Multiterminal Network Tomography

1. Multiterminal Network Tomography Laura Felicia Matusevich (Texas A&M) and F. Alberto Grünbaum (UC Berkeley)

2. Networks • Let G be a directed graph without loops

3. Networks • Let G be a directed graph without loops

4. Networks • Let G be a directed graph without loops and without multiple directed edges.

5. Networks • Let G be a directed graph without loops and without multiple directed edges.

6. Networks • Let G be a directed graph without loops and without multiple directed edges.

7. Networks • Let G be a directed graph without loops and without multiple directed edges. • Sources will be called incoming terminals

8. Networks • Let G be a directed graph without loops and without multiple directed edges. • Sources will be called incoming terminals, sinks will be called outgoing terminals

9. Networks • Let G be a directed graph without loops and without multiple directed edges. • Sources will be called incoming terminals, sinks will be called outgoing terminals, all other vertices will be called intermediate terminals.

10. Networks • Let G be a directed graph without loops and without multiple directed edges. • Sources will be called incoming terminals, sinks will be called outgoing terminals, all other vertices will be called intermediate terminals. • G is called a multiterminal network.

11. An Example

12. Another Example

13. The Inverse Problem • Assign a number to each arrow of G, the probability that a message goes through that arrow.

14. The Inverse Problem • Assign a number to each arrow of G, the probability that a message goes through that arrow.

15. The Inverse Problem • Assign a number to each arrow of G, the probability that a message goes through that arrow. • Goal: to recover these numbers from measurements at the incoming and outgoing terminals.

16. Why is this tomography?

17. Why is this tomography?

18. Why is this tomography?

19. Why is this tomography?

20. Why is this tomography?

21. Why is this tomography?

22. Why is this tomography?

23. Why is this tomography?

24. Why is this tomography?

25. Why is this tomography?

26. Why is this tomography?

27. Why is this tomography?

28. An Example

29. An Example • Consider the numbers the probability that a message goes from source to sink in steps.

30. An Example • Consider the numbers

31. An Example • Consider the numbers

32. An Example • Consider the numbers the probability that a message goes from source to sink in steps.

33. An Example • Consider the numbers the probability that a message goes from source to sink in steps. • These numbers determine

34. Observations • The numbers contain all the information.

35. Observations • The numbers contain all the information. • We can determine three quantities, when we’d like six.

36. Observations • The numbers contain all the information. • We can determine three quantities, when we’d like six. • More reasonable to measure:

37. Observations • The numbers contain all the information. • We can determine three quantities, when we’d like six. • More reasonable to measure: the moments of travel time.

38. Another Example

39. Another Example Get everything from

40. Another Example Get everything from explicitly!

41. Plan for the rest of the talk

42. Plan for the rest of the talk • Explain how to solve the “4pixelman” problem. (Grünbaum and Patch; Grünbaum and M.)

43. Plan for the rest of the talk • Explain how to solve the “4pixelman” problem. (Grünbaum and Patch; Grünbaum and M.) F.A. Grünbaum and S. Patch, The use of Grassman identities for inversion of a general model in diffuse tomography, Proceedings of the Lapland Conference on Inverse Problems, Saariselka, Finland, June 1992. F.A. Grünbaum and S. Patch , Simplification of a general model in diffuse tomography, in “Inverse Problems in Scattering and Imaging”, M. A. Fiddy (ed.), Proc. SPIE 176, 744–754. S. Patch, Consistency conditions in Diffuse Tomography, Inverse problems, 10, 1 , (1994) , 199–212. S. Patch, Recursive recovery of a family of Markov transition probabilities from boundary value data, J. Math. Phys. 36(7) (July 1995), 3395–3412. S. Patch, A recursive algorithm for diffuse planar tomography, Chapter 20 in “Discrete Tomography: Foundations, Algorithms, and Applications”, G. Herman and A. Kuba (eds.), Birkhauser, Boston, 1999. F.A. Grünbaum and L.F. Matusevich, Explicit inversion formulas for a model in diffuse tomography, Advances in Applied Mathematics, 29, (2002), 172-183. F.A. Grünbaum and L.F. Matusevich , A nonlinear inverse problem inspired by 3-dimensional diffuse tomography, Int. J. Imaging Technology, 12 (2002), 198–203. F. A. Grünbaum and L.F. Matusevich, A network tomography problem related to the hypercube, Contemporary Mathematics, (2004) 362 189-197.

44. Plan for the rest of the talk • Explain how to solve the “4pixelman” problem. (Grünbaum and Patch; Grünbaum and M.) • Construct a large class of networks for which this method works.

45. Plan for the rest of the talk • Explain how to solve the “4pixelman” problem. (Grünbaum and Patch; Grünbaum and M.) • Construct a large class of networks for which this method works. F. A. Grünbaum and L. F. Matusevich, An identification problem for multiterminal networks, solving for the transit matrix from input-output data, Preprint.

46. Setting up the equations

47. Setting up the equations • Construct four matrices:

48. Setting up the equations • Construct four matrices:

49. Setting up the equations • Construct four matrices:

50. Setting up the equations • Construct four matrices: