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Processing HARDI Data to Recover Crossing Fibers. Maxime Descoteaux PhD student Advisor: Rachid Deriche Odyssée Laboratory, INRIA/ENPC/ENS, INRIA Sophia-Antipolis, France. Plan of the talk. Introduction of HARDI data Spherical Harmonics Estimation of HARDI

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processing hardi data to recover crossing fibers

Processing HARDI Data to Recover Crossing Fibers

Maxime Descoteaux

PhD student

Advisor: Rachid Deriche

Odyssée Laboratory, INRIA/ENPC/ENS,

INRIA Sophia-Antipolis, France

plan of the talk

Plan of the talk

Introduction of HARDI data

Spherical Harmonics Estimation of HARDI

Q-Ball Imaging and ODF Estimation

Multi-Modal Fiber Tracking

brain white matter connections
Brain white matter connections

Short and long association fibers in the right hemisphere

([Williams-etal97])

cerebral anatomy
Cerebral Anatomy

Radiations of the corpus callosum ([Williams-etal97])

diffusion mri recalling the basics
Diffusion MRI: recalling the basics
  • Brownian motion or average PDF of water molecules is along white matter fibers
  • Signal attenuation proportional to average diffusion

in a voxel

[Poupon, PhD thesis]

classical dti model
Classical DTI model

DTI

-->

  • Brownian motion P of water molecules can be described by a Gaussian diffusion

process characterized by rank-2 tensor D (3x3 symmetric positive definite)

Diffusion MRI signal : S(q)

Diffusion profile : qTDq

limitation of classical dti
Limitation of classical DTI
  • DTI fails in the presence of many principal directions of different fiber bundles within the same voxel
  • Non-Gaussian diffusion process

True diffusion

profile

DTI diffusion

profile

[Poupon, PhD thesis]

high angular resolution diffusion imaging hardi
High Angular Resolution Diffusion Imaging (HARDI)

162 points

252 points

  • N gradient directions
  • We want to recover fiber crossings

Solution: Process all discrete noisy samplings on the sphere using high order formulations

[Wedeen, Tuch et al 1999]

our contributions
Our Contributions
  • New regularized spherical harmonic estimation of the HARDI signal
  • New approach for fast and analytical ODF reconstruction in Q-Ball Imaging
  • New multi-modal fiber tracking algorithm
sketch of the approach
Sketch of the approach

Data on the sphere

For l = 6,

C = [c1, c2 , …, c28]

Spherical harmonic

description of data

ODF

ODF

spherical harmonic estimation of the signal

Spherical Harmonic Estimation of the Signal

Description of discrete data on the sphere

Physically meaningful spherical harmonic basis

Regularization of the coefficients

spherical harmonics formulation
Spherical harmonicsformulation
  • Orthonormal basis for complex functions on the sphere
  • Symmetric when order l is even
  • We define a real and symmetric modified basis Yj such that the signal

[Descoteaux et al. MRM 56:2006]

spherical harmonics sh coefficients
Spherical Harmonics (SH) coefficients
  • In matrix form, S = C*B

S : discrete HARDI data 1 x N

C : SH coefficients 1 x R = (1/2)(order + 1)(order + 2)

B : discrete SH, Yj(R x N

(N diffusion gradients and R SH basis elements)

  • Solve with least-square

C = (BTB)-1BTS

[Brechbuhel-Gerig et al. 94]

regularization with the laplace beltrami b
Regularization with the Laplace-Beltrami ∆b
  • Squared error between spherical function F and its smooth version on the sphere ∆bF
  • SH obey the PDE
  • We have,
minimization regularization
Minimization & -regularization
  • Minimize

(CB - S)T(CB - S) + CTLC

=>

C = (BTB + L)-1BTS

  • Find best  with L-curve method
  • Intuitively,  is a penalty for having higher order terms in the modified SH series

=> higher order terms only included when needed

effect of regularization
Effect of regularization

[Descoteaux et al., MRM 06]

 = 0

With Laplace-Beltrami regularization

fast analytical odf estimation

Fast Analytical ODF Estimation

Q-Ball Imaging

Funk-Hecke Theorem

Fiber detection

q ball imaging qbi tuch mrm04
ODF can be computed directly from the HARDI signal over a single ball

