3D Surface Parameterization3D Surface Parameterization

Olga Sorkine, May 2005Olga Sorkine, May 2005

Part OnePart One

Parameterization and PartitionParameterization and Partition

Some slides borrowed from Pierre Alliez and Craig Gotsman

What is a parameterization?What is a parameterization?

S S R R3 3 - given surface - given surface

D D R R22 - parameter domain - parameter domain

ss : D : D S 1-1 and onto S 1-1 and onto

),(

),(

),(

),(

vuz

vuy

vux

vus

Example – flattening the earthExample – flattening the earth

Isoparametric curves Isoparametric curves on the surfaceon the surface

One parameter fixed, one varies:One parameter fixed, one varies:

Family 1 (varying u): Family 1 (varying u): LLv0 v0 ((uu)) = = ss((uu, v, v00))

Family 2 (varying v): Family 2 (varying v): MMu0 u0 ((vv)) = = ss((vv00, , vv))

Analytic example:Analytic example:

Parameters: Parameters: u = x, v = yu = x, v = y

D = D = [[––1,1]1,1][[––1,1]1,1]. .

z = zz = z((x,yx,y)) = – = –((xx22+y+y22))

ss((x,yx,y)) = = ((x, y, zx, y, z((x,yx,y))))

-1

1

h

Another example:Another example:

Parameters: Parameters: , , hh

D = D = [0,[0,]][[––1,1]1,1]

xx((, h, h)) = cos = cos(())

yy((, h, h)) = h = h

zz((, h, h)) = sin = sin(())

Triangular MeshTriangular Mesh

• Standard Standard discretediscrete 3D surface representation 3D surface representation in Computer Graphics – piecewise linearin Computer Graphics – piecewise linear

• Mesh GeometryMesh Geometry: list of vertices (3D points of : list of vertices (3D points of the surface)the surface)

• Mesh Connectivity or TopologyMesh Connectivity or Topology: description : description of the facesof the faces

Triangular MeshTriangular Mesh

Triangular MeshTriangular Mesh

Mesh RepresentationMesh Representation

GeometryGeometry::vv1 1 – (x – (x11, y, y11, z, z11))

vv2 2 – (x– (x22, y, y22, z, z22))

vv3 3 – (x– (x33, y, y33, z, z33))......vvn n – (x– (xnn, y, ynn, z, znn))

TopologyTopology::Triangle listTriangle list

{v{v11, v, v22, v, v33}}......{v{vkk, v, vll, v, vmm}}

vv11

vv22vv33

vvnn

Mesh ParameterizationMesh Parameterization

• Uniquely defined by mapping mesh Uniquely defined by mapping mesh vertices to the parameter domain:vertices to the parameter domain:

UU : {v : {v11, …, v, …, vnn}} D D R R22

UU(v(vii) = () = (uuii, v, vii))

• No two edges cross in the plane (in No two edges cross in the plane (in DD))

Mesh parameterization Mesh parameterization mesh embedding mesh embedding

Mesh parameterizationMesh parameterization

Parameter domainD R2

Mesh surface

S R3

EmbeddingEmbedding

UU

ParameterizatioParameterizatio

nn ss

s = U -1

Mesh parameterizationMesh parameterization

Mesh parameterizationMesh parameterization

ss and and UU are piecewise-linear are piecewise-linear

Linear inside each mesh triangleLinear inside each mesh triangle

In 2D In 3DUU

ss

A mapping between two triangles is a unique affine

mapping

A B

C

P

Barycentric coordinatesBarycentric coordinates

, , , , , ,

, , , , , ,

, , denotes the (signed) area of the triangle

P B C P C A P A BP A B C

A B C A B C A B C

Mapping triangle to triangleMapping triangle to triangless

p1p2

p3

q1 q2

q3

3321

212

321

131

321

32

,,

,,

,,

,,

,,

,,)( q

ppp

ppq

ppp

ppq

ppp

pp pppps

• Only topological disks can be embeddedOnly topological disks can be embedded

• Other topologies must be “cut” or partitionedOther topologies must be “cut” or partitioned

Non-simple domainsNon-simple domains

CuttingCutting

Applications of parameterizationApplications of parameterization

• Texture mappingTexture mapping

• Surface resampling (remeshing)Surface resampling (remeshing)– Mesh compressionMesh compression– Multiresolution analysisMultiresolution analysis

Using parameterization, we can Using parameterization, we can operate on the 3D surface as if it operate on the 3D surface as if it were flatwere flat

Texture mappingTexture mapping

Texture mappingTexture mapping

Texture mappingTexture mapping

Remeshing Remeshing

Remeshing Remeshing

Remeshing Remeshing

parameterizationparameterization

resamplingresampling

RemeshingRemeshing

Remeshing examplesRemeshing examples

More remeshing examplesMore remeshing examples

Bad parameterization…Bad parameterization…

Distortion measuresDistortion measures

• Angle preservationAngle preservation

• Area preservationArea preservation

• StretchStretch

• etc...etc...

Bad parameterizationBad parameterization

Better…Better…

Distortion minimizationDistortion minimization

Kent et al ‘92 Floater 97 Sander et al ‘01

Texture map

Resampling problemsResampling problems

Cat mesh Distortingembedding

Resamplingon regular grid

Dealing with distortion and Dealing with distortion and non-disk topologynon-disk topology

ProblemsProblems: :

1) Parameterization of complex surfaces 1) Parameterization of complex surfaces introduces introduces distortion. distortion.

2) Only topological disk can be embedded.2) Only topological disk can be embedded.

SolutionSolution: : partitionpartition and/or and/or cutcut the mesh into the mesh into several patches, parameterize each patch several patches, parameterize each patch independently.independently.

PartitionPartition

Introducing seams (cuts)Introducing seams (cuts)

Introducing seams (cuts)Introducing seams (cuts)

Introducing seams (cuts)Introducing seams (cuts)

Introducing seams (cuts)Introducing seams (cuts)

Partition – problemsPartition – problems

• Discontinuity of parameterizationDiscontinuity of parameterization

• Visible artifacts in texture mappingVisible artifacts in texture mapping

• Require special treatmentRequire special treatment– Vertices along seams have several (u,v) Vertices along seams have several (u,v)

coordinatescoordinates– Problems in mip-mappingProblems in mip-mapping

Make seams short and hide them

Piecewise continuous Piecewise continuous parameterizationparameterization

SummarySummary

• “ “Good” parameterization = non-distortingGood” parameterization = non-distorting– Angles and area preservationAngles and area preservation– Continuous param. of complex surfaces cannot Continuous param. of complex surfaces cannot

avoid distortion.avoid distortion.

• “ “Good” partition/cut:Good” partition/cut:– Large patches, minimize seam lengthLarge patches, minimize seam length– Align seams with features (=hide them)Align seams with features (=hide them)

End of Part OneEnd of Part One