

Half-Edge Data StructureHalf-Edge Data StructureVertex ListVertex List Face ListFace List Edge ListEdge List

..V0..

..V1..

..V2..

..V3....

E0 E2 E1E5 E4 E3

.

.

V0 V1, F0, E1 E2, E5 ..

V1 V0, F1, E3 E4, E0 ..

E0

E3

E4

E2

E1 V0

V1

F0F1

E5

Winged-Edge Data Winged-Edge Data StructureStructure

Vertex ListVertex List Face ListFace List Edge ListEdge List..V0.., E0..V1.., E0..V2.., E1..V3.., E4

.

.

E0E4..

V0 V1, F0 F1, E1 E2, E3 E4 ..

E0

E3

E4E2

E1V0

V1

F0

F1Edge is oriented based on one of its faces: left or right

Mesh SimplificationMesh Simplification

courtesy H. HoppeIdeally, the change is incremental

Nominal FrameworkNominal Framework Pre-processingPre-processing

Create and store levels in a data-Create and store levels in a data-structurestructure

Rendering timeRendering time Decide the appropriate detailDecide the appropriate detail

Coarse or fine-grainedCoarse or fine-grained Output count or approximation errorOutput count or approximation error

Traverse DS to generate that detailTraverse DS to generate that detail RenderRender

Competing GoalsCompeting Goals Computational EfficiencyComputational Efficiency

Some pre-processingSome pre-processing Storage EfficiencyStorage Efficiency QualityQuality

Error metricsError metrics screen space screen space vsvs object space object space Geometry, attributes, appearanceGeometry, attributes, appearance

Global Global vsvs local optimization local optimization local may be fasterlocal may be faster global may generate closer approximationsglobal may generate closer approximations

AttributesAttributes ColorColor Normal VectorsNormal Vectors Texture CoordinatesTexture Coordinates CurvatureCurvature Material PropertiesMaterial Properties

Topology ConsiderationsTopology Considerations Manifold Manifold vsvs non-manifold non-manifold Topology preservationTopology preservation

important in some areasimportant in some areas if topology is changed, then we can:if topology is changed, then we can:

close holes in objectsclose holes in objects join disconnected componentsjoin disconnected components

Shape and attribute Shape and attribute appearanceappearance more more important than topologyimportant than topology

Possible Algorithm?Possible Algorithm? Subdivide space into cellsSubdivide space into cells Choose a vertex to represent each Choose a vertex to represent each

cellcell For each pair of cellsFor each pair of cells

If two vertices in two cells have an edgeIf two vertices in two cells have an edge Connect the representative vertices Connect the representative vertices

Borel, Rossignac [1993]Borel, Rossignac [1993]

Reduce Geometry CountReduce Geometry Count Remove a vertexRemove a vertex

That leave a holeThat leave a hole Retriangulate the holeRetriangulate the hole

Schroeder et al., 1992Schroeder et al., 1992

Simple OperationSimple Operation Collapse edges, one at a timeCollapse edges, one at a time

vvll vvrr

vvtt

vvss

ecol(vecol(vs s ,v,vt t , , vvss ))’’

vvssvvll vvrr’’

SimplificationSimplification

13,54613,546 500500 152152 150150

MM00MM11MM175175

ecolecol00ecolecoliiecolecoln-1n-1

M=MM=Mnn^̂

courtesy H. Hoppe

Inverse?Inverse?

vvll vvrr

v’v’tt

v’v’ssvvss

vvll vvrr

vspl(vvspl(vs s ,v,vl l ,v,vr r , , vvss ,,vvtt ,…),…)’’ ’’parametersparameters

ReconstructionReconstruction

150150

MM00 MM11

vsplvspl00

152152

MM175175

500500

… … vsplvsplii … …

13,54613,546

vsplvspln-1n-1

MMnn=M=M̂̂

Progressive Mesh (PM)Progressive Mesh (PM) representation representation

MMnn=M=M̂̂

courtesy H. Hoppe

Continuous LODContinuous LODFrom PM, extract From PM, extract MMii of any size of any size

