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DW-MRI, Tractography,
and Connectivity: what
Machine Learning can do?


     Ting-Shuo Yo




   Max Planck Institute
   for Human Cognitive and Brain Sciences
   Leipzig, Germany


       Max Planck Institute for Human Cognitive and Brain Sciences
Where the story begins
●   Diffusion Weighted MRI (DWI) is a newly
    developed MR scanning protocol, which can
    detect the movement/displacement of water
    molecules in tissues.
●   So far, the techniques used in DWI analysis are
    mostly deterministic and mechanical. The
    stochastic approaches (ML related) can bring
    new insights to this field.


                                                                                  2

                               Max Planck Institute for Human Cognitive and Brain Sciences
Outline
●   MPG/MPIs
●   A brief introduction of DWI
●   What DWI can do
●   A comparison of different tractography algorithms
●   What ML can do in DWI




                                                                                     3

                                  Max Planck Institute for Human Cognitive and Brain Sciences
Outline
●   MPG/MPIs
        –   Max Planck Society
        –   Objective and Organization
        –   MPI - CBS
●   A brief introduction of DWI
●   What DWI can do
●   A comparison of different tractography algorithms
●   What ML can do in DWI
                                                                                     4

                                  Max Planck Institute for Human Cognitive and Brain Sciences
The Max Planck Society
●   The Max Planck Society for the
    Advancement of Science is an independent,
    non-profit research organization.
●   In particular, the Max Planck Society takes up
    new and innovative and interdisciplinary
    research areas that German universities are
    not in a position to accommodate or deal with
    adequately.


                                                     5
The Max Planck Institutes
●   The research institutes
    of the Max Planck
    Society perform basic
    research in the interest
    of the general public in
    the natural sciences,
    life sciences, social
    sciences, and the
    humanities.
●   Currently there are 81
    MPIs.

                                     6
7

Max Planck Institute for Human Cognitive and Brain Sciences
The MPI for CBS




                  8
Outline
●   MPG/MPIs
●   An Introduction of DWI tractography
       –   Local modelling
       –   Fibre tracking
●   What DWI can do
●   A comparison of different tractography algorithms
●   What ML can do in DWI

                                                                                  9

                               Max Planck Institute for Human Cognitive and Brain Sciences
Diffusion Weighted MRI
●   MRI can detect the
    movement of water
    molecules.
●   The movement is
    constrained by the
    neural fibers.




                                                                           10

                         Max Planck Institute for Human Cognitive and Brain Sciences
Diffusion Weighted MRI
●   By posing a gradient magnetic field, the
    displacement in the corresponding direction
    can be measured.




                                                                                11

                              Max Planck Institute for Human Cognitive and Brain Sciences
Tractography          (1)




       ●   Local modelling:
       ➢   Reconstruct the fibre
           orientation within each voxel




                                                              12

            Max Planck Institute for Human Cognitive and Brain Sciences
Tractography              (2)




●   Diffusion propagator
    –   Diffusion Tensor (DT)
    –   Multiple compartment models
    –   Persistent Angular Structure (PAS)

●   Fibre Orientation Distribution Function
    –   Spherical Deconvolution



                                                                                     13

                                   Max Planck Institute for Human Cognitive and Brain Sciences
Tractography             (3)




●   Fiber tracking:
➢   Reconstruct fibre tracts by
    integrating the reconstructed
    local information




                                                                                      14

                                    Max Planck Institute for Human Cognitive and Brain Sciences
Tractography          (4)




●   Streamline approach
    –   Deterministic
    –   Probabilistic

●   Optimization for a larger region
    –   Spin tracking
    –   Gibbs tracking



                                                                                  15

                                Max Planck Institute for Human Cognitive and Brain Sciences
Tractography          (5)




●   Deterministic tracking
    –   At each step, only
        consider the most likely
        direction
         ●   Curvature threshold
         ●   Step size
         ●   Interpolation
         ●   ......




