Barycentric CoordinatesBarycentric  Coordinatesonon 

SurfacesSurfaces

Yaron Lipman Thomas FunkhouserR if R tPrinceton UniversityRaif RustamovDrew University

Motivation

Barycentric coordinates good for:y ginterpolationshadingdeformationBezier surfacesparameterizationinterior distance image cloningshape retrievalfi i lfinite elements

Goal

Define barycentric coordinateson surfaces:

generalize planarintrinsicfast to compute

Definition on the planep

1v

2v 5v2v 5vp

verticesof comb affine =p

3v 4v

Challenges on surfacesg

b l h ll f ip belongs to convex hull of vertices

Challenges on surfacesg

Mobius, Alfeld et al, Cabral et al Spherical triangles

Ju et al Spherical convex

L t l S h i l llLanger et al Spherical all

Challenges on surfacesg

involves Cartesian coords

not intrinsic

Our approachpp

Our approachpp

Riemannian

Outline

IntroductionIntroductionRiemannian Center of MassConstructionConstructionComputationP iPropertiesResultsApplications

Usual center of mass

Euclidean distance

Riemannian center of mass

Geodesic distance

Karcher’s theorem

If th i t t “t f ” f h th th If the points are not “too far” from each other, the Riemannian center of mass is unique

has a unique minimumhas a unique minimum,which is the unique zero of

Construction

Construction

Construction

Construction

Computation Summaryp y

• Pick a kind of planar baryc coords• E.g. Mean Value Coords

• Compute the gradient polygon at each pointE l f l • Explicit formula exists

• Surface baryc coords == planar baryc coords wrt gradient polygon

Computation of gradientsp g

inverse exponential mapinverse exponential map

Schmidt et al. [2006]

Computation of gradientsp g

Schmidt et al. [2006]

Intuitiveness

I point towards v1. My length is equal to

geod. dist. from p to v1

Intuitiveness

I am at the I point towards v1.

My length is equal to geod. dist. from p to v1

“correct” distance and direction wrt p

Discrete settingg

Discrete settingg

Discrete settingg

Discrete settingg

Pentagon Need five instances of

“single source, all destinations”

Propertiesp

Defining properties g p pLagrangianPartition of unity Riemannian center of mass

Unique reconstruction from coordinatesDue to Karcher’s theoremDue to Karcher s theorem

Planar reproductionIf surface is plane, get planar coordinates backp , g p

Similarity invarianceSmoothness

Propertiesp

Edge linearityEdge linearity

Propertiesp

Isometry invariancey

isometry invariance + unique reconstruction = = isometry map can be reconstructedy p

Results

Darkest red point – vertex wrt which coords are pcomputedDark blue – small valueDark red – large valueEqually spaced isolines

Planar baryc coords = =Mean Value Coordinates

Effect of surface shapep

Varietyy

Varietyy

Effect of planar coordinatep

Mean Value Coordinates Maximum Entropy Coordinates

Timing (seconds)g ( )

Application: Interpolationpp p

Application: Decal mappingpp pp g

Local parameterizations – used for texturingLocal parameterizations used for texturingWe use the same idea as in image warping

Application: Decal mappingpp pp g

App: Correspondence Refinementpp p

Based on isometry reconstruction propertyCorrespondence is exact, if true isometry

App: Correspondence Refinementpp p

App: Correspondence Refinementpp p

Summaryy

Definition and construction of Definition and construction of

barycentric coordinates on surfaces:

properly generalize existing planar coordinates insensitive to isometric deformationsinsensitive to isometric deformationseasy to implement, and fast to compute

Future work

Better Karcher’s theoremBetter Karcher s theoremOther distances instead of geodesic inUniqueness of c.m. for larger polygons?Uniqueness of c.m. for larger polygons?

More empirical studies of various choices ofDistanceDistancePlanar barycentric coordinates

Further applications of surface coordinatesFurther applications of surface coordinates

Thank youy

SoftwareSoftwareSzymon Rusinkiewicz for Trimesh2Danil Kirsanov for Exact GeodesicDanil Kirsanov for Exact Geodesic

3D models Daniela GiorgiDaniela GiorgiAIM@SHAPE

Raison d'êtreRaison d êtreRemy of Ratatouille whose posing for [Joshi et al 2007] got me interested in barycentric coordinatesg y

Thank youy