1. Stochastic Systems Group Curve Sampling and Conditional Simulation Ayres Fan John W. Fisher III Alan S. Willsky MIT Stochastic Systems Group

2. Stochastic Systems Group Outline 1.Overview 2.Curve evolution 3.Markov chain Monte Carlo 4.Curve sampling 5.Conditional simulation

3. Stochastic Systems Group Overview Curve evolution attempts to find a curve C (or curves C i ) that best segment an image (according to some model) Goal is to minimize an energy functional E(C) (view as a negative log likelihood) Find a local minimum using gradient descent

4. Stochastic Systems Group Sampling instead of optimization Draw multiple samples from a probability distribution p (e.g., uniform, Gaussian) Advantages: –Naturally handles multi-modal distributions –Avoid local minima –Higher-order statistics (e.g., variances) –Conditional simulation

5. Stochastic Systems Group Outline 1.Overview 2.Curve evolution 3.Markov chain Monte Carlo 4.Curve sampling 5.Conditional simulation

6. Stochastic Systems Group Planar curves A curve is a function We wish to minimize an energy functional with a data fidelity term and regularization term: This results in a gradient flow: We can write any flow in terms of the normal:

7. Stochastic Systems Group Euclidean curve shortening flow Let’s examine E is minimized when C is short Gradient flow should be in the direction that minimizes the curve length the fastest Use Euler-Lagrange and we see:

8. Stochastic Systems Group Level-Set Methods A curve is a function (infinite dimensional) A natural implementation approach is to use marker points on the boundary (snakes) –Reinitialization issues –Difficulty handling topological change Level set methods instead evolve a surface (one dimension higher than our curve) whose zeroth level set is the curve (Sethian and Osher)

9. Stochastic Systems Group Embedding the curve Force level set  to be zero on the curve Chain rule gives us

10. Stochastic Systems Group References Snakes: Active contour models, Kass, Witkin, Terzopoulos 1987 Geodesic active contours, Caselles, Kimmel, Sapiro 1995 Region competition, Zhu and Yuille, 1996 Mumford-Shah approach, Tsai, Yezzi, Willsky 2001 Active contours without edges, Chan and Vese, 2001

11. Stochastic Systems Group Examples-I

12. Stochastic Systems Group Examples-II

13. Stochastic Systems Group Examples-III

14. Stochastic Systems Group Examples-IV

15. Stochastic Systems Group Examples-V

16. Stochastic Systems Group Outline 1.Overview 2.Curve evolution 3.Markov chain Monte Carlo 4.Curve sampling 5.Conditional simulation

17. Stochastic Systems Group General MAP Model For segmentation: –x is a curve –y is the observed image (can be vector) –S is a shape model –Data model usually IID given the curve We wish to sample from p(x|y;S), but cannot do so directly

18. Stochastic Systems Group Markov Chain Monte Carlo Class of sampling methods that iteratively generate candidates based on previous iterate (forming a Markov chain) Examples include Gibbs sampling, Metropolis-Hastings Instead of sampling from p(x|y;S), sample from a proposal distribution q and keep samples according to an acceptance rule a

19. Stochastic Systems Group Metropolis algorithm Metropolis algorithm: 1.Start with x 0 2.At time t, generate candidate  t (given x t-1 ) 3.Set x t =  t with probability min(1, p(  t )/p(x t-1 )), otherwise x t = x t-1 4.Go back to 2

20. Stochastic Systems Group Asymptotic Convergence Two main requirements to ensure iterates converge to samples: 1)q spans support of p This means that any element with non-zero probability must be able to be generated by a sequence of samples from q 2)Detailed balance is achieved

21. Stochastic Systems Group Metropolis-Hastings Metropolis update rule only results in detailed balance when q is symmetric: When this is not true, we need to evaluate the Hastings ratio: and keep a candidate with probability min(1, r)

22. Stochastic Systems Group Outline 1.Overview 2.Curve evolution 3.Markov chain Monte Carlo 4.Curve sampling 5.Conditional simulation

23. Stochastic Systems Group Basic model Modified Chan-Vese energy functional We can view data model as being iid inside and outside the curve with different Gaussian distributions

24. Stochastic Systems Group MCMC Curve Sampling Generate perturbation on the curve: Sample by adding smooth random fields:  controls the degree of smoothness in field Note for portions where f is negative, shocks may develop (so called prairie fire model) Implement using white noise and circular convolution

25. Stochastic Systems Group Coverage/Detailed balance It is easy to show we can go from any curve C 1 to any other curve C 2 (shrink to a point) For detailed balance, we need to compute probability of generating C´ from C (and vice versa) If we assume that the normals are approximately the same, then we see f = f ´ (symmetric)

26. Stochastic Systems Group Initial results

27. Stochastic Systems Group Smoothness issues While detailed balance assures asymptotic convergence, may need to wait a very long time In this case, smooth curves have non-zero probability under q, but are very unlikely to occur Can view accumulation of perturbations as (h is the smoothing kernel, n i is white noise) Solution: make q more likely to move towards high-probability regions of p