                                                                                     16

                                   Max Planck Institute for Human Cognitive and Brain Sciences
Tractography               (6)




●   Probabilistic tracking
    –   Perform deterministic tracking for multiple times
    –   Allow uncertainty at each step




                                                                                       17

                                     Max Planck Institute for Human Cognitive and Brain Sciences
Tractography              (7)




●   Probabilistic tracking and tractogram
    –   Probability of connection




                                                                                      18

                                    Max Planck Institute for Human Cognitive and Brain Sciences
Tractography           (8)




●   Optimization for a larger region
    –   Spin tracking
    –   Gibbs tracking




                                                    From Kreher et al. 2008
                                                                                  19

                                Max Planck Institute for Human Cognitive and Brain Sciences
Outline
●   MPG/MPIs
●   A brief introduction of DWI
●   What DWI can do
        –   To reveal anatomical structure in white matter
        –   To construct the general brain network
        –   In vivo
●   A comparison of different tractography algorithms
●   What ML can do in DWI
                                                                                       20

                                     Max Planck Institute for Human Cognitive and Brain Sciences
White matter structure from DWI
●   Product of tractography




                                                                                21

                              Max Planck Institute for Human Cognitive and Brain Sciences
Brain Network from DWI
●   Hagmann 2008




                                                                       22

                     Max Planck Institute for Human Cognitive and Brain Sciences
What DWI can do
●   fMRI shows "where" is working.
       –   The "nodes" in a graph/network
●   DWI shows the structure of the fiber bundles.
       –   The “edges" in a graph/network
       –   With further analysis, can also show "strength of
            edges".
●   The brain network:
       –   The amount of nodes: 10^2
       –   The amount of edges: 10^3
                                                                                      23

                                    Max Planck Institute for Human Cognitive and Brain Sciences
Outline
●   MPG/MPIs
●   A brief introduction of DWI
●   What DWI can do
●   A comparison of different tractography
    algorithms
        –   Selected algorithms
        –   Procedure
        –   Results
●   What ML can do in DWI
                                                                                    24

                                  Max Planck Institute for Human Cognitive and Brain Sciences
Selected Algorithms




                                                              25

            Max Planck Institute for Human Cognitive and Brain Sciences
Procedure




                                                         26

       Max Planck Institute for Human Cognitive and Brain Sciences
Results   (1)




                                                        27

      Max Planck Institute for Human Cognitive and Brain Sciences
Results   (2)




                                                        28

      Max Planck Institute for Human Cognitive and Brain Sciences
Results   (3)




                                                        29

      Max Planck Institute for Human Cognitive and Brain Sciences
Results   (4)




                                                        30

      Max Planck Institute for Human Cognitive and Brain Sciences
Results   (5)




                                                        31

      Max Planck Institute for Human Cognitive and Brain Sciences
Results   (6)




                                                        32

      Max Planck Institute for Human Cognitive and Brain Sciences
Results   (7)




                                                        33

      Max Planck Institute for Human Cognitive and Brain Sciences
Results   (8)




                                                        34

      Max Planck Institute for Human Cognitive and Brain Sciences
Quick Summary
●   More connections
     –   Local models which allow multiple fibres
     –   Probabilistic tracking
●   Consistent patterns across methods
     –   Strong connections within a lobe
     –   Strong connections to corpus callosum
     –   Weak trans-callosum connections


                                                                                    35

                                  Max Planck Institute for Human Cognitive and Brain Sciences
Results   (9)




                                                        36

      Max Planck Institute for Human Cognitive and Brain Sciences
Outline
●   MPG/MPIs
●   A brief introduction of DWI
●   What DWI can do
●   A comparison of different tractography algorithms
●   What ML can do in DWI
        –   Local model reconstruction
        –   Fiber tracking
        –   Further application
                                                                                      37

                                    Max Planck Institute for Human Cognitive and Brain Sciences
ML in DWI

●   Local modeling: deconvolution approach
       –   Assume the signals are convolution of neural
            fibers and noises.
       –   Need to “learn" the deconvolution kernel from
            data defined as "one single fiber".
       –   So far only GLM (2nd order polynomial) is used.
       –   More sophisticated kernel methods can be used.



