Barycentric Coordinates · Mobius, Alfeld et al, Cabral et al Spherical triangles Ju et al...

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Barycentric Coordinates Barycentric Coordinates on on Surfaces Surfaces Yaron Lipman Thomas Funkhouser R ifR t Princeton University Raif Rus t amov Drew University

Transcript of Barycentric Coordinates · Mobius, Alfeld et al, Cabral et al Spherical triangles Ju et al...

Page 1: Barycentric Coordinates · Mobius, Alfeld et al, Cabral et al Spherical triangles Ju et al Spherical convex L t lLanger et al Sh i l llSpherical all. Challenges on surfaces involves

Barycentric CoordinatesBarycentric  Coordinatesonon 

SurfacesSurfaces

Yaron Lipman Thomas FunkhouserR if R tPrinceton UniversityRaif RustamovDrew University

Page 2: Barycentric Coordinates · Mobius, Alfeld et al, Cabral et al Spherical triangles Ju et al Spherical convex L t lLanger et al Sh i l llSpherical all. Challenges on surfaces involves

Motivation

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

Page 3: Barycentric Coordinates · Mobius, Alfeld et al, Cabral et al Spherical triangles Ju et al Spherical convex L t lLanger et al Sh i l llSpherical all. Challenges on surfaces involves

Goal

Define barycentric coordinateson surfaces:

generalize planarintrinsicfast to compute

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Definition on the planep

1v

2v 5v2v 5vp

verticesof comb affine =p

3v 4v

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Challenges on surfacesg

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

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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

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Challenges on surfacesg

involves Cartesian coords

not intrinsic

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Our approachpp

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Our approachpp

Riemannian

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Outline

IntroductionIntroductionRiemannian Center of MassConstructionConstructionComputationP iPropertiesResultsApplications

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Usual center of mass

Euclidean distance

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Riemannian center of mass

Geodesic distance

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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

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Construction

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Construction

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Construction

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Construction

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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

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Computation of gradientsp g

inverse exponential mapinverse exponential map

Schmidt et al. [2006]

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Computation of gradientsp g

Schmidt et al. [2006]

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Intuitiveness

I point towards v1. My length is equal to

geod. dist. from p to v1

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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

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Discrete settingg

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Discrete settingg

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Discrete settingg

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Discrete settingg

Pentagon Need five instances of

“single source, all destinations”

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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

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Propertiesp

Edge linearityEdge linearity

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Propertiesp

Isometry invariancey

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

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Results

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

Planar baryc coords = =Mean Value Coordinates

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Effect of surface shapep

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Varietyy

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Varietyy

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Effect of planar coordinatep

Mean Value Coordinates Maximum Entropy Coordinates

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Timing (seconds)g ( )

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Application: Interpolationpp p

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Application: Decal mappingpp pp g

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

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Application: Decal mappingpp pp g

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App: Correspondence Refinementpp p

Based on isometry reconstruction propertyCorrespondence is exact, if true isometry

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App: Correspondence Refinementpp p

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App: Correspondence Refinementpp p

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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

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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

Page 44: Barycentric Coordinates · Mobius, Alfeld et al, Cabral et al Spherical triangles Ju et al Spherical convex L t lLanger et al Sh i l llSpherical all. Challenges on surfaces involves

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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Thank youy


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