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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


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



  • 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


DTI diffusion


[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



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

Illustration of the funk radon transform frt

FRT single ball



Illustration of the Funk-Radon Transform (FRT)

Diffusion Signal

Funk hecke theorem
Funk-Hecke Theorem single ball

[Funk 1916, Hecke 1918]

Recalling funk radon integral

Funk-Hecke ! single ball Problem: Delta function is discontinuous at 0 !

Recalling Funk-Radon integral

Solving the fr integral trick using a delta sequence

Funk-Hecke formula single ball

Delta sequence


Solving the FR integral:Trick using a delta sequence

Final analytical odf expression in sh coefficients
Final Analytical ODF expression in SH coefficients single ball

  • 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 single ball

[Campbell et al.

NeuroImage 05]


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 fasciculus

Extract ODF maxima

Extension to streamline FACT

Streamline tracking
Streamline Tracking fasciculus

  • 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 fasciculus

Principal tensor direction


ODF maxima

Limitations of DTI-FACT

Dti fact tracking
DTI-FACT Tracking fasciculus


Dti fact odf maxima
DTI-FACT + ODF maxima fasciculus


Principal odf fact tracking
Principal ODF FACT Tracking fasciculus


Multi modal odf fact
Multi-Modal ODF FACT fasciculus


Dti fact tracking1
DTI-FACT Tracking fasciculus


Principal odf fact tracking1
Principal ODF FACT Tracking fasciculus


Multi modal odf fact1
Multi-Modal ODF FACT fasciculus


Dti fact tracking2
DTI-FACT Tracking fasciculus



Dti fact tracking3
DTI-FACT Tracking fasciculus



Very low FA




Lower FA thresh

Principal odf fact tracking2
Principal ODF FACT Tracking fasciculus



Multi modal fact tracking
Multi-modal FACT Tracking fasciculus



Summary fasciculus

Signal S on the sphere

Spherical harmonic

description of S

Multi-Modal tracking


Fiber directions

Contributions advantages
Contributions & advantages fasciculus

  • 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 fasciculus

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 fasciculus

  • 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 fasciculus

  • Odyssée Tools Available

    • ODF Estimation, GFA Estimation

    • Odyssée Visualization

    • + more ODF and DTI applications…


  • Used by:

    • CMRR, University of Minnesota, USA

    • Hopital Pitié-Salpétrière, Paris

Thank you
Thank you! fasciculus

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.


-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 fasciculus

  • SH

  • SH PDE

  • Real

  • Modified basis

Trick to solve the fr integral
Trick to solve the FR integral fasciculus

  • Use a delta sequence n approximation of the delta function  in the integral

    • Many candidates: Gaussian of decreasing variance

  • Important property

N is a delta sequence
fasciculusn is a delta sequence




Nice trick
Nice trick! fasciculus



Funk hecke theorem1
Funk-Hecke Theorem fasciculus

  • 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 fasciculus

z = 1000


[Tuch; MRM04]

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

Funk-Radon ~= ODF

  • Funk-Radon Transform

  • True ODF

Synthetic data experiment
Synthetic Data Experiment fasciculus

  • Multi-Gaussian model for input signal

  • Exact ODF

Field of synthetic data

55 fasciculus crossing

b = 3000

Field of Synthetic Data

b = 1500

SNR 15

order 6

90 crossing

Odf evaluation
ODF evaluation fasciculus

Tuch reconstruction vs analytic reconstruction
Tuch reconstruction vs fasciculusAnalytic reconstruction

Analytic ODFs

Tuch ODFs

Difference: 0.0356 +- 0.0145

Percentage difference: 3.60% +- 1.44%


Human brain
Human Brain fasciculus

Analytic ODFs

Tuch ODFs

Difference: 0.0319 +- 0.0104

Percentage difference: 3.19% +- 1.04%


Time complexity
Time Complexity fasciculus

  • 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 fasciculus

  • 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 fasciculus

  • Tuch -> O(XYZNk)

  • Our analytic QBI -> O(XYZNR)

  • Factor ~15 speed up

Estimation of the adc

Estimation of the ADC fasciculus

Characterize multi-fiber diffusion

High order anisotropy measures

Apparent diffusion coefficient

Diffusion MRI signal : S(g) fasciculus

Apparent Diffusion Coefficient

ADC profile : D(g) = gTDg

In the hardi literature
In the HARDI literature… fasciculus

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 fasciculus

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 fasciculus

  • Generalization of the ADC,

    rank-2 D(g) = gTDg


General tensor

Independent elements

Dk of the tensor

[Ozarslan et al., mrm 2003]

Summary of algorithm
Summary of algorithm fasciculus


Harmonic (SH)


High Order

Diffusion Tensor



from linear-regression

Modified SH basis Yj

Least-squares with




C = (BTB + L)-1BTX

D = M-1C

[Descoteaux et al. MRM 56:2006]