                                                                                      38

                                    Max Planck Institute for Human Cognitive and Brain Sciences
ML in DWI
●   Fiber tracking
        –   Speed up the optimization process.
        –   Different fiber reconstruction method.


●   Probabilistic modeling of fiber tracts




                                                                                       39

                                     Max Planck Institute for Human Cognitive and Brain Sciences
MICCAI'09 Fiber Cup
●   6 datasets:
    –   3 of resolution 3x3x3mm (image size: 64x64x3) and
        3 b-values (650, 1500 and 2000)
    –   3 of resolution 6x6x6mm (image size: 64x64x1) and
        3 b-values (650, 1500, 2650)
●   Participants have to return one single fiber per
    spatial position selected.



                                                                                    40

                                  Max Planck Institute for Human Cognitive and Brain Sciences
MICCAI'09 Fiber Cup




                                                              41

            Max Planck Institute for Human Cognitive and Brain Sciences
A Very Brief Review of Tractography
●   Local modeling
●   Fiber tracking




                                                                       42

                     Max Planck Institute for Human Cognitive and Brain Sciences
Why are we doing this?
●   Streamline-based tractography:
    –   Each simulation (a fiber) is a possible trajectory in
        the given vector field.
         ●   What is the probability of one given fiber?
         ●   How to select the most representative fibers?




                                                                                            43

                                          Max Planck Institute for Human Cognitive and Brain Sciences
Probability of a Fiber Tract                                   (1)



●   Fiber tract, t = { x1, x2, ...., xl }
●   P(t) = P( x1, x2, ...., xl )




                                                                                        44

                                      Max Planck Institute for Human Cognitive and Brain Sciences
Probability of a Fiber Tract                                                (2)



●   Conditional Probability and Joint Probability
     –   P(A|B) = P(A,B) / P(B)
     –   P(A,B) = P(A|B) P(B)

●   P(t) = P( x1, x2, ...., xl )
         = P(xl| x1, ...., xl-1) P(x1, ...., xl-1)
         = P(xl| x1, ...., xl-1) P(xl-1|x1, ...., xl-2) P(x1, ...., xl-2)
         = P(xl| x1, ...., xl-1) P(xl-1|x1, ...., xl-2) ......P(x2|x1) P(x1)


                                                                                                       45

                                                     Max Planck Institute for Human Cognitive and Brain Sciences
Probability of a Fiber Tract                                              (3)



●
    Assumption: fiber tracking is a 1st order Markov
    process
    –   P(xi| x1, ...., xi-1) = P(xl|xi-1)
    –   P(t) = P( x1, x2, ...., xl )
             = P(xl| x1, ...., xl-1) P(xl-1|x1, ...., xl-2) ......P(x2|x1) P(x1)
             = P(xl|xl-1) P(xl-1|xl-2) ......P(x2|x1) P(x1)
                          l−1
             =   P  x 1 ∏ P  x i1∣x i 
                          i=1



                                                                                                     46

                                                   Max Planck Institute for Human Cognitive and Brain Sciences
Probability of a Fiber Tract                                          (4)



●   How do we define P(xi+1|xi) and P(xi) ?
    –   C: connection probability map
    –   P(xi) ~ C(xi)
    –   P(xi+1|xi) ~ C(xi+1|xi) ~ C(xi+1,xi)


                        l−1
    P t=P  x1  ∏ P  x i1∣x i 
                        i=1




                                                                                                 47

                                               Max Planck Institute for Human Cognitive and Brain Sciences
Finite State Automata                         (1)



●   Each step of fiber tracking can lead to next
    middle point or the terminal point.




