1. Smoothing 3D Meshes using Markov Random Fields
Vedrana Andersen

2. Overview
Aim: To investigate the use of Markov Random Fields (MRF) for formulating priors on 3D surfaces represented as triangle meshes
Focus on: Mesh smoothing, feature-preserving mesh smoothing (preserving surface ridges)
Vertex process and edge process

3. MRF THEORY Markov Random Fields
Random fields, sites, labels, labeling
Markov Random Fields, Markovianity, neighborhood system and cliques
Markov-Gibbs equivalence
Gibbs Random Fields (Gibbs distribution)

4. MESH SMOOTHING Smoothness Prior
Mesh smoothing: sites – vertices, labels – spatial positions of vertices
Absolute mean curvature: dihedral angle and edge length
Potential of the smoothness prior

5. MESH SMOOTHING Smoothness Prior
Potential assigned to all 4-cliques of vertices

6. MRF THEORY MAP-MRF Labeling
Maximum a-posteriori solution
Bayes rule
In terms of energy

7. MESH SMOOTHING Likelihood Function
Input mesh – underlying surface corrupted by noise
Noise: isotropic Gaussian
Likelihood energy

8. MESH SMOOTHING Optimization
Metropolis sampler with simulated annealing
Sampling – new candidate positions
Metropolis criterion
Cooling scheme

9. MESH SMOOTHING Optimization
Optimization parameters:
Sampling step size
Initial temperature and annealing constant
Modeling parameter:
Weight of the data term

10. MESH SMOOTHING Results
10x10x10 cube corrupted with Gaussian noise (σ=0.3 average edge length)
Very slow cooling, 500 iterations, (Fig. 7.3)

11. MESH SMOOTHING Results
Monitoring the potentials and the number of updates over time

12. MESH SMOOTHING Results
Smoothing with a zero temperature, iterations, (Fig. 7.5)

13. MESH SMOOTHING Discussion
Convergence, monitoring
Sensitive to the size of the sampling step
Optimization? Annealing?
What is a smooth mesh?

14. MESH SMOOTHING Alternative Formulations
Original formulation – dihedral angles and edge lengths
Quadratic and square-root potentials

15. MESH SMOOTHING Alternative Formulations
Angle based potential – indifferent
Quadratic – over-smoothing
Square root – feature preserving

16. MESH SMOOTHING Alternative Formulations
Results of using curvature-based, quadratic and square-root potential, 300 iterations (Fig. 7.15)

17. MESH SMOOTHING Alternative Formulations
Feature preserving –
thresholded smoothness
potential, implicit edge
labeling
Conclusion: Control achieved by the choice of the smoothness potential

18. FEATURE DETECTION Edge Labeling
Detecting feature edges – ridges of the underlying surface
Idea: To use edge labels as weights for smoothing process
Based on:
edge sharpness,
neighborhood support.

19. FEATURE DETECTION Edge Labeling
continuous or discrete,
deterministic or stochastic.

20. FEATURE DETECTION Ridge Sharpness
Sharpness threshold Ф0
Deterministic case:
thresholding,
cut-off function.

21. FEATURE DETECTION Ridge Sharpness
Stochastic case (MRF framework, assigning sharpness potential to 1-edge cliques of edges):
linear potential
alternative: difference from cut-off function

22. FEATURE DETECTION Neighborhood Support
Support threshold θ0
Idea: Presence of sharp edges in the neighborhood influences labeling

23. FEATURE DETECTION Neighborhood Support
Stochastic case
discrete linear formulation
alternative: penalizing label differences along a line

24. FEATURE DETECTION Edge Labeling Results
Labeling edges of the fandisk model, (Fig. 9.2)

25. FEATURE DETECTION Two Questions
Neighborhood support – does it help?
If not, edge sharpness – is it needed?
Relevant for coupled models

26. FEATURE-PRESERVING MESH SMOOTHING Coupled Model
Using edge labels as weights
Potentials contributing to the total posterior energy:
smoothness,
likelihood,
edge labeling (sharpness and neighborhood support)

27. FEATURE-PRESERVING MESH SMOOTHING Coupled Model
Minimizing total posterior energy, or…
Alternating between vertex process and edge process
Independent cooling schemes
Ordering of vertex and edge process

28. FEATURE-PRESERVING MESH SMOOTHING Results
Smoothing the noisy cube, noise 0.2 average edge length, (Fig. 11.5)

29. FEATURE-PRESERVING MESH SMOOTHING Results
Reconstructing the fandisk model, noise 0.2 average edge length, (Fig. 11.7)

30. FEATURE-PRESERVING MESH SMOOTHING Discusion
Two questions revisited:
Neighborhood support?
Edge labeling?
Control vs. automation

31. FEATURE-PRESERVING MESH SMOOTHING Possible Improvements
Sampling (shape, adaptive step)
Optimization (deterministic?)
Larger neighborhood for edge labeling
Surface to surface smoothing
Mesh optimization
Future work:
Piecewise-quadratic surfaces

32. Thank you!