Integral over the perpendicular equator = Funk-Radon Transform

Q-Ball Imaging (QBI) [Tuch; MRM04]

ODF ->

[Tuch; MRM04]

~= ODF

funk hecke theorem
Funk-Hecke Theorem

[Funk 1916, Hecke 1918]

final analytical odf expression in sh coefficients
Final Analytical ODF expression in SH coefficients
  • Fast: speed-up factor of 15 with classical QBI
  • Validated against ground truth and classical QBI

[Descoteaux et al. ISBI 06 & HBM 06]

biological phantom
Biological phantom

[Campbell et al.

NeuroImage 05]

T1-weigthed

Diffusion tensors

corpus callosum corona radiata superior longitudinal fasciculus
Corpus callosum - corona radiata - superior longitudinal fasciculus

FA map + diffusion tensors

ODF + maxima

multi modal fiber tracking

Multi-Modal Fiber Tracking

Extract ODF maxima

Extension to streamline FACT

streamline tracking
Streamline Tracking
  • FACT: Fiber Assignment by Continuous Tracking
    • Follow principal eigenvector of diffusion tensor
    • Stop if FA < thresh and if curving angle > 

typically thresh = 0.15 and = 45 degrees)

  • Limited and incorrect in regions of fiber crossing
  • Used in many clinical applications

[Mori et al, 1999

Conturo et al, 1999,

Basser et al 2000]

limitations of dti fact

Classical DTI

Principal tensor direction

HARDI

ODF maxima

Limitations of DTI-FACT
dti fact tracking2
DTI-FACT Tracking

start->

start->

dti fact tracking3
DTI-FACT Tracking

start->

start->

Very low FA

threshold

<-start

start->

Lower FA thresh

summary
Summary

Signal S on the sphere

Spherical harmonic

description of S

Multi-Modal tracking

ODF

Fiber directions

contributions advantages
Contributions & advantages
  • Regularized spherical harmonic (SH) description of the signal
  • Analytical ODF reconstruction
    • Solution for all directions in a single step
    • Faster than classical QBI by a factor 15
  • SH description has powerful properties
    • Easy solution to : Laplace-Beltrami smoothing, inner products, integrals on the sphere
    • Application for sharpening, deconvolution, etc…
contributions advantages1
Contributions & advantages

4) Tracking using ODF maxima = Generalized FACT algorithm

=> Overcomes limitations of FACT from DTI

  • Principal ODF direction
    • Does not follow wrong directions in regions of crossing
  • Multi-modal ODF FACT
    • Can deal with fanning, branching and crossing fibers
perspectives
Perspectives
  • Multi-modal tracking in the human brain
  • Tracking with geometrical information from locally supporting neighborhoods
    • Local curvature and torsion information
    • Better label sub-voxel configurations like bottleneck, fanning, merging, branching, crossing
  • Consider the full diffusion ODF in the tracking and segmentation
    • Probabilistic tracking from full ODF

[Savadjiev & Siddiqi et al. MedIA 06], [Campbell & Siddiqi et al. ISBI 06]

brainvisa anatomist
BrainVISA/Anatomist
  • Odyssée Tools Available
    • ODF Estimation, GFA Estimation
    • Odyssée Visualization
    • + more ODF and DTI applications…
  • http://brainvisa.info/
  • Used by:
      • CMRR, University of Minnesota, USA
      • Hopital Pitié-Salpétrière, Paris
thank you
Thank you!

Thanks to collaborators:

C. Lenglet, M. La Gorce, E. Angelino, S. Fitzgibbons, P. Savadjiev,

J. Campbell, B. Pike, K. Siddiqi, A. Andanwer

Key references:

-Descoteaux et al, ADC Estimation and Applications, MRM 56, 2006.-Descoteaux et al, A Fast and Robust ODF Estimation Algorithm in Q-Ball Imaging, ISBI 2006.

