1. A Hidden Polyhedral Markov Field Model for Diffusion MRI Alexey Koloydenko Division of Statistics Nottingham University, UK

2. Diffusion MRI Club @ Nottingham  Statistics Prof. I. Dryden, Diwei Zhou  Academic Radiology Prof. D. Auer, Dr. P. Morgan  Clinical Neurology Dr. C. Tench

3. Diffusion Magnetic Resonance Imaging (DMRI) RightLeft Back Front Bottom

4. DMRI  Probes matter in predefined directions  by measuring distribution of X, displacement of water molecules within a material over a fixed time interval.  Material microstructure determines p(x), pdf of distribution of X.  Measurements of certain features of p(x) reveal material microstructure.

5. D. Alexander “An Introduction to computational diffusion MRI: The diffusion tensor and beyond”, 2006

6. Toward diagnosis of white matter diseases  Stroke, epilepsy, multiple sclerosis,…  Dominant directions of particle displacements dominant fibre directions  Developmental and pathological conditions of the brain integrity and organization of white matter fibres

7. Main models  DMRI signal S = magnetization of all contributing spins: – signal with no diffusion-weighting  Diffusion Tensor (DT) MRI assumption: t - diffusion time, D – diffusion tensor

8. Statistical models Assuming independent  Additive Gaussian noise  Multiplicative Gaussian noise

9. Putting things together: Single DT MRI Models   Estimate from  Estimation: (constrained) NLLS & LLS  Acceptable results for regions with single dominant fibre direction

10. Example

11. Courtesy of D. Zhou (Gray matter) 0<FA<1 (White matter)

12. Handling crossing fibres Multiple Tensors D(k)  Assuming  Parameter estimation is difficult: NLLS is sensitive to initialization

13. Solutions and work-arounds  Inter-voxel dependence  Further constraints on individual tensors (e.g. cylindrical )  Bayesian approach  Application dependent other  Revision of (assumptions underlying derivation of)

14. Hidden MRF on a hemi-polyhedron

15. Example   “Halving”

16.   Hidden layer: indicates component “responsible for”  Conditioned on assume independent or

17. Hidden MRF Invariant under symmetry group of

18. Estimation  Parameters: are currently nuisance  Algorithms: EM, VT - Viterbi Training (Extraction)  Current choice – VT. Simpler, computationally stable, …

19. VT  Choose  Obtain to maximize  Obtain to maximize  Repeat until  Small scale/ exhaust search  Small scale/ - numerically, - single DT - easy

20. Current efforts  Truncated hemi-icosahedron, |V|/2=30  Comparative analysis (with traditional parametric and Bayesian approaches)  Interpretation of the hidden layer

21. Other (non DMRI) issues  EM?  What if N is large? 1. Viterbi alignment on multidimensional lattices. a.) Variable state Viterbi algorithm (R. Gray & J. Li, `00) b.) Annealing (S. Geman & D. Geman, `84) 2. Estimation of, MCMC (L. Younes, `91)

22. References  D. Alexander “An Introduction to computational diffusion MRI: The diffusion tensor and beyond”, Chapter in "Visualization and image processing of tensor fields" editted by J.Weickert and H.Hagen, Springer 2006  J. Li, A. Najmi, R. Gray, "Image classification by a two dimensional hidden Markov model," IEEE Transactions on Signal Processing, 48(2):517-33, February 2000.  D. Joshi, J. Li, J. Wang, "A computationally efficient approach to the estimation of two- and three-dimensional hidden Markov models," IEEE Transactions on Image Processing, 2005  L. Younes, Maximum likelihood estimation for Gibbs fields. Spatial Statistics and Imaging: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference, A. Possolo (editor), Lecture Notes-Monograph Series, Institute of Mathematical Statistics, Hayward, California (1991)Maximum likelihood estimation for Gibbs fields.  S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,'' IEEE Trans. Pattern Anal. Mach. Intell., 6, 721-741, 1984  A. Koloydenko “A Hidden Polyhedral MRF model for Diffusion MRI data” in preparation