Higher Order Cartesian Tensor Representation of Orientation Distribution Functions (ODFs)

Transcription

1 Higher Order Cartesian Tensor Representation of Orientation Distribution Functions (ODFs) Yonas T. Weldeselassie (Ph.D. Candidate) Medical Image Computing and Analysis Lab, CS, SFU

2 DT-MR Imaging Introduction A medical imaging technique that produces in vivo images of biological tissues weighted with the local micro-structural characteristics of water diffusion. 2

3 DT-MR Imaging Image Acquisition At each voxel, With the presence of G acquire S i With the absence of G acquire S 0 3

4 Reconstruction DTI Model 2 nd Order Cartesian Tensor model Tanner-Stejskal Equation Result 3D image where at each voxel the direction of water diffusion is locally modeled by a Gaussian probability density function whose covariance matrix is a second order 3 3 symmetric positive definite matrix (diffusion tensor). 4

5 DT-MR Image Visualization Ellipsoidal Visualization Spectral decomposition of diffusion tensors D = V Λ V T, V = (v 1 v 2 v 3 ) Eigenvectors matrix Λ = diag(λ 1, λ 2, λ 3 ) Eigenvalues matrix Real and positive eigenvalues Eigenvalues measure amount of diffusion Eigenvectors measure orientation of diffusion Can be visualized as ellipsoids 5

6 Feature Extraction What information is contained in Diffusion Tensors? Anisotropy measure Diffusivity measure Neural connectivity 6

7 Working with Diffusion Tensor Images Segmentation Cluster similar tensors together Interpolation Question: How to measure similarity of diffusion tensors? Combine nearby diffusion tensors to interpolate at a given point Registration Segmentation, Interpolation and Registration Question: How to combine diffusion tensors? Align diffusion tensor images by minimizing the dissimilarity of diffusion tensors at every point Question: How to measure dissimilarity of diffusion tensors? Solution: Device Mathematical Tensor Distance Metric! 7

8 Diffusion Tensor Distance Metric Tensor Distance: Euclidean Consider diffusion tensors as vectors in R 9 Taking symmetry into account, why not R 6 8

9 Diffusion Tensor Distance Metric Tensor Distance: Euclidean Short comings: Norm of a matrix that is not necessarily a diffusion tensor Assumes the space of diffusion tensors forms vector space Interpolation results to tensor swelling effect 9

10 Motivation Diffusion Tensor Distance Metric Tensor Distance: J-divergence Diffusion of water molecule Brownian motion Brownian motion Gaussian distribution Diffusion Tensors Covariance of Gaussian distributions Therefore Similarity measure between Gaussian distributions Similarity measure between Diffusion Tensors 10

11 Motivation Diffusion Tensor Distance Metric Tensor Distance: Log-Euclidean Diffusion Tensors Symmetric Positive Definite Matrices (Do not form vector space) Matrix Logarithm Transformation Symmetric Matrices (Form vector space) Euclidean Distance 11

12 Motivation Diffusion Tensor Distance Metric Tensor Distance: Riemannian Metric The exponential map that maps symmetric matrices with Euclidean distance to symmetric positive definite matrices with a geodesic distance INCREASES DISTANCES Only for commuting matrices does the exponential map carry the line segment joining the logs of matrices to the geodesic joining the matrices 12

13 Contribution DT-MR Images Segmentation Graph Cuts Interactive, globally optimal and polynomial time solvable segmentation technique extended for DT-MR images using tensor similarity measures for connectivity information 13

14 Contribution Adaptive Seeding DT-MR Images Visualization with Streamtubes Automatic generating of seed points Regions with dissimilar tensors more seed points Culling without having to compute unnecessary streamtubes Faster visualization 14

15 Clinical Application Cortico-Striatal White Matter Tractography for Parkinson's Disease Changes in cortex far from primary sites of pathology in PD are investigated. In addition to FA and MD, vector field properties of eigenvectors are studied Controls PDs Mean (Standard deviation) p-value FA (0.058) (0.045) 0.06 MD (0.011) x (0.051) x Div (0.072) x (0.033) x Reduced FA (p = 0.06) and reduced MD (p = 0.03) are obtained in PD compared to the control subjects. Moreover increased Div (p = 0.10) is observed in PD compared to the control subjects. 15

