1. Random Walks for Vector Field Denoising João Paixão, Marcos Lage, Fabiano Petronetto, Alex Laier, Sinésio Pesco, Geovan Tavares, Thomas Lewiner, Hélio Lopes Matmidia Laboratory – Department of Mathematics PUC–Rio – Rio de Janeiro, Brazil

2. Motivation Vector Fields in Science and Engineering Flow in an artificial heart Flow patterns in a tube University of Cambridge (2009)

3. Motivation Noise in vector data-acquisition Flow around a live swimming fish (Yoshida et al 2004)

4. Problem

5. Problem:Noise Denoising

6. Gaussian Filtering E.g. 5x5 Gaussian Filter

7. Limitations Feature Destruction

8. Limitations Feature Destruction

9. Random Walks on the Graph Feature

10. Previous Work Smolka et al. 2001 Random Walk for Image Enhancement

11. Previous Work Sun et al. 2007 Mesh Denoising

12. Random Walks for Vector Fields What we want -Meshless -Feature-preserving What do we need -Graph -Probabilities that avoid crossing features

13. How to build the graph

14. Feature Functions Direction Magnitude

15. Feature Functions Direction Magnitude Other feature functions in the paper!

16. Probabilities is the neighborhood of vector i. 3 3 4 4 2 2 1 Probability from vector i to vector j

17. Time to walk A B

18. A B

19. A B

20. A B

21. A B

22. - the probability of going from node A to node B after n steps A B

23. Random Walk Filtering Weighted Average of Random Walk Probabilities

24. Feature-preserving Discontinuity

25. Simple Example

27. Granular Flow

28. Gaussian FilteringRandom Walk Filtering

29. Particle Image Velocimetry

30. GaussianRandom Walk Particle Image Velocimetry

31. Landslide

36. Summary -Feature Preserving -Meshless -Interpretative -Flexible -Easy to implement

37. Limitations -Number of parameters -Dependency in them

38. Future Works - 3D vector field denoising algorithm

39. Thank you for your attention