                                                                                  48

                                Max Planck Institute for Human Cognitive and Brain Sciences
Finite State Automata                     (2)




          t={x 1 , ... , x l }
                                   l−1
          P t=P0  x l  ∏ 1−P 0  x i 
                                   i=1



                                                                   49

                 Max Planck Institute for Human Cognitive and Brain Sciences
Finite State Automata                                 (3)



●   How to define P0?
    –   # of fibers in the neighboring voxels, NB(x)
    –   (1-P0(xi)) ~ C(NB(xi))
                                                    P 0  x=1−C  x k

    –   C(NB(xi))~ C(xi)                                   K = 20, 10, 5

                     l−1
     P t=P0  x l  ∏ 1−P 0  x i 
                     i=1

             l −1
    P t≃∏ 1−1−C  xi k
             i=1
                                                                                            50

                                          Max Planck Institute for Human Cognitive and Brain Sciences
Finite State Automata                                    (4)



●   Likelihood and Log-likelihood
                       l−1
       P t=P0  x l  ∏ 1−P 0  x i 
                       i=1

              l −1
       P t≃∏ 1−1−C  xi       k

              i=1


              l−1                           l−1
       L t≃∑ ln 1−1−C  x i k ≃∑ −1−C  xi k
              i=1                           i=1

                                  Approximation with 1st order Taylor's expansion

                                                                                              51

                                            Max Planck Institute for Human Cognitive and Brain Sciences
Entropy of a Fiber Tract                                  (1)



●   Entropy
                       l
              H t =∑ C  x i ⋅lnC  x i 
                      i=1

●   Can be seen as the log-likelihood of
               l                                l

              ∑ C  xi ⋅lnC  xi =ln ∏ C  x i              C xi 
                                                                            
              i=1                             i =1




                                                                                         52

                                       Max Planck Institute for Human Cognitive and Brain Sciences
Fiber Cup Results              (2)



Max. Entropy       Max. Likelihood




                                                                        53

                      Max Planck Institute for Human Cognitive and Brain Sciences
ML in DWI
●   Connectivity based clustering
       –   Brain parcellation
       –   Brain tissue is mostly
            continuous without clear
            segmentation, how to
            define regions on it?
       –   Perform clustering based
            on the connectivity
            matrices.



                                                                                        54

                                      Max Planck Institute for Human Cognitive and Brain Sciences
Leipzig, Germany   Saclay, Gif-sur-Yvette, France

A. Anwander        M. Descoteaux
T.R. Knösche       P. Fillard
T. Yo              C. Poupon


                                                                          55

                        Max Planck Institute for Human Cognitive and Brain Sciences
Questions




                                                         56

       Max Planck Institute for Human Cognitive and Brain Sciences
Doing what the brain does - how
   computers learn to listen




                                                                    57

                  Max Planck Institute for Human Cognitive and Brain Sciences
Thank You




                                                  58

Max Planck Institute for Human Cognitive and Brain Sciences

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Diffusion MRI, Tractography,and Connectivity: what machine learning can do?

