A Hidden Polyhedral Markov Field Model for Diffusion MRI

Alexey KoloydenkoDivision of StatisticsNottingham University, UK

Diffusion MRI Club @ Nottingham

Statistics Prof. I. Dryden, Diwei Zhou Academic Radiology Prof. D. Auer, Dr. P. Morgan Clinical Neurology Dr. C. Tench

Diffusion Magnetic Resonance Imaging (DMRI)

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t

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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.

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

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

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

Statistical models

Assuming independent Additive Gaussian noise

Multiplicative Gaussian noise

Putting things together:Single DT MRI Models

Estimate

from Estimation: (constrained) NLLS & LLS Acceptable results for regions with single

dominant fibre direction

Example

Courtesy of D. Zhou

(Gray matter) 0<FA<1 (White matter)

Handling crossing fibresMultiple Tensors D(k)

Assuming

Parameter estimation is difficult: NLLS is sensitive to initialization

Solutions and work-arounds

Inter-voxel dependence Further constraints on individual

tensors (e.g. cylindrical ) Bayesian approach Application dependent other Revision of (assumptions underlyingderivation of)

1 2 3

Hidden MRF on a hemi-polyhedron

Example

“Halving”

Hidden layer: indicates component

“responsible for” Conditioned on assume independent or

Hidden MRF

Invariant under symmetry group of

Estimation Parameters: are currently nuisance

Algorithms: EM, VT - Viterbi Training (Extraction)

Current choice – VT. Simpler, computationally stable, …

VT Choose Obtain to

maximize Obtain to

maximize

Repeat until

Small scale/ exhaust search

Small scale/ - numerically, -

single DT - easy

Current efforts Truncated hemi-icosahedron, |V|/2=30

Comparative analysis (with traditional parametric and Bayesian approaches)

Interpretation of the hidden layer

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)

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)

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