1. Markov Random Fields Tomer Michaeli Graduate Course 048926

2. Wiener Filter for 1D GMRF

3. Wiener Filter for 2D GMRF OriginalBlur, small noiseDeblurredSevere noiseDenoised

4. Gaussian vs. Weak Spring Potentials

5. Distribution of Derivatives in Natural Images

6. Robust Non-Convex Potentials “Lorentzian”

7. Robust Non-Convex Potentials

8. Lorentzian potentials

9. Robust Non-Convex Potentials Input Quadratic Potentials (Gaussian Prior) Robust Potentials (Hyper-Laplacian Prior)

10. Unstable Reconstruction with Non-Convex Potentials

12. Robust Convex Potentials

13. Symmetric Bound Surrogate

14. MRF-Examples

15. Pairwise Cliques Manually chosen potentials

16. Weak spring Gaussian noisedenoising- weak spring potential

17. Hyper Laplacian Gaussian Prior ( ) Levin et. al ( )Hyper Laplacian ( ) Laplacian Prior ( )

18. Hyper Laplacian Gaussian Prior ( ) Levin et. al ( )Hyper Laplacian ( ) Laplacian Prior ( )

19. Larger Cliques Learned potentials

20. Fields of Experts Learned models Square 3x3 cliques Diamond shaped 5x5 cliques Square 7x7 cliques Cliques structures

21. Fields of Experts- Denoising

22. Fields of Experts- Inpainting

23. Shrinkage Fields

24. Gibbs sampling example: Bivariate normal distribution Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

25. Gibbs sampling example: Bivariate normal distribution

26. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Binary Denoising Before After Image represented as binary discrete variables. Some proportion of pixels randomly changed polarity.

27. Segmentation

28. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Multi-label Denoising Before After Image represented as discrete variables representing intensity. Some proportion of pixels randomly changed according to a uniform distribution.

29. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Denoising Goal Observed DataUncorrupted Image

30. Most of the pixels stay the same Observed image is not as smooth as original Now consider pdf over binary images that encourages smoothness – Markov random field 30 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Denoising Goal Observed DataUncorrupted Image

31. 31 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Markov random fields Normalizing constant (partition function) Cost function Returns any number Subset of variables (clique) Relationship

32. 32Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Smoothing Example

33. 33Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Smoothing Example

34. 34Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Smoothing Example Samples – mostly smooth

35. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Max-Flow Problem Goal: To push as much ‘flow’ as possible through the directed graph from the source to the sink. Cannot exceed the (non-negative) capacities c ij associated with each edge.

36. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Saturated Edges When we are pushing the maximum amount of flow: There must be at least one saturated edge on any path from source to sink (otherwise we could push more flow) The set of saturated edges hence separate the source and sink

37. Graph Construction One node per pixel (here a 3x3 image) Edge from source to every pixel node Edge from every pixel node to sink Reciprocal edges between neighbours Note that in the minimum cut EITHER the edge connecting to the source will be cut, OR the edge connecting to the sink, but NOT BOTH (unnecessary). Which determines whether we give that pixel label 1 or label 0. Now a 1 to 1 mapping between possible labelling and possible minimum cuts Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

38. Graph Construction Now add capacities so that minimum cut, minimizes our cost function Unary costs U(0), U(1) attached to links to source and sink. Either one or the other is paid. Pairwise costs between pixel nodes as shown. Why? Easiest to understand with some worked examples. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

39. Example 1 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 39

40. Example 2 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 40

41. Example 3 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 41

42. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 42

43. Denoising Results Original Pairwise costs increasing Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

44. Convex vs. non-convex costs Quadratic Convex Submodular Truncated Quadratic Not Convex Not Submodular Potts Model Not Convex Not Submodular Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

45. What is wrong with convex costs? Pay lower price for many small changes than one large one Result: blurring at large changes in intensity Observed noisy imageDenoised result Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 45

46. Denoisi ng Results: Alpha Expansi on Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

47. Background subtraction Applications

48. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Grab cut Applications

49. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Stereo vision Applications

50. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Shift-map image editing Applications

51. 51Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

52. Shift-map image editing Applications