28. Stochastic Systems Group Adding mean force We can add deterministic elements to f (i.e., a mean to q) The average behavior should then be to move towards higher-probability areas of p In the limit, setting f to be the gradient flow of the energy functional results in always accepting the perturbation We keep the random field but have a non-zero mean:

29. Stochastic Systems Group Detailed balance redux Before we just assumed that q was symmetric and used Metropolis updates Now we need to evaluate q and use Metropolis- Hastings updates Probability of going from C to C´ is the probability of generating f (which is Gaussian) and the reverse is the probability of f ´ (also Gaussian)

30. Stochastic Systems Group Detailed balance continued Relationship between f and f ´ complicated due to the fact that the normal function changes Various levels of exactness –Assume –Infinitesimal approximation (ignore tangential) –Trace along (technical issues)

31. Stochastic Systems Group New results

32. Stochastic Systems Group Further research We would like q to naturally generate smooth curves –Different structure for the perturbation Speed always an issue for MCMC approaches –Multiresolution perturbations –Parameterized perturbations (efficient basis) –Hierarchical models How to properly use samples?

33. Stochastic Systems Group Outline 1.Overview 2.Curve evolution 3.Markov chain Monte Carlo 4.Curve sampling 5.Conditional simulation

34. Stochastic Systems Group User Information In many problems, the model admits many reasonable solutions Currently user input largely limited to initialization We can use user information to reduce the number of reasonable solutions –Regions of inclusion or exclusion –Partial segmentations Can help with both convergence speed and accuracy Interactive segmentation

35. Stochastic Systems Group Conditional simulation With conditional simulation, we are given the values on a subset of the variables We then wish to generate sample paths that fill in the remainder of the variables (e.g., simulating Brownian motion)

36. Stochastic Systems Group Simulating curves Say we are given C s, a subset of C (with some uncertainty associated with it) We wish to sample the unknown part of the curve C u One way to view is as sampling from: Difficulty is being able to evaluate the middle term as theoretically need to integrate p(C)

37. Stochastic Systems Group Simplifying Cases Under special cases, evaluation of is tractable: 1.When C is low-dimensional (can do analytical integration or Monte-Carlo integration) 2.When C s is assumed to be exact 3.When p(C) has special form (e.g., independent) 4.When willing to approximate

38. Stochastic Systems Group Basic model in 3D Energy functional with surface area regularization: With a slice-based model, we can write the regularization term as:

39. Stochastic Systems Group Zero-order hold approximation Approximate volume as piecewise-constant “cylinders”: Then we see that the surface areas are: We see terms related to the curve length and the difference between neighboring slices Upper bound to correct surface area

40. Stochastic Systems Group Overall regularization term Adding everything together results in: self potentials edge potentials

41. Stochastic Systems Group Graphical Models Key Markov property: Graph Separation Conditional Independence

42. Stochastic Systems Group 2.5D Approach In 3D world, natural (or built-in) partition of volumes into slices Assume Markov relationship among slices Then have local potentials (e.g., PCA) and edge potentials (coupling between slices) Naturally lends itself to local Metropolis- Hastings approach (iterating over the slices)

43. Stochastic Systems Group 2.5D Model We can model this as a simple chain structure with pairwise interactions This admits the following factorization:

44. Stochastic Systems Group Partial segmentations Assume that we are given segmentations of every other slice We now want to sample surfaces conditioned on the fact that certain slices are fixed Markovianity tells us that c 2 and c 4 are independent conditioned on c 3

45. Stochastic Systems Group Log probability for c 2 We can then construct the probability for c 2 conditioned on its neighbors using the potential functions defined previously:

46. Stochastic Systems Group Results Neighbor segmentations Our result (cyan) with expert (green)

47. Stochastic Systems Group Larger spacing Apply local Metropolis-Hastings algorithm where we sample on a slice-by-slice basis Theory shows that asymptotic convergence is unchanged Unfortunately larger spacing similar to less regularization Currently have issues with poor data models that need to be resolved

48. Stochastic Systems Group Full 3D In medical imaging, have multiple slice orientations (axial, sagittal, coronal) In seismic, vertical and horizontal shape structure expected With a 2.5D approach, this introduces complexity to the graph structure

49. Stochastic Systems Group Incorporating perpendicular slices c  is now coupled to all of the horizontal slices c  only gives information on a subset of each slice Specify edge potentials as, e.g.:

50. Stochastic Systems Group Other extensions Additional features can be added to the base model: –Uncertainty on the expert segmentations –Shape models (semi-local) –Exclusion/inclusion regions –Topological change (through level sets)

51. Stochastic Systems Group Applications Currently looking for applications to demonstrate utility of curve sampling –Multi-modal distributions –Morphology changes –Confidence levels

52. Stochastic Systems Group Interface How do we present the information to the user? –Pixel or curve statistics? –Aggregate statistics or present samples? How do we receive information from the user? –Readily incorporated into simulation model –Easy to specify