1. CS175 2003 1 CS 175 – Week 7 Parameterization Boundary, Non-Linear, and Global Methods

2. CS175 2003 2 Overview choosing boundaries non-linear methods most isometric stretch minimization angle-based flattening hierarchical solvers parameterizing closed surfaces

3. CS175 2003 3 Choosing Boundaries convex boundary chord-length parameterization along circle n-gon (vertices = feature points) non-convex boundary projection into least squares plane interactive automatic (natural bdy conditions)

4. CS175 2003 4 Most Isometric Param’s measure distortion per triangle singular values of linear map matrix condition number minimize non-linear energy rational quadratic function minimize e.g. with Gauss-Seidel- like iterations

5. CS175 2003 5 Stretch Minimization MIPS neglect stretch use different measure also based on singular values punishes stretch highly non-linear

6. CS175 2003 6 Angular Based Flattening consider the problem in terms of angles quadratic energy non-linear constraints solve with Lagrangian multipliers and Newton method reconstruct parameterization from one edge and all the angles

7. CS175 2003 7 Hierarchical Solvers create coarse mesh hierarchy solve problem on coarsest level insert vertices of next finer level good initial value

8. CS175 2003 8 Hierarchical Solvers

9. CS175 2003 9 Closed Surfaces find appropriate parameter domain triangle mesh with few triangles topologically equivalent solve global problem