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

CS 175 – Week 7 Parameterization Boundary, Non-Linear, and Global Methods. Overview. choosing boundaries non-linear methods most isometric stretch minimization angle-based flattening hierarchical solvers parameterizing closed surfaces. Choosing Boundaries. convex boundary

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

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  1. CS 175 – Week 7ParameterizationBoundary, Non-Linear, and Global Methods

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

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

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

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

  8. Hierarchical Solvers

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

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