  • 1. DW-MRI, Tractography, and Connectivity: what Machine Learning can do? Ting-Shuo Yo Max Planck Institute for Human Cognitive and Brain Sciences Leipzig, Germany Max Planck Institute for Human Cognitive and Brain Sciences
  • 2. Where the story begins ● Diffusion Weighted MRI (DWI) is a newly developed MR scanning protocol, which can detect the movement/displacement of water molecules in tissues. ● So far, the techniques used in DWI analysis are mostly deterministic and mechanical. The stochastic approaches (ML related) can bring new insights to this field. 2 Max Planck Institute for Human Cognitive and Brain Sciences
  • 3. Outline ● MPG/MPIs ● A brief introduction of DWI ● What DWI can do ● A comparison of different tractography algorithms ● What ML can do in DWI 3 Max Planck Institute for Human Cognitive and Brain Sciences
  • 4. Outline ● MPG/MPIs – Max Planck Society – Objective and Organization – MPI - CBS ● A brief introduction of DWI ● What DWI can do ● A comparison of different tractography algorithms ● What ML can do in DWI 4 Max Planck Institute for Human Cognitive and Brain Sciences
  • 5. The Max Planck Society ● The Max Planck Society for the Advancement of Science is an independent, non-profit research organization. ● In particular, the Max Planck Society takes up new and innovative and interdisciplinary research areas that German universities are not in a position to accommodate or deal with adequately. 5
  • 6. The Max Planck Institutes ● The research institutes of the Max Planck Society perform basic research in the interest of the general public in the natural sciences, life sciences, social sciences, and the humanities. ● Currently there are 81 MPIs. 6
  • 7. 7 Max Planck Institute for Human Cognitive and Brain Sciences
  • 8. The MPI for CBS 8
  • 9. Outline ● MPG/MPIs ● An Introduction of DWI tractography – Local modelling – Fibre tracking ● What DWI can do ● A comparison of different tractography algorithms ● What ML can do in DWI 9 Max Planck Institute for Human Cognitive and Brain Sciences
  • 10. Diffusion Weighted MRI ● MRI can detect the movement of water molecules. ● The movement is constrained by the neural fibers. 10 Max Planck Institute for Human Cognitive and Brain Sciences
  • 11. Diffusion Weighted MRI ● By posing a gradient magnetic field, the displacement in the corresponding direction can be measured. 11 Max Planck Institute for Human Cognitive and Brain Sciences
  • 12. Tractography (1) ● Local modelling: ➢ Reconstruct the fibre orientation within each voxel 12 Max Planck Institute for Human Cognitive and Brain Sciences
  • 13. Tractography (2) ● Diffusion propagator – Diffusion Tensor (DT) – Multiple compartment models – Persistent Angular Structure (PAS) ● Fibre Orientation Distribution Function – Spherical Deconvolution 13 Max Planck Institute for Human Cognitive and Brain Sciences
  • 14. Tractography (3) ● Fiber tracking: ➢ Reconstruct fibre tracts by integrating the reconstructed local information 14 Max Planck Institute for Human Cognitive and Brain Sciences
  • 15. Tractography (4) ● Streamline approach – Deterministic – Probabilistic ● Optimization for a larger region – Spin tracking – Gibbs tracking 15 Max Planck Institute for Human Cognitive and Brain Sciences
  • 16. Tractography (5) ● Deterministic tracking – At each step, only consider the most likely direction ● Curvature threshold ● Step size ● Interpolation ● ...... 16 Max Planck Institute for Human Cognitive and Brain Sciences
  • 17. Tractography (6) ● Probabilistic tracking – Perform deterministic tracking for multiple times – Allow uncertainty at each step 17 Max Planck Institute for Human Cognitive and Brain Sciences
  • 18. Tractography (7) ● Probabilistic tracking and tractogram – Probability of connection 18 Max Planck Institute for Human Cognitive and Brain Sciences
