Parameterization locally looks like Euclidian space of collection - PDF document

2-Manifold What makes for a smooth manifold? Parameterization  locally looks like Euclidian space of  collection of charts Meshes Meshes  mutually compatible ll ibl on their overlaps  form an atlas Parameterizations are key CS 176 Winter 2011 CS 176 Winter 2011 1 2 Parameterizations Task Find map from mesh to R 2 What is a parameterization?  function from some region  ⊂ R 2  identify region on mesh and find to the embedded surface M ⊂ R 3 map to R 2 n et al. 2002 Image from Desbrun  what’s different? Image from Sander et al. 2002  existence of solutions when mapping to convex region CS 176 Winter 2011 CS 176 Winter 2011 3 4 First Attempt Parameterizations Applications Tutte, 1963  texture mapping  make planar vertex the barycenter of its neighbors  piecewise linear interpolation of interpolation of  fix boundary  fix boundary attributes across mesh  convex!  resampling original centroid mapping  simulation  needs surface as function CS 176 Winter 2011 CS 176 Winter 2011 5 6

Conventions Computing it How do we denote things? Solution of linear system  we are looking for 2 scalar valued  input: functions defined over the surface  boundary: convex K-gon  edge weights: d h  we will quietly ignore the dimensionality of u and x vectors  though it will be obvious from context CS 176 Winter 2011 CS 176 Winter 2011 7 8 Existence Setting the Weights Solvable? How to choose “barycenter”  must show:  Tutte, “uniform”  i.e., no redundancy in equations  minimizes square lengths (springs)  what solver? h t l ?  system is large, (non-)symmetric  iterative  Jacobi, Gauss-Seidel  CG, bi-CG CS 176 Winter 2011 CS 176 Winter 2011 9 10 Setting the Weights Setting the Weights How to choose the “barycenter” Floater weights  Floater, “shape preserving”  barycentric coords?? R 3 R 2  geodesic polar map Scaled angles Lengths CS 176 Winter 2011 CS 176 Winter 2011 11 12

How well does it Work? Smoothing Connection Denoising surfaces uniform inverse distance weighted shape preserving  move towards centroid in R 3 original salt dome  connection with Laplacian sample dataset smoothing smoothing  Taubin, 1995  Desbrun, 1999  Guskov, 1999 sample approximations based on parameterization all images from Floater, 1997 CS 176 Winter 2011 CS 176 Winter 2011 13 14 Laplace Equation Laplace Discretization Why is this connection useful? Taubin, 1995  measures smoothness  uniformity assumption  second derivatives (gradient squared)  over “time”, minimizes it “ti ” i i i it Question  smoothing of both  how to discretize Laplace Umbrella  geometry Operator  parameterization CS 176 Winter 2011 CS 176 Winter 2011 15 16 Laplace Discretization Mean Curvature Flow With collaborators, in 1999 With collaborators, in 1999  Laplace-Beltrami  also used implicit time stepping for stability  mean curvature flow  take shape into account t k h i t t original Umbrella flow Mean curvature flow CS 176 Winter 2011 CS 176 Winter 2011 17 18

Measuring Distortion Harmonic Map Energy of a map Properties  discrete harmonic  area minimization leads to minimal surfaces: soap bubbles  Eck, Polthier, Desbrun domain range CS 176 Winter 2011 CS 176 Winter 2011 19 20 Good Measures Variational Approach Comparison of methods Find parameterization which minimizes discrete energies  from Desbrun et al., 2002  DHP (harmonic)  sensitivity to mesh shape Images from Desbrun et al., 2002  symmetric linear system… (nice)  need to fix boundary still Kent et al., 1992 Floater, 1997 Sander et al., 2001 DAP DCP CS 176 Winter 2011 CS 176 Winter 2011 21 22 DHP Boundaries Properties How to fix them?  used to build discrete conformal map  Dirichlet boundary  note that coefficients can be negative  interpolation (bad!) (bad!)  e.g., k-gon on circle with relative k i l ith l ti boundary edge length preserved  keeps angles but can result in very  “unnatural” large area distortion  Neumann boundary  match derivatives on boundary CS 176 Winter 2011 CS 176 Winter 2011 23 24

Natural Boundary Computational Issues Area gradient Computing charts  match on  face clustering boundary  all manner of issues… Neumann Neumann Dirichlet Dirichlet  boundary conditions b d diti  map to particular boundary? which?? j Linear system solvers i  iterative for large, sparse systems! k CS 176 Winter 2011 CS 176 Winter 2011 25 26 Recent Results More formal def. of conformality  see Schr ö der et al. Non linearity creeps in  but much stronger properties! CS 176 Winter 2011 27