MM00 vsplvspl00 vsplvspl11 vsplvspli-1i-1 vsplvspln-1n-1

MMii

3,478 faces?3,478 faces?3,4783,478

MM00 MMnn=M=M̂̂MMii

courtesy H. Hoppe

Vertex CorrespondenceVertex CorrespondenceMMnn MM00

MMccMMff

vv11vv22vv33vv44vv55vv66vv77vv88

vv11vv22vv33

MMf-1f-1

vv11vv22vv33vv44vv55vv66vv77

ecolecol

MMf-2f-2

vv11vv22vv33vv44vv55vv66

ecolecol ecolecol

courtesy H. Hoppe

Space OverheadSpace Overhead

Attrib deltas:Attrib deltas:vvtt - v - vss

vvs s - v- vss

……

’’’’vvll vvrr

v’v’tt

v’v’ss

vspl(vvspl(vs s ,v,vl l ,v,vr r , , vvss ,,vvtt ,…),…)’’ ’’

vvssvvll vvrr

Topology:Topology: Index of vIndex of vss Pair of index offsetsPair of index offsets

Selective RefinementSelective Refinement

MM00 vsplvspl00 vsplvspl11 vsplvspli-1i-1 vsplvspln-1n-1

courtesy H. HoppeWhat’s the problem?What’s the problem?

Parent-Child Parent-Child CorrespondenceCorrespondence

vvss

vvttvvuu

vsplitvsplit

vv22

Vertex HierarchyVertex Hierarchyvsplvspl00MM00 vsplvspl11 vsplvspl22 vsplvspl33 vsplvspl44 vsplvspl55

vv11 vv33MM00

vv1010 vv1111

vsplvspl33

vv11 vv22

vv44 vv55

vsplvspl00

vv88 vv99

vsplvspl22

vv33

vv66 vv77

vsplvspl11

vv55

vv1212 vv1313

vsplvspl44

vv1010

vsplvspl55

vv1414 vv1515

vv66

PM:PM:

MMnn

[Xia & Varshney 96]

MM00

vsplvspl22

vv1111

vv11 vv22

vv44 vv88 vv99

vv33

vv77

vv55

vv1212 vv1313

vv1010

vv1414 vv1515

vv66

vv22

Selective RefinementSelective Refinementvsplvspl00MM00 vsplvspl11 vsplvspl33 vsplvspl44 vsplvspl55

vv11 vv33MM00

vv1010 vv1111

vsplvspl33

vv11 vv22

vv44 vv55

vsplvspl00

vv66 vv77

vsplvspl11

vv55

vv1212 vv1313

vsplvspl44

vv1010

selectively refined meshselectively refined mesh

vv88 vv99

vsplvspl22

vv33

vv88 vv99

vsplvspl22

vv33

Restrictions?Restrictions?

DependenciesDependencies

vsplitvsplit

ecolecol vsplitvsplit legal if legal if vvss, , vvll , and , and vvrr present present ecolecol legal if local neighbors present legal if local neighbors present

vvll vvrr

v’v’tt

v’v’ss

vvssvvll vvrr

[Xia & Varshney 96]

ConsistencyConsistencyvsplitvsplit

ecolecol

ffn1n1

ffn0n0

vvss

ffn3n3

ffn2n2

vsplitvsplit legal if: legal if:vvss is active is activeffn0n0,f,fn1n1,f,fn2n2,f,fn3n3 are active are active

ecolecol legal if: legal if:v’v’ss,v’,v’tt are active are activeffn0n0,f,fn1n1,f,fn2n2,f,fn3n3 are adjacent are adjacent

ffn1n1

ffn0n0v’v’tt

ffn3n3

ffn2n2

v’v’uu

[Hoppe 97]

Rendering AlgorithmRendering Algorithm Start with the active Start with the active frontfront in previous in previous

frame.frame. Exploits frame coherenceExploits frame coherence

For each vertex, decided For each vertex, decided refinerefine or or coarsencoarsen If legal, perform operation, otherwise:If legal, perform operation, otherwise:

Make vertex split legal by generating necessary Make vertex split legal by generating necessary verticesvertices

Leave Edge collapses aloneLeave Edge collapses alone Amortize?Amortize?