  • 19. Tractography (8) ● Optimization for a larger region – Spin tracking – Gibbs tracking From Kreher et al. 2008 19 Max Planck Institute for Human Cognitive and Brain Sciences
  • 20. Outline ● MPG/MPIs ● A brief introduction of DWI ● What DWI can do – To reveal anatomical structure in white matter – To construct the general brain network – In vivo ● A comparison of different tractography algorithms ● What ML can do in DWI 20 Max Planck Institute for Human Cognitive and Brain Sciences
  • 21. White matter structure from DWI ● Product of tractography 21 Max Planck Institute for Human Cognitive and Brain Sciences
  • 22. Brain Network from DWI ● Hagmann 2008 22 Max Planck Institute for Human Cognitive and Brain Sciences
  • 23. What DWI can do ● fMRI shows "where" is working. – The "nodes" in a graph/network ● DWI shows the structure of the fiber bundles. – The “edges" in a graph/network – With further analysis, can also show "strength of edges". ● The brain network: – The amount of nodes: 10^2 – The amount of edges: 10^3 23 Max Planck Institute for Human Cognitive and Brain Sciences
  • 24. Outline ● MPG/MPIs ● A brief introduction of DWI ● What DWI can do ● A comparison of different tractography algorithms – Selected algorithms – Procedure – Results ● What ML can do in DWI 24 Max Planck Institute for Human Cognitive and Brain Sciences
  • 25. Selected Algorithms 25 Max Planck Institute for Human Cognitive and Brain Sciences
  • 26. Procedure 26 Max Planck Institute for Human Cognitive and Brain Sciences
  • 27. Results (1) 27 Max Planck Institute for Human Cognitive and Brain Sciences
  • 28. Results (2) 28 Max Planck Institute for Human Cognitive and Brain Sciences
  • 29. Results (3) 29 Max Planck Institute for Human Cognitive and Brain Sciences
  • 30. Results (4) 30 Max Planck Institute for Human Cognitive and Brain Sciences
  • 31. Results (5) 31 Max Planck Institute for Human Cognitive and Brain Sciences
  • 32. Results (6) 32 Max Planck Institute for Human Cognitive and Brain Sciences
  • 33. Results (7) 33 Max Planck Institute for Human Cognitive and Brain Sciences
  • 34. Results (8) 34 Max Planck Institute for Human Cognitive and Brain Sciences
  • 35. Quick Summary ● More connections – Local models which allow multiple fibres – Probabilistic tracking ● Consistent patterns across methods – Strong connections within a lobe – Strong connections to corpus callosum – Weak trans-callosum connections 35 Max Planck Institute for Human Cognitive and Brain Sciences
  • 36. Results (9) 36 Max Planck Institute for Human Cognitive and Brain Sciences
  • 37. Outline ● MPG/MPIs ● A brief introduction of DWI ● What DWI can do ● A comparison of different tractography algorithms ● What ML can do in DWI – Local model reconstruction – Fiber tracking – Further application 37 Max Planck Institute for Human Cognitive and Brain Sciences
  • 38. ML in DWI ● Local modeling: deconvolution approach – Assume the signals are convolution of neural fibers and noises. – Need to “learn" the deconvolution kernel from data defined as "one single fiber". – So far only GLM (2nd order polynomial) is used. – More sophisticated kernel methods can be used. 38 Max Planck Institute for Human Cognitive and Brain Sciences
  • 39. ML in DWI ● Fiber tracking – Speed up the optimization process. – Different fiber reconstruction method. ● Probabilistic modeling of fiber tracts 39 Max Planck Institute for Human Cognitive and Brain Sciences
  • 40. MICCAI'09 Fiber Cup ● 6 datasets: – 3 of resolution 3x3x3mm (image size: 64x64x3) and 3 b-values (650, 1500 and 2000) – 3 of resolution 6x6x6mm (image size: 64x64x1) and 3 b-values (650, 1500, 2650) ● Participants have to return one single fiber per spatial position selected. 40 Max Planck Institute for Human Cognitive and Brain Sciences
