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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

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

Introduction

Background

Analytic ODF reconstruction

Results

Discussion


Introduction

Introduction

Cerebral anatomy

Basics of diffusion MRI


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]


Background

Background

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

FRT single ball

->

ODF

Illustration of the Funk-Radon Transform (FRT)

Diffusion Signal


Funk radon odf

z = 1 single ball

z = 1000

J0(2z)

[Tuch; MRM04]

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

Funk-Radon ~= ODF

  • Funk-Radon Transform

  • True ODF


My contributions
My Contributions single ball

  • 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 single ball

Spherical harmonic description

Funk-Hecke Theorem


Sketch of the approach
Sketch of the approach single ball

S in Q-space

Physically meaningful

spherical harmonic

basis

For l = 6,

C = [c1, c2 , …, c28]

Spherical harmonic

description of S

Analytic solution using

Funk-Hecke formula

ODF


Spherical harmonics formulation
Spherical harmonics single ballformulation

  • 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 single ball

  • 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 ∆ single ballb

  • Squared error between spherical function F and its smooth version on the sphere ∆bF

  • SH obey the PDE

  • We have,


Minimization regularization
Minimization & single ball 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 single ball

  • For any ()

S = [d1, d2, …, dN]

For l = 6,

C = [c1, c2 , …, c28]


Funk hecke theorem

Funk-Hecke Theorem single ball

Solve the Funk-Radon integral

Delta sequence


Funk hecke theorem1
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


Trick to solve the fr integral
Trick to solve the FR integral single ball

  • 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 single ball

Delta sequence

=>

Solving the FR integral


Final analytic odf expression
Final Analytic ODF expression single ball

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


Time complexity
Time Complexity single ball

  • 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 single ball

  • 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 single ball

Synthetic dataBiological rat spinal chords phantom

Human brain



Synthetic data experiment1
Synthetic Data Experiment single ball

  • Multi-Gaussian model for input signal

  • Exact ODF


Strong agreement
Strong Agreement single ball

Multi-Gaussian model with SNR 35

Average difference

between exact ODF

and estimated ODF

b-value


Field of synthetic data

55 single ball crossing

b = 3000

Field of Synthetic Data

b = 1500

SNR 15

order 6

90 crossing


Real data experiment

Real Data Experiment single ball

Biological phantom

Human Brain


Biological phantom
Biological phantom single ball

[Campbell et al.

NeuroImage 05]

T1-weigthed

Diffusion tensors


Tuch reconstruction vs analytic reconstruction
Tuch reconstruction vs single ballAnalytic reconstruction

Analytic ODFs

Tuch ODFs

Difference: 0.0356 +- 0.0145

Percentage difference: 3.60% +- 1.44%

[INRIA-McGill]


Human brain
Human Brain single ball

Analytic ODFs

Tuch ODFs

Difference: 0.0319 +- 0.0104

Percentage difference: 3.19% +- 1.04%

[INRIA-McGill]


Genu of the corpus callosum frontal gyrus fibers
Genu of the corpus callosum - frontal gyrus fibers single ball

FA map + diffusion tensors

ODFs


Corpus callosum corona radiata superior longitudinal
Corpus callosum - corona radiata - superior longitudinal single ball

FA map + diffusion tensors

ODFs


Corona radiata diverging fibers longitudinal fasciculus
Corona radiata diverging fibers - longitudinal fasciculus single ball

FA map + diffusion tensors

ODFs



Summary
Summary single ball

S in Q-space

Physically meaningful

spherical harmonic

basis

Spherical harmonic

description of S

Analytic solution using

Funk-Hecke formula

ODF

Fiber directions


Advantages of our approach
Advantages of our approach single ball

  • 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 single ball

  • 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? single ball

  • 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! single ball

Key references:

  • http://www-sop.inria.fr/odyssee/team/ 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
single balln is a delta sequence

1)

2)

=>


Nice trick
Nice trick! single ball

3)

=>


Spherical harmonics
Spherical Harmonics single ball

  • SH

  • SH PDE

  • Real

  • Modified basis


Funk hecke theorem2
Funk-Hecke Theorem single ball

  • 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 single ball

rank-2 tensor

HARDI

ODF reconstruction

Limitations of classical DTI


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