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Analytic ODF Reconstruction and Validation in Q-Ball Imaging. Maxime Descoteaux 1 Work done with E. Angelino 2 , S. Fitzgibbons 2 , R. Deriche 1 1. Projet Odyssée, INRIA Sophia-Antipolis, France 2. Physics and Applied Mathematics, Harvard University, USA. McGill University, Jan 18th 2006.

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Analytic ODF Reconstruction and Validation in Q-Ball Imaging

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Analytic odf reconstruction and validation in q ball imaging

Analytic ODF Reconstruction and Validation in Q-Ball Imaging

Maxime Descoteaux1

Work done with E. Angelino2, S. Fitzgibbons2, R. Deriche1

1. Projet Odyssée, INRIA Sophia-Antipolis, France

2. Physics and Applied Mathematics, Harvard University, USA

McGill University, Jan 18th 2006

Plan of the talk

Plan of the talk



Analytic ODF reconstruction





Cerebral anatomy

Basics of diffusion MRI

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

Principal direction of dti

Principal direction of DTI

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

Q-Space Imaging

Q-Ball Imaging

High angular resolution diffusion imaging hardi

High Angular Resolution Diffusion Imaging (HARDI)

162 points

642 points

  • N gradient directions

  • We want to recover fiber crossings

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

High order reconstruction

High Order Reconstruction

  • We seek a spherical function that has maxima that agree with underlying fibers

Diffusion profile

Diffusion Orientation

Distribution Function (ODF)

Fiber distribution

Diffusion orientation distribution function odf

Diffusion Orientation Distribution Function (ODF)

  • Method to reconstruct the ODF

  • Diffusion spectrum imaging (DSI)

    • Sample signal for many q-ball and many directions

    • Measured signal = FourierTransform[P]

    • Compute 3D inverse fourier transform -> P

    • Integrate the radial component of P -> ODF

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]

[Tuch; MRM04]

Illustration of the funk radon transform frt




Illustration of the Funk-Radon Transform (FRT)

Diffusion Signal

Funk radon odf

z = 1

z = 1000


[Tuch; MRM04]

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

Funk-Radon ~= ODF

  • Funk-Radon Transform

  • True ODF

My contributions

My Contributions

  • The Funk-Radon can be solved ANALITICALLY

    • Spherical harmonics description of the signal

    • One step matrix multiplication

  • Validation against ground truth evidence

    • Rat phantom

    • Knowledge of brain anatomy

  • Validation and Comparison against Tuch reconstruction

[collaboration with McGill]

Analytic odf reconstruction

Analytic ODF Reconstruction

Spherical harmonic description

Funk-Hecke Theorem

Sketch of the approach

Sketch of the approach

S in Q-space

Physically meaningful

spherical harmonic


For l = 6,

C = [c1, c2 , …, c28]

Spherical harmonic

description of S

Analytic solution using

Funk-Hecke formula


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. SPIE-MI 06]

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 m = (1/2)(order + 1)(order + 2)

    B : discrete SH, Yj(m x N

    (N diffusion gradients and m 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

Sh description of the signal

SH description of the signal

  • For any ()

S = [d1, d2, …, dN]

For l = 6,

C = [c1, c2 , …, c28]

Funk hecke theorem

Funk-Hecke Theorem

Solve the Funk-Radon integral

Delta sequence

Funk hecke theorem1

Funk-Hecke Theorem

[Funk 1916, Hecke 1918]

Recalling funk radon integral

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

Recalling Funk-Radon integral

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

(if time, proof)

Solving the fr integral

Funk-Hecke formula

Delta sequence


Solving the FR integral

Final analytic odf expression

Final Analytic ODF expression

(if time bigO analysis with Tuch’s ODF reconstruction)

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

Validation and results

Validation and Results

Synthetic dataBiological rat spinal chords phantom

Human brain

Synthetic data experiment

Synthetic Data Experiment

Synthetic data experiment1

Synthetic Data Experiment

  • Multi-Gaussian model for input signal

  • Exact ODF

Strong agreement

Strong Agreement

Multi-Gaussian model with SNR 35

Average difference

between exact ODF

and estimated ODF


Field of synthetic data

55 crossing

b = 3000

Field of Synthetic Data

b = 1500

SNR 15

order 6

90 crossing

Real data experiment

Real Data Experiment

Biological phantom

Human Brain

Biological phantom

Biological phantom

[Campbell et al.

NeuroImage 05]


Diffusion tensors

Tuch reconstruction vs analytic reconstruction

Tuch reconstruction vsAnalytic reconstruction

Analytic ODFs

Tuch ODFs

Difference:0.0356 +- 0.0145

Percentage difference:3.60% +- 1.44%


Human brain

Human Brain

Analytic ODFs

Tuch ODFs

Difference:0.0319 +- 0.0104

Percentage difference:3.19% +- 1.04%


Genu of the corpus callosum frontal gyrus fibers

Genu of the corpus callosum - frontal gyrus fibers

FA map + diffusion tensors


Corpus callosum corona radiata superior longitudinal

Corpus callosum - corona radiata - superior longitudinal

FA map + diffusion tensors


Corona radiata diverging fibers longitudinal fasciculus

Corona radiata diverging fibers - longitudinal fasciculus

FA map + diffusion tensors


Discussion conclusion

Discussion & Conclusion



S in Q-space

Physically meaningful

spherical harmonic


Spherical harmonic

description of S

Analytic solution using

Funk-Hecke formula


Fiber directions

Advantages of our approach

Advantages of our approach

  • Analytic ODF reconstruction

    • Discrete interpolation/integration is eliminated

  • Solution for all directions is obtained in a single step

  • Faster than Tuch’s numerical approach

  • Output is a spherical harmonic description which has powerful properties

Spherical harmonics properties

Spherical harmonics properties

  • Can use funk-hecke formula to obtain analytic integrals of inner products

    • Funk-radon transform, deconvolution

  • Laplacian is very simple

    • Application to smoothing, regularization, sharpening

  • Inner product

    • Comparison between spherical functions

What s next

What’s next?

  • Tracking fibers!

  • Can it be done properly from the diffusionODF?

  • Can we obtain a transformation between the input signal and the fiber ODF using spherical harmonics

Thank you

Thank you!

Key references:

  • Maxime.Descoteaux/index.en.html

  • Tuch D. Q-Ball Imaging, MRM 52, 2004

    Thanks to:

    P. Savadjiev, J. Campbell, B. Pike, K. Siddiqi

N is a delta sequence

n is a delta sequence




Nice trick

Nice trick!



Spherical harmonics

Spherical Harmonics

  • SH

  • SH PDE

  • Real

  • Modified basis

Funk hecke theorem2

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]

Limitations of classical dti

Classical DTI

rank-2 tensor


ODF reconstruction

Limitations of classical DTI

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