  • 41. MICCAI'09 Fiber Cup 41 Max Planck Institute for Human Cognitive and Brain Sciences
  • 42. A Very Brief Review of Tractography ● Local modeling ● Fiber tracking 42 Max Planck Institute for Human Cognitive and Brain Sciences
  • 43. Why are we doing this? ● Streamline-based tractography: – Each simulation (a fiber) is a possible trajectory in the given vector field. ● What is the probability of one given fiber? ● How to select the most representative fibers? 43 Max Planck Institute for Human Cognitive and Brain Sciences
  • 44. Probability of a Fiber Tract (1) ● Fiber tract, t = { x1, x2, ...., xl } ● P(t) = P( x1, x2, ...., xl ) 44 Max Planck Institute for Human Cognitive and Brain Sciences
  • 45. Probability of a Fiber Tract (2) ● Conditional Probability and Joint Probability – P(A|B) = P(A,B) / P(B) – P(A,B) = P(A|B) P(B) ● P(t) = P( x1, x2, ...., xl ) = P(xl| x1, ...., xl-1) P(x1, ...., xl-1) = P(xl| x1, ...., xl-1) P(xl-1|x1, ...., xl-2) P(x1, ...., xl-2) = P(xl| x1, ...., xl-1) P(xl-1|x1, ...., xl-2) ......P(x2|x1) P(x1) 45 Max Planck Institute for Human Cognitive and Brain Sciences
  • 46. Probability of a Fiber Tract (3) ● Assumption: fiber tracking is a 1st order Markov process – P(xi| x1, ...., xi-1) = P(xl|xi-1) – P(t) = P( x1, x2, ...., xl ) = P(xl| x1, ...., xl-1) P(xl-1|x1, ...., xl-2) ......P(x2|x1) P(x1) = P(xl|xl-1) P(xl-1|xl-2) ......P(x2|x1) P(x1) l−1 = P  x 1 ∏ P  x i1∣x i  i=1 46 Max Planck Institute for Human Cognitive and Brain Sciences
  • 47. Probability of a Fiber Tract (4) ● How do we define P(xi+1|xi) and P(xi) ? – C: connection probability map – P(xi) ~ C(xi) – P(xi+1|xi) ~ C(xi+1|xi) ~ C(xi+1,xi) l−1 P t=P  x1  ∏ P  x i1∣x i  i=1 47 Max Planck Institute for Human Cognitive and Brain Sciences
  • 48. Finite State Automata (1) ● Each step of fiber tracking can lead to next middle point or the terminal point. 48 Max Planck Institute for Human Cognitive and Brain Sciences
  • 49. Finite State Automata (2) t={x 1 , ... , x l } l−1 P t=P0  x l  ∏ 1−P 0  x i  i=1 49 Max Planck Institute for Human Cognitive and Brain Sciences
  • 50. Finite State Automata (3) ● How to define P0? – # of fibers in the neighboring voxels, NB(x) – (1-P0(xi)) ~ C(NB(xi)) P 0  x=1−C  x k – C(NB(xi))~ C(xi) K = 20, 10, 5 l−1 P t=P0  x l  ∏ 1−P 0  x i  i=1 l −1 P t≃∏ 1−1−C  xi k i=1 50 Max Planck Institute for Human Cognitive and Brain Sciences
  • 51. Finite State Automata (4) ● Likelihood and Log-likelihood l−1 P t=P0  x l  ∏ 1−P 0  x i  i=1 l −1 P t≃∏ 1−1−C  xi  k i=1 l−1 l−1 L t≃∑ ln 1−1−C  x i k ≃∑ −1−C  xi k i=1 i=1 Approximation with 1st order Taylor's expansion 51 Max Planck Institute for Human Cognitive and Brain Sciences
  • 52. Entropy of a Fiber Tract (1) ● Entropy l H t =∑ C  x i ⋅lnC  x i  i=1 ● Can be seen as the log-likelihood of l l ∑ C  xi ⋅lnC  xi =ln ∏ C  x i  C xi   i=1 i =1 52 Max Planck Institute for Human Cognitive and Brain Sciences
  • 53. Fiber Cup Results (2) Max. Entropy Max. Likelihood 53 Max Planck Institute for Human Cognitive and Brain Sciences
  • 54. ML in DWI ● Connectivity based clustering – Brain parcellation – Brain tissue is mostly continuous without clear segmentation, how to define regions on it? – Perform clustering based on the connectivity matrices. 54 Max Planck Institute for Human Cognitive and Brain Sciences
  • 55. Leipzig, Germany Saclay, Gif-sur-Yvette, France A. Anwander M. Descoteaux T.R. Knösche P. Fillard T. Yo C. Poupon 55 Max Planck Institute for Human Cognitive and Brain Sciences
  • 56. Questions 56 Max Planck Institute for Human Cognitive and Brain Sciences
  • 57. Doing what the brain does - how computers learn to listen 57 Max Planck Institute for Human Cognitive and Brain Sciences
  • 58. Thank You 58 Max Planck Institute for Human Cognitive and Brain Sciences