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On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function. Luke Bloy 1 , Ragini Verma 2 The Section of Biomedical Image Analysis University of Pennsylvania Department of Bioengineering 1 Department of Radiology 2. Diffusion Tensor Imaging. - PowerPoint PPT Presentation
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On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution FunctionLuke Bloy1, Ragini Verma2
The Section of Biomedical Image AnalysisUniversity of PennsylvaniaDepartment of Bioengineering1
Department of Radiology2
Diffusion Tensor Imaging
DTI model is incapable of representing multiple orientations
Diffusion imaging rests on the assumption that the diffusion process correlates with the underlying tissue structure.
Diffusion Orientation Distribution Function
ODF: Approximates the radial projection of the diffusion propagator.
It essentially describes the probability that a water molecule will diffuse in a certain direction.
Its maxima have been shown to correspond with principle directions of the underlying diffusion process.
How to find the maxima of Orientation Distribution Function? Existing Methods:
Optimization Methods Spherical Newton’s method Powell’s method Need to ensure convergence Need to avoid small local maxima
Finite Difference Method Accuracy is limited by Mesh Size (accuracy of 4 degrees
requires 1280 mesh points
NEEDS REFS
Computing Maxima of the Diffusion Orientation Distribution Function
Our method:
•Represent ODF as symmetric Cartesian tensor
•Compute the stationary points of the ODF from a system of polynomial equations
• Classify the stationary points using the local curvature of the ODF graph into principle directions, secondary maxima, minima and inflection points.
Equivalence of Real Spherical Harmonic Expansion and Symmetric Cartesian Tensors
Real Symmetric Spherical Functions
Real Spherical Harmonics
Real Symmetric Cartesian Tensors
Orientation Distribution Function as a Cartesian Tensor
In spherical coordinates the from of Funk-Radon transform allows a the computation of the ODF RSH expansion in terms of the RSH expansion of the MRI signal.
Since M is a change of basis matrix it is invertible and the ODF tensor can be computed
Stationary points
Stationary points are points on the sphere ( ) where the derivative of the ODF vanishes. Using the tensor representation of the ODF, they are solutions to the following system of equations.
t is a solution to an lth order polynomial
Use the method of resultants to solve for v and u.
Classification of Stationary PointsUse the principle curvatures (k1, k2) to
classify each stationary point:
Minima Inflection Points
Principle Directions Secondary Maxima
Stationary points of the Orientation Distribution Function
Diffusion ODF reconstructions from simulated fiber populations performed with a rank 4 tensor. Red lines indicate principle directions, Blue minima, Black lines saddle points and green lines indicate secondary maxima.
One Fiber Two Fibers Three Fibers
Affect of Signal to Noise on Principle Direction Calculation
Single Tensor Model
b = 3000 sec /cm2
64 gradient direction
50 DWI signals, each with randomly chosen principle direction, at each SNR
SNR: 5,10,15,25,35,45
Angular Error = true calcPD ,PD )
Application to Clinically Feasible Data
3Tesla scanner
64 Gradient Directions
Single average
Scan time ~ 8 min
B = 1000 sec /cm2
CC : Corpus CollosumSCR : Superior Corona RadiataALIC : Anterior Limb of the Internal Capsule
Future Work
Implementation within fiber tracking framework
Investigation of geometric features (mean curvature/Gauss curvature) of the ODF surface as measures of diffusion anisotropy
Thanks…
Computing Principle Curvatures
Limitations of Diffusion Tensor ImagingSingle Single
FibersFibersMultiple Multiple
FibersFibers
Behrens et al, Neuroimage, 34 (1) Behrens et al, Neuroimage, 34 (1) 20072007
As many as 1/3 of white Matter voxels may be effected .
DTI model is incapable of representing multiple orientations
Real Spherical Harmonics of Even Order
Images of the first few?
Symmetric Cartesian Tensors
Ref Max.
Relationship between Anisotropy and Mean Curvature
Single tensor model
mean diffusivity of 700 mm2/sec
SNR = 35
Red line = absence of noise
False Positives Rates
SNR # of PDS
10 65%
15 92%
>25 100%
Equivalence of Real Spherical Harmonic Expansion and Symmetric Cartesian TensorsReal Symmetric Spherical
Functions Real Symmetric Cartesian TensorsReal Spherical
Harmonics