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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
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
Outline
IntroductionIntroductionRiemannian Center of MassConstructionConstructionComputationP iPropertiesResultsApplications
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
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
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
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
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
Application: Decal mappingpp pp g
Local parameterizations – used for texturingLocal parameterizations used for texturingWe use the same idea as in image warping
App: Correspondence Refinementpp p
Based on isometry reconstruction propertyCorrespondence is exact, if true isometry
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
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