-http://www-sop.inria.fr/odyssee/team/Maxime.Descoteaux/index.en.html

-Ozarslan et al., Generalized tensor imaging and analytical relationships between diffusion tensor and HARDI, MRM 2003.

-Tuch, Q-Ball Imaging, MRM 52, 2004

spherical harmonics
Spherical Harmonics
  • SH
  • SH PDE
  • Real
  • Modified basis
trick to solve the fr integral
Trick to solve the FR integral
  • Use a delta sequence n approximation of the delta function  in the integral
    • Many candidates: Gaussian of decreasing variance
  • Important property
funk hecke theorem1
Funk-Hecke Theorem
  • Key Observation:
    • Any continuous function f on [-1,1] can be extended to a continous function on the unit sphere g(x,u) = f(xTu), where x, u are unit vectors
  • Funk-Hecke thm relates the inner product of any spherical harmonic and the projection onto the unit sphere of any function f conitnuous on [-1,1]
funk radon odf

z = 1

z = 1000

J0(2z)

[Tuch; MRM04]

(WLOG, assume u is on the z-axis)

Funk-Radon ~= ODF
  • Funk-Radon Transform
  • True ODF
synthetic data experiment
Synthetic Data Experiment
  • Multi-Gaussian model for input signal
  • Exact ODF
field of synthetic data

55 crossing

b = 3000

Field of Synthetic Data

b = 1500

SNR 15

order 6

90 crossing

tuch reconstruction vs analytic reconstruction
Tuch reconstruction vsAnalytic reconstruction

Analytic ODFs

Tuch ODFs

Difference: 0.0356 +- 0.0145

Percentage difference: 3.60% +- 1.44%

[INRIA-McGill]

human brain
Human Brain

Analytic ODFs

Tuch ODFs

Difference: 0.0319 +- 0.0104

Percentage difference: 3.19% +- 1.04%

[INRIA-McGill]

time complexity
Time Complexity
  • Input HARDI data |x|,|y|,|z|,N
  • Tuch ODF reconstruction:

O(|x||y||z| N k)

(8N) : interpolation point

k := (8N)

  • Analytic ODF reconstruction

O(|x||y||z| N R)

R := SH elements in basis

time complexity comparison
Time Complexity Comparison
  • Tuch ODF reconstruction:
    • N = 90, k = 48 -> rat data set

= 100 , k = 51 -> human brain

= 321, k = 90 -> cat data set

  • Our ODF reconstruction:
    • Order = 4, 6, 8 -> m = 15, 28, 45

=> Speed up factor of ~3

time complexity experiment
Time complexity experiment
  • Tuch -> O(XYZNk)
  • Our analytic QBI -> O(XYZNR)
  • Factor ~15 speed up
estimation of the adc

Estimation of the ADC

Characterize multi-fiber diffusion

High order anisotropy measures

in the hardi literature
In the HARDI literature…

2 class of high order ADC fitting algorithms:

  • Spherical harmonic (SH) series

[Frank 2002, Alexander et al 2002, Chen et al 2004]

  • High order diffusion tensor (HODT)

[Ozarslan et al 2003, Liu et al 2005]

high order diffusion tensor hodt generalization
High order diffusion tensor (HODT) generalization

Rank l = 2 3x3

D = [ Dxx Dyy Dzz Dxy Dxz Dyz ]

Rank l = 4 3x3x3x3

D = [ Dxxxx Dyyyy Dzzzz Dxxxy Dxxxz Dyzzz Dyyyz Dzzzx Dzzzy Dxyyy Dxzzz Dzyyy Dxxyy Dxxzz Dyyzz ]

tensor generalization of adc
Tensor generalization of ADC
  • Generalization of the ADC,

rank-2 D(g) = gTDg

rank-l

General tensor

Independent elements

Dk of the tensor

[Ozarslan et al., mrm 2003]

summary of algorithm
Summary of algorithm

Spherical

Harmonic (SH)

Series

High Order

Diffusion Tensor

(HODT)

HODT D

from linear-regression

Modified SH basis Yj

Least-squares with

-regularization

M

transformation

C = (BTB + L)-1BTX

D = M-1C

[Descoteaux et al. MRM 56:2006]

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