Implicit Fairing of Irregular Meshes using Diffusion and

Curvature Flow

What are we doing?

• Fairness: low variation in curvature.

• Move vertices to achieve.– Mesh topology stays the same.

Overview

• How do we move vertices?– Explicit integration– Implicit integration

• What direction do we move vertices?– Simple Laplacian– Scale-dependant Laplacian– Curvature

How do we move vertices?

Integration Intuition

• X’ = -a·X (a > 0)

• Exact solution: e-a·t

– Give or take a scale factor.

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.

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.

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.

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…

Where do we move vertices?

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.

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.

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.

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.

Summary of Operators

• Simple Laplacian– Mangles irregular meshes.

• Scale-dependant Laplacian– Better results.– Still slides.

• Curvature– Best shape preservation.– Minimal sliding.– Ugly weights.

That Shrinking Thing…

• Calculating mesh volume is easy.

• Rescale the mesh to preserve the volume.

• Amplifies low frequencies.– Only if the mesh shrinks.

Summary

• Classic ideas, new applications.– Implicit integration.– Lowpass filtering as smoothing.– Scaling a mesh.

• Better operators for smoothing.– Scale-dependant Laplacian.– Mean curvature.

Where to go from here?

• Anisotropic smoothing.

• Statistical techniques.– One step smoothing.– Bilateral mesh denoising.

Fair well.