A Hidden Polyhedral Markov Field Model for Diffusion

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

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

Diffusion Magnetic Resonance Imaging (DMRI) Front Right Left Bottom Back

DMRI o o 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.

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

Toward diagnosis of white matter diseases o o o 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

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

Statistical models Assuming independent o Additive Gaussian noise o Multiplicative Gaussian noise

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

Example

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

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

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

Hidden MRF on a hemi-polyhedron

Example o o “Halving”

o o o Hidden layer: indicates component “responsible for” Conditioned on assume independent or

Hidden MRF Invariant under symmetry group of

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

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

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

Other (non DMRI) issues o o 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)

References o o o 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) 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