At each frame apply only a fraction of the At each frame apply only a fraction of the eligible opseligible ops

previous meshprevious mesh

vv55vv1010 vv1111 vv44 vv88 vv99

vv77vv1212 vv1313

vv11 vv22 vv33

Rendering AlgorithmRendering Algorithm

MM00

vv66

vv1414 vv1515

vv1212 vv1313vv1212

vv1010 vv1111vv1010 vv1111 vv44vv44

vv66

vv1414 vv1515

vv66

vv1414 vv1515

vv88 vv99

vv33

vv1515

vv77vv77

vv88vv88 vv99vv99

new meshnew mesh

dependencydependency

vv1111vv1010 vv44

vv1414 vv1515

vv77

vv88 vv99

LOD AlgorithmsLOD Algorithms Simplification operatorSimplification operator Where should it be appliedWhere should it be applied

No optimizationNo optimization e.g., uniform grid cellse.g., uniform grid cells

Greedy optimizationGreedy optimization Sort edge collapses by errorSort edge collapses by error

Re-insert modified edges after each stepRe-insert modified edges after each step Lazy optimizationLazy optimization

Re-insert a modified edge only when in frontRe-insert a modified edge only when in front Local Local vsvs Global optimization Global optimization

Error MetricError Metric Preserve Preserve appearanceappearance::

Geometric shapeGeometric shape Scalar fieldsScalar fields (e.g. color) (e.g. color) Discontinuity curvesDiscontinuity curves

E e e dA e dLshape scalars disc ( ) ( )face areas disc. edges

pointspoints pointspoints

courtesy H. Hoppe

Measuring ErrorMeasuring Error Geometric errorGeometric error

DistanceDistance between the original and between the original and simplified surface?simplified surface?

Volume Volume between the surfaces?between the surfaces? Visual errorVisual error

Color, normal, & texture distortionColor, normal, & texture distortion Silhouettes, background illuminationSilhouettes, background illumination SemanticsSemantics Many othersMany others

Measuring Geometric Measuring Geometric ErrorError

Surface-surfaceSurface-surface Hausdorff distanceHausdorff distance

Vertex-surface vs Vertex-surface vs Vertex-planeVertex-plane Vertex-vertexVertex-vertex Average distance or Max distanceAverage distance or Max distance

Quadric ErrorQuadric Error [Garland & Heckbert] 1998[Garland & Heckbert] 1998 Measure error by deviation from shapeMeasure error by deviation from shape

Vertices are at intersection of planesVertices are at intersection of planes

Do not collapse edgesDo not collapse edges Merge verticesMerge vertices

v’v’tt

v’v’ss

Quadric Error MetricQuadric Error Metric Plane equation for a face:Plane equation for a face:

0 :p DCzByAx

1zyx

DCBAT vp

Distance to vertex Distance to vertex vv : :

Distance to vertex Distance to vertex vv : :

)(

2 )()(vplanesp

T vpv

)(

))((vplanesp

TT vppv

)(

)(vplanesp

TT vppv vppvvplanesp

TT

)(

Quadric Error (cont’d)Quadric Error (cont’d)

Sum over all planes intersecting at vSum over all planes intersecting at v Call it Call it Q, the quadric errorQ, the quadric error

vvQQvvvv TT ))((

2

2

2

2

DCDBDADCDCBCACBDBCBABADACABA

ppT

Maintain quadric Q for every vertexMaintain quadric Q for every vertex Assign Edge QuadricsAssign Edge Quadrics

Q1 Q2

v2v1

Sort edges based on quadric errorSort edges based on quadric error Need position of resulting vertexNeed position of resulting vertex

One of original vertices or mid-point?One of original vertices or mid-point?