16 Tensor Distance and Anisotropy Measures Novel Decomposition of Tensor Distance to Shape and Orientation Distances Motivation Existing tensor distance metrics do not elucidate Contribution of shape dissimilarity Contribution of orientation dissimilarity Existing tensor distance metrics are not useful Interpolation/regularization of tensors using only shape Interpolation/regularization of tensors using only orientation Could existing diffusion anisotropy measures be computed more appropriately using shape distance measure? 16

17 Tensor Distance and Anisotropy Measures Shape Distance Shape Distance Independent of orientation Given D 1 = V 1 Λ 1 V 1T and D 2 = V 2 Λ 2 V T 2 d SH (D 1, D 2 ) = d(d 1, D 2 ) when V 1 = V 2 Λ 1 = diag(λ 1,λ 2,λ 3 ) and Λ 1 = diag(μ 1,μ 2,μ 3 ) 17

18 Tensor Distance and Anisotropy Measures Motivation Shape Anisotropy (SA) Look at Fractional anisotropy (distance from closest isotropy) Aim Derived using Euclidean norms D iso is not at shortest geodesic Find the closest anisotropy using tensor distance 18

19 Tensor Distance and Anisotropy Measures Shape Anisotropy (SA) Comparison of SA with FA (Qualitative) 19

20 Tensor Distance and Anisotropy Measures Tissue type discrimination with SA Can SA better detect tissue groups than existing shape and anisotropy measures? 20

21 Higher Order Tensors Higher Order Tensor Representation of ODFs Shortcomings of 2 nd Order Tensors 21

22 Higher Order Tensors Higher Order Tensor Representation of ODFs Shortcomings of DTI Model (2 nd Order) It can only resolve a single fiber direction within each voxel 22

23 Higher Order Tensors Higher Order Tensor Representation of ODFs Shortcomings of DTI Model (2 nd Order) It can only resolve a single fiber direction within each voxel Yet, human cerebral white matter possesses considerable intravoxel structure at the millimeter resolution typical of MRI 23

24 Higher Order Tensors Higher Order Tensor Representation of ODFs Shortcomings of DTI Model (2 nd Order) It can only resolve a single fiber direction within each voxel Yet, human cerebral white matter possesses considerable intravoxel structure at the millimeter resolution typical of MRI Thus, DTI model is inadequate for resolving neural architecture in regions with complex fiber patterns 24

25 Higher Order Tensors Why Higher Order Tensor? Shortcomings of DTI Model (2 nd Order) It can only resolve a single fiber direction within each voxel Yet, human cerebral white matter possesses considerable intravoxel structure at the millimeter resolution typical of MRI Thus, DTI model is inadequate for resolving neural architecture in regions with complex fiber patterns 25

26 Higher Order Tensors (HOT) Why Higher Order Tensor? Shortcomings of DTI Model (2 nd Order) It can only resolve a single fiber direction within each voxel Yet, human cerebral white matter possesses considerable intravoxel structure at the millimeter resolution typical of MRI Thus, DTI model is inadequate for resolving neural architecture in regions with complex fiber patterns af, arcuate fasciculus; mog, middle occipital gyrus; or, optic radiation; os, occipital sulcus; scc, splenium of the corpus callosum; sog, superior occipital gyrus; ta, tapetum 26

27 Higher Order Tensors (HOT) Diffusivity Function The diffusivity d(g) is related to the DW-MR signal by HOT Model (ADC) DW-MR signal is antipodal symmetry, therefore the tensor is fully symmetric For 4 th order tensors, instead of 3 4 = 81, we get only 15 unique entries 27

28 Higher Order Tensors (HOT) Diffusivity Function Parameterization of Diffusivity Function 28

29 Higher Order Tensors (HOT) Diffusivity Function Parameterization of Diffusivity Function 29

30 Higher Order Tensors (HOT) Results Diffusivity Function 30

31 But then... Higher Order Tensors Higher Order Tensor ADCs and Deviation Angle Error 31

32 Higher Order Tensor ODFs Higher Order Tensor Representation of ODFs Use Standard Multi-fiber Reconstruction... 32

33 Higher Order Tensor ODFs Results Higher Order Tensor Representation of ODFs 33

34 Contribution... Higher Order Tensor ODFs Higher Order Tensor Representation of ODFs 34