Harmonic Coordinates for Character Articulation PIXAR

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1 Harmonc Coordnates for Character Artculaton PIXAR Pushkar Josh Mark Meyer Tony DeRose Bran Green Tom Sanock

2 We have a complex source mesh nsde of a smpler cage mesh We want vertex deformatons appled to the cage to be appled approprately to the source mesh

3 Mean Value Coordnates Ju, Schaefer, Warren, 2005 p = g( p) C g p ( p) C New object vertces Mean value coordnate Weghtng functons Deformed cage vertces

4 g To compute for each nteror pont p: ( p) Consder each pont x on the boundary Multply f ( x) by the recprocal dstance from x to p Average over all x From the Ju paper: g( p) x = wxv (, ) f( xds ) x wxvds (, ) p p where wxp (, ) = x 1 p and S p s the unt sphere centered at p

5 g To compute for each nteror pont p: ( p) Consder each pont x on the boundary Multply f ( x) by the recprocal dstance from x to p Average over all x Good thngs: Topologcal flexblty n desgnng the cage (any closed tr-mesh) Deformatons are smooth Functons are lnear, so no poppng p x Bad thngs: Does not respect the vsblty of x from p If a cage vertex has a negatve weght assocated wth t, then the object vertex and cage vertex wll move n opposte drectons

6 Mean Value Coordnate Feld Desred Coordnate Feld + -

7 Instead, let s average over all Brownan paths leavng p Ths wll consder the vsblty of x from p Essental for any concave mesh Interestngly enough Ths s the same as solvng Laplace s equaton Porf, Stone 1978 & Bass 1995 p x Δ h( p) = 0 p Interor( C) Solve for every cage vertex p Let us frst approach thngs n two dmensons

8 Boundary condtons: Let p denote a pont on the boundary C of C Then: h( p) = φ ( p), for all p C where φ ( p) s the pecewse lnear functon such that φ ( C ) = δ, j j

9 Propertes: Interpolaton ( ) = δ, h C j j Smoothness The functons h p are smooth n the nteror of the cage Non-negatvty ( ) 0 ( ) h p for all p C Interor localty Interor localty holds f we have the non-negatvty property and no nteror extrema Lnear reproducton Gven an arbtrary f(p), the coordnate functons can be used to defne: [ ]( ) ( ) ( ) H f p = h p f C Ths s the no poppng condton

10 Propertes: Affne nvarance h ( p ) = 1 for all p C Generalzaton of barycentrc coordnates ( ) h p s the barycentrc coordnate of p wth respect to C

11 Interpolaton: ( ) = φ ( C ) = δ, h C j j j Smoothness: Away from the boundary, harmonc coordnates are solutons, so they are smooth n the cage nteror On the boundary, they are only as smooth as the boundary condtons Non-negatvty: Harmonc functons acheve extreme at ther boundares Boundary values are restrcted to [0,1] So nteror values are restrcted to [0,1] Interor localty: Harmonc functons possess no nteror extrema

12 Lnear reproducton: Ths holds for everywhere on the boundary of C, by defnton: [ ]( ) = ( ) ( ) = φ ( ) ( ) H f p h p f C p f C Snce f(p) s lnear, second dervatves vansh, e: 2 = f( p) 0 and f(p) satsfes Laplace s equaton on the nteror of C Snce H[f](p) s a lnear combnaton of harmonc functons, t also satsfes Laplace s equaton Use proof by nducton to generalze to any n-dmenson

13 Results Cage Vertces Object Vertces Grd resoluton Solve tme Pose tme Soluton sze (MB) Error: < Total footprnt: < 90MB

14 Future Work: Compute the harmonc coordnates for each cage vertex ndependently and n parallel Better solvers (currently usng MultGrd) Octrees Localze re-solves

Barycentric Coordinates. From: Mean Value Coordinates for Closed Triangular Meshes by Ju et al.

Barycentric Coordinates. From: Mean Value Coordinates for Closed Triangular Meshes by Ju et al. Barycentrc Coordnates From: Mean Value Coordnates for Closed Trangular Meshes by Ju et al. Motvaton Data nterpolaton from the vertces of a boundary polygon to ts nteror Boundary value problems Shadng Space