Transcript Implicit Fairing of Irregular Meshes using Diffusion and

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
– 2 > a·dt > 1
–
a·dt > 2
OK.
Damped oscillation.
BOOM!
• Take small steps to stay stable.
Implicit Integration
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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.