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Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow

This overview discusses the techniques used to smooth irregular meshes, including explicit and implicit integration methods, and different operators such as the simple Laplacian, scale-dependent Laplacian, and curvature-based smoothing. It also covers the concept of mesh scaling and the benefits and limitations of each approach.

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Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow

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  1. Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow

  2. What are we doing? • Fairness: low variation in curvature. • Move vertices to achieve. • Mesh topology stays the same.

  3. Overview • How do we move vertices? • Explicit integration • Implicit integration • What direction do we move vertices? • Simple Laplacian • Scale-dependant Laplacian • Curvature

  4. How do we move vertices?

  5. Integration Intuition • X’ = -a·X (a > 0) • Exact solution: e-a·t • Give or take a scale factor.

  6. Explicit Integration • X(t+dt) = (1-a·dt)·X(t) • 1 > a·dt > 0 OK. • 2 > a·dt > 1 Damped oscillation. • a·dt > 2 BOOM! • Take small steps to stay stable.

  7. Implicit Integration • X(t+dt) = X(t)-a·dt·X(t+dt) • X(t+dt)·(1+a·dt) = X(t) • X(t+dt) = X(t)·(1+a·dt)-1 • Higher computational cost. • Better stability. • Huge time steps are possible. • … but accuracy will suffer.

  8. Generic Smoothing Eq. • Explicit • X(t+dt) = (I+lam·dt·M)·X(t) • Implicit • X(t+dt)·(I-lam·dt·M) = X(t) • Inverting matrices sucks. (So don’t do it…) • Sparsity works for us.

  9. When is Implicit Faster? • Small edges require small steps for explicit integration. • Sparsity allows faster implicit integration. • Accuracy seems to be OK for big time steps. • Can only take big steps with implicit…

  10. Where do we move vertices?

  11. Simple Laplacian • Move a vertex towards a weighted sum of its neighbors. • Equal weighting. • Assumption: it’s a regular mesh. • Mangles the mesh if it’s not. • Shrinkage. • Sliding.

  12. Scale-dependant Laplacian • Move a vertex towards a weighted sum of its neighbors. • Weighted by inverse edge length. • Less shape distortion. • Less sliding. • A little more computation is required.

  13. Curvature • Move a vertex along the surface normal, based on the magnitude of the curvature. • A.k.a. Move a vertex towards a weighted sum of its neighbors. • But with a funky weight… • Not normalized, either. • Sliding should be negligible.

  14. Laplacian & Signal Processing • Raising L to a higher power... • Steeper roll off. (Higher order filter.) • Denser matrix. (Less sparse.) • L2 is a good balance. • Combinations of L and L2 allow resonant filters, etc.

  15. Summary of Operators • Simple Laplacian • Mangles irregular meshes. • Scale-dependant Laplacian • Better results. • Still slides. • Curvature • Best shape preservation. • Minimal sliding. • Ugly weights.

  16. That Shrinking Thing… • Calculating mesh volume is easy. • Rescale the mesh to preserve the volume. • Amplifies low frequencies. • Only if the mesh shrinks.

  17. Summary • Classic ideas, new applications. • Implicit integration. • Lowpass filtering as smoothing. • Scaling a mesh. • Better operators for smoothing. • Scale-dependant Laplacian. • Mean curvature.

  18. Where to go from here? • Anisotropic smoothing. • Statistical techniques. • One step smoothing. • Bilateral mesh denoising.

  19. Fair well.

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