a hidden polyhedral markov field model for diffusion mri alexey koloydenko division of statistics...
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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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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