Barycentric Coordinates Interpolation Barycentric given data at - PDF document

Barycentric Coordinates Interpolation Barycentric  given data at sites, interpolate Coordinates smoothly and intuitively “in between”  easy over simplices: linear  easy over simplices: linear  more general shapes needed  morphing, shape deformation, attribute interpolation, physical modeling, and on and on and on CS 176 Winter 2011 CS 176 Winter 2011 1 2 Basic Principles Basic Setup: Cage Data on boundary; extend  as affine combination: if then  desirables  constant precision:  linear precision:  from now on, we’ll focus on  convex: many choices; concave: few… Weber/Ben-Chen/Gotsman’s method CS 176 Winter 2011 CS 176 Winter 2011 3 4 Planar Case Properties Treat everything in complex plane Complex barycentric interpolation  tools from complex analysis…  preserve similarities  given source and target polygon  distinction with real coeffs?  preserve affine transformation iff ffi f i iff  not actually desirable… why?  desirables: CS 176 Winter 2011 CS 176 Winter 2011 5 6

Real vs. Complex Trade-off Must give up something…  not interpolating anymore How to find such functions?  study continuous setting  Affine transform not quite pleasing… CS 176 Winter 2011 CS 176 Winter 2011 7 8 Not An Interpolation Cauchy Formula Visualize one coordinate function Holomorphic functions  Cauchy kernel:  integral version of mean value theorem  recover value from average of boundary  Cauchy coordinates: CS 176 Winter 2011 CS 176 Winter 2011 9 10 Resulting Formulas Only need data on boundary!  apply to polygon  define f linearly along edge  grind out integrals… CS 176 Winter 2011 CS 176 Winter 2011 11 12

Examples Properties Best holomorphic function?  it doesn’t interpolate; is it “best”?  closest to given boundary data  stick with C j but use “virtual” poly. ti k ith C b t “ i t l” l  minimize functional to find best poly. CS 176 Winter 2011 CS 176 Winter 2011 13 14 Szegö Coordinates Solution Optimize “fit” Pseudo inverse  size is number of vertices (small)  need C j on boundary…  define through limit  letting H be sampling operator:  do point collocation (sample boundary) LSQ problem matrix with sampled C j as columns CS 176 Winter 2011 CS 176 Winter 2011 15 16 Example Visualization CS 176 Winter 2011 CS 176 Winter 2011 17 18

Comparison Another Comparison Cauchy-Green vs Szego CS 176 Winter 2011 CS 176 Winter 2011 19 20 Visualization Point-to-Point Simplify UI  just specify landmarks  underconstrain (typically)  add fairness constraint dd CS 176 Winter 2011 CS 176 Winter 2011 21 22 P2P Coordinates Example Joint minimization  point collocation… CS 176 Winter 2011 CS 176 Winter 2011 23 24

Visualization Example CS 176 Winter 2011 CS 176 Winter 2011 25 26 Video CS 176 Winter 2011 27