21 QQQ

Quadrics Based Quadrics Based SimplificationSimplification

vvQQvvvv TT ))((

Optimal Vertex Optimal Vertex PlacementPlacement

Minimize Q to calculate optimal Minimize Q to calculate optimal coordinates for placing new vertexcoordinates for placing new vertex

1000

1000

1

34332313

24232212

14131211

qqqq

qqqq

qqqq

V

Boundary PreservationBoundary Preservation Label boundariesLabel boundaries Form boundary plane perpendicular Form boundary plane perpendicular

to faceto face Convert planes into quadricsConvert planes into quadrics Weighted sum of quadricsWeighted sum of quadrics

Scale border plane quadric higherScale border plane quadric higher

Preventing Mesh Preventing Mesh InversionInversion

Preventing foldovers:Preventing foldovers:

Adjacent face normals should not flipAdjacent face normals should not flip Disallow foldover, or simply weight Disallow foldover, or simply weight

heavilyheavily

10

54

6

13

2

9 A

78

10

54

63

28

9 A

merge

Quadric Error MetricQuadric Error Metric Pros:Pros:

Reasonably fastReasonably fast Good fidelity even for drastic reductionGood fidelity even for drastic reduction Robust -- handles non-manifold surfacesRobust -- handles non-manifold surfaces Aggregation -- can merge objectsAggregation -- can merge objects

Cons:Cons: Introduces non-manifold surfacesIntroduces non-manifold surfaces User controls virtual-edge collapse thresholdUser controls virtual-edge collapse threshold

Increased running time due to search for potential pairsIncreased running time due to search for potential pairs Correct value varies with model densityCorrect value varies with model density

Needs extension to handle color (7x7 matrices)Needs extension to handle color (7x7 matrices)

Result – Bunny ModelResult – Bunny Model

69,451 triangles

1,000 triangles

100 triangles1.4% of original

size0.14% of original size

courtesy Garland, Heckbert

Result – Terrain ModelResult – Terrain Model

199,114 faces

999 faces (46 secs)

courtesy Garland, Heckbert

Method ComparisonMethod Comparison

Original model

Uniform Vertex Clustering

Edge Contractions

Pair Contractions

(4,204 faces)

(262 faces)

(250 faces)

(250 faces)courtesy Garland, Heckbert

Image-Driven Image-Driven SimplificationSimplification

[Lindstrom & Turk 2000][Lindstrom & Turk 2000] Compare simplifications Compare simplifications

to original to original via imagesvia images Collapse edgeCollapse edge Render from many viewsRender from many views Evaluate difference (Evaluate difference (need need

image metricimage metric)) ““Unrender” and retryUnrender” and retry Pick cheapest and applyPick cheapest and apply

Image-Driven Image-Driven SimplificationSimplification

Pros: Pros: Preserves appearancePreserves appearance Does not need to trade off geometric Does not need to trade off geometric

error against attribute errorerror against attribute error Shading artifacts accounted forShading artifacts accounted for Side effects:Side effects:

Drastically simplifies invisible regions!Drastically simplifies invisible regions!

Image-Driven Image-Driven SimplificationSimplification

Cons:Cons: VeryVery slow slow Still many examples that breaks it:Still many examples that breaks it:

Hard to know how many images is enoughHard to know how many images is enough Hard to know how to evaluate images (RMS Hard to know how to evaluate images (RMS

vs vs Bolin-Meyer)Bolin-Meyer)

Screen Space ErrorScreen Space Error Depends on location and orientation Depends on location and orientation

of error vectorof error vector We don’t even store error vectorsWe don’t even store error vectors

Screen Space ErrorScreen Space Error

θ

y/z

y,zy+ε sin θz+ε cos θ ε

εs

y+ε sin θz+ε cos θ y/zεs =

Eye

ImagePlane

Screen Space ErrorScreen Space Error

)2

tan(2 22

z

drzdp

r

LOD

eye

viewingplane

p

d

Appearance PreservationAppearance Preservation

Preserve three appearance Preserve three appearance attributes:attributes: Surface PositionSurface Position Surface CurvatureSurface Curvature Material ColorMaterial Color

Each may require different samplingEach may require different sampling

Normals Normals UndersampledUndersampled

13,433 triangles 1,749 triangles10 pixels of surface deviation

Normals Properly Normals Properly SampledSampled

13,433 triangles 1,749 triangles, 10 pixels of deviation

v1, c1, n1

v = vertex coordinate = (x,y,z)c = color = (r,g,b)n = normal = (nx,ny,nz)

v2, c2, n2

v3, c3, n3

RecallRecall

Filters surface Filters surface position, colors, position, colors, and normalsand normals

