CS CS 175 – Week 7 Parameterization Boundary, Non-Linear, and Global Methods

CS Overview choosing boundaries non-linear methods most isometric stretch minimization angle-based flattening hierarchical solvers parameterizing closed surfaces

CS 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)

CS 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

CS Stretch Minimization MIPS neglect stretch use different measure also based on singular values punishes stretch highly non-linear

CS 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

CS Hierarchical Solvers create coarse mesh hierarchy solve problem on coarsest level insert vertices of next finer level good initial value

CS Hierarchical Solvers

CS Closed Surfaces find appropriate parameter domain triangle mesh with few triangles topologically equivalent solve global problem