Harmonc Coordnates for Character Artculaton PIXAR Pushkar Josh Mark Meyer Tony DeRose Bran Green Tom Sanock
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
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
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
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
Mean Value Coordnate Feld Desred Coordnate Feld + -
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
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
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
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
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
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
Results Cage Vertces 325 112 39 27 Object Vertces 9775 8019 269 136 Grd resoluton 5 5 4 5 Solve tme 57.4 17.6 5.85 0.83 Pose tme 0.111 0.026 0.0001 0.0007 Soluton sze (MB) 9.2 3.7 0.32 0.048 Error: < 0.005 Total footprnt: < 90MB
Future Work: Compute the harmonc coordnates for each cage vertex ndependently and n parallel Better solvers (currently usng MultGrd) Octrees Localze re-solves