Must filter all Must filter all three three equallyequally

texture map

normal map

v1, t1

v2, t2

v3, t3

v = vertex coordinate = (x,y,z)t = texture coordinate = (u,v)c = color = (r,g,b)n = normal vector = (nx,ny,nz)

c1

c2c3

n1

n2n3

Decoupled Decoupled RepresentationRepresentation

Decoupled ApproachDecoupled Approach

Simplification filters surface Simplification filters surface position and texture coordinatesposition and texture coordinates

Color and normal attributes filtered Color and normal attributes filtered per-pixel (mip-mapping, etc.)per-pixel (mip-mapping, etc.)

Sample Normal MapSample Normal Map

polygonal surface patch normal map

xx

Texture Deviation MetricTexture Deviation Metricmesh Mmesh Mii mesh Mmesh Mi+1i+1

2D texture domain2D texture domain

(i+1)(i+1)stst edge collapse edge collapse

XXii XXi+1i+1

ei,i+1(x) = || xi+1 - xi ||

Ei,i+1 = max ei,i+1(x) xP

P

New Texture CoordinatesNew Texture Coordinates

Invalid texnew Valid choices lie in convex kernel

Texture Deviation ErrorTexture Deviation Error

X

||))((||)( 10 XTTXXE ii

Texture Space

APS Levels-of-detailAPS Levels-of-detail

courtesy J. Cohen

7,809 tris

3,905 tris

1,951 tris

975 tris

488 tris

TerrainsTerrains Uniform array of height valuesUniform array of height values

May encode in raster formats (DEM, May encode in raster formats (DEM, GeoTIFF)GeoTIFF)

Store Z instead of colorStore Z instead of color Easy to interpolate to find elevationsEasy to interpolate to find elevations Easy view cullingEasy view culling There also are TINsThere also are TINs

Triangulated Irregular NetworksTriangulated Irregular Networks

TINsTINs

Fewer polygons neededFewer polygons needed Adaptively sampledAdaptively sampled Precisely model maxima, minima, Precisely model maxima, minima,

ridges, valleys, overhangs, cavesridges, valleys, overhangs, caves

LOD Hierarchy LOD Hierarchy StructuresStructures

QuadTree Hierarchy

BinTree Hierarchy

QuadtreesQuadtrees Each quad is actually two trianglesEach quad is actually two triangles Produces cracks and T-junctionsProduces cracks and T-junctions But simpleBut simple

BintreesBintrees

Binary Triangle tree Binary Triangle tree Right triangular irregular networks Right triangular irregular networks

(RTIN)(RTIN) longest edge bisectionlongest edge bisection

Easier to avoid cracks and T-Easier to avoid cracks and T-junctionsjunctions

Neighbor must be at adjacent levelsNeighbor must be at adjacent levels

Cracks and T-JunctionsCracks and T-Junctions

Avoid cracks:Avoid cracks: Convert cracks into T-junctionsConvert cracks into T-junctions Fill cracks with extra trianglesFill cracks with extra triangles

Avoid T-junctions:Avoid T-junctions: Continue to simplify ...Continue to simplify ...

Bin-tree and T-junctionBin-tree and T-junction

Terrain as a PyramidTerrain as a Pyramid

Coarsest Level

Finest Level

•Terrain as mipmap pyramid

•LOD using nested grids

[Losasso & Hoppe 2005]

Geometry Clip MapsGeometry Clip Maps Transition regionTransition region

Insert “0 area” trianglesInsert “0 area” triangles Damp refined vertices as they get close Damp refined vertices as they get close

to the coarser levelto the coarser level

Geometry Clip MapsGeometry Clip Maps Transition regionTransition region

Insert 0 area trianglesInsert 0 area triangles Damp refined vertices as they get close Damp refined vertices as they get close

to the coarser levelto the coarser level

Texture Mapping PMTexture Mapping PM [Sander et al, 2001][Sander et al, 2001]