3d surface parameterization olga sorkine, may 2005
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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
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