3D Surface Parameterization Olga Sorkine, May 2005

Part One Parameterization and Partition Some slides borrowed from Pierre Alliez and Craig Gotsman

What is a parameterization? S  R 3 - given surface S  R 3 - given surface D  R 2 - parameter domain D  R 2 - parameter domain s : D  S 1-1 and onto s : D  S 1-1 and onto

Example – flattening the earth

Isoparametric curves on the surface One parameter fixed, one varies: One parameter fixed, one varies: Family 1 (varying u): L v0 (u) = s(u, v 0 ) Family 1 (varying u): L v0 (u) = s(u, v 0 ) Family 2 (varying v): M u0 (v) = s(v 0, v) Family 2 (varying v): M u0 (v) = s(v 0, v)

Analytic example: Parameters: u = x, v = y D = [ – 1,1]  [ – 1,1]. z = z(x,y) = –(x 2 +y 2 ) s(x,y) = (x, y, z(x,y))

1   h Another example: Parameters: , h D = [0,  ]  [ – 1,1] x( , h) = cos(  ) y( , h) = h z( , h) = sin(  )

Triangular Mesh Standard discrete 3D surface representation in Computer Graphics – piecewise linearStandard discrete 3D surface representation in Computer Graphics – piecewise linear Mesh Geometry: list of vertices (3D points of the surface) Mesh Geometry: list of vertices (3D points of the surface) Mesh Connectivity or Topology: description of the faces Mesh Connectivity or Topology: description of the faces

Triangular Mesh

Mesh Representation Geometry: v 1 – (x 1, y 1, z 1 ) v 2 – (x 2, y 2, z 2 ) v 3 – (x 3, y 3, z 3 )... v n – (x n, y n, z n ) Topology: Triangle list {v 1, v 2, v 3 }... {v k, v l, v m } v1v1v1v1 v2v2v2v2 v3v3v3v3 vnvnvnvn

Mesh Parameterization Uniquely defined by mapping mesh vertices to the parameter domain: Uniquely defined by mapping mesh vertices to the parameter domain: U : {v 1, …, v n }  D  R 2 U(v i ) = (u i, v i ) No two edges cross in the plane (in D ) No two edges cross in the plane (in D ) Mesh parameterization  mesh embedding

Mesh parameterization Parameter domain D  R 2 Mesh surface S  R 3 Embedding U Parameterization s s = U -1

Mesh parameterization

s and U are piecewise-linear Linear inside each mesh triangle In 2DIn 3D U s A mapping between two triangles is a unique affine mapping

A B C P Barycentric coordinates

Mapping triangle to triangle s p1p1 p2p2 p3p3 q1q1 q2q2 q3q3

Only topological disks can be embedded Only topological disks can be embedded Other topologies must be “cut” or partitioned Other topologies must be “cut” or partitioned

Non-simple domains

Cutting

Applications of parameterization Texture mappingTexture mapping Surface resampling (remeshing)Surface resampling (remeshing) –Mesh compression –Multiresolution analysis Using parameterization, we can operate on the 3D surface as if it were flat

Texture mapping

Remeshing

Remeshing

Remeshing parameterization resampling

Remeshing

Remeshing examples

More remeshing examples

Bad parameterization…

Distortion measures Angle preservationAngle preservation Area preservationArea preservation StretchStretch etc...etc...

Bad parameterization

Better…

Distortion minimization Kent et al ‘92Floater 97Sander et al ‘01 Texture map

Resampling problems Cat meshDistorting embedding Resampling on regular grid

Dealing with distortion and non-disk topology Problems: 1) Parameterization of complex surfaces introduces distortion. 2) Only topological disk can be embedded. Solution: partition and/or cut the mesh into several patches, parameterize each patch independently.

Partition

Introducing seams (cuts)

Partition – problems Discontinuity of parameterization Discontinuity of parameterization Visible artifacts in texture mapping Visible artifacts in texture mapping Require special treatment Require special treatment –Vertices along seams have several (u,v) coordinates –Problems in mip-mapping Make seams short and hide them

Piecewise continuous parameterization

Summary “Good” parameterization = non-distorting “Good” parameterization = non-distorting –Angles and area preservation –Continuous param. of complex surfaces cannot avoid distortion. “Good” partition/cut: “Good” partition/cut: –Large patches, minimize seam length –Align seams with features (=hide them)

End of Part One