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, France2. Physics and Applied Mathematics, Harvard University, USA

McGill University, Jan 18th 2006

Plan of the talk

Introduction

Background

Analytic ODF reconstruction

Results

Discussion

Introduction

Cerebral anatomy

Basics of diffusion MRI

Short and long association fibers in the right hemisphere

([Williams-etal97])

Brain white matter connections

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

Cerebral Anatomy

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

Diffusion profile : qTDqDiffusion MRI signal : S(q)

• Brownian motion P of water molecules can be described by a Gaussian diffusion process characterized by rank-2 tensor D (3x3 symmetric positive definite)

DTI-->

Principal direction of DTI

• DTI fails in the presence of many principal directions of different fiber bundles within the same voxel

• Non-Gaussian diffusion process

Limitation of classical DTI

[Poupon, PhD thesis]True diffusion

profileDTI diffusion

profile

Background

High Angular Resolution Diffusion Imaging

Q-Space Imaging

Q-Ball Imaging

…

High Angular Resolution Diffusion Imaging (HARDI)

• N gradient directions • We want to recover fiber crossings

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

162 points 642 points

High Order Reconstruction

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

Diffusion profileFiber distribution Diffusion OrientationDistribution 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

[Tuch; MRM04]

Illustration of the Funk-Radon Transform (FRT)

Diffusion Signal

FRT->

ODF

• Funk-Radon Transform

• True ODF

Funk-Radon ~= ODF

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

J0(2z)

z = 1 z = 1000

[Tuch; MRM04]

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

Spherical harmonic description

Funk-Hecke Theorem

Sketch of the approachS in Q-space

Spherical harmonic

description of S

ODF

Physically meaningfulspherical harmonicbasis

Analytic solution usingFunk-Hecke formula

For l = 6,

C = [c1, c2 , …, c28]

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

• In matrix form, S = C*BS : discrete HARDI data 1 x NC : SH coefficients 1 x m = (1/2)(order + 1)(order + 2)

B : discrete SH, Yjm 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

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

• SH obey the PDE

• We have,

Minimization & regularization

• Minimize (CB - S)T(CB - S) + CTLC

=>C = (BTB + L)-1 BTS

• 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

For l = 6,

C = [c1, c2 , …, c28]

S = [d1, d2, …, dN]

SH description of the signal

• For any ()

Funk-Hecke Theorem

Solve the Funk-Radon integral

Delta sequence

Funk-Hecke Theorem

[Funk 1916, Hecke 1918]

Recalling Funk-Radon integral

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

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)

Funk-Hecke formula

Solving the FR integral

=>

Delta sequence

Final Analytic ODF expression

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

Time Complexity

• Input HARDI data |x|,|y|,|z|,N• Tuch ODF reconstruction:

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

(8N) : interpolation point

k := (8N) • Analytic ODF reconstruction

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

R := SH elements in basis

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

Synthetic dataBiological rat spinal chords phantom

Human brain

Synthetic Data Experiment

Synthetic Data Experiment

• Multi-Gaussian model for input signal

• Exact ODF

Strong Agreement

b-value

Average differencebetween exact ODFand estimated ODF

Multi-Gaussian model with SNR 35

Field of Synthetic Data

90 crossing

b = 1500SNR 15order 6

55 crossingb = 3000

Real Data Experiment

Biological phantom

Human Brain

Biological phantom

T1-weigthed Diffusion tensors

[Campbell et al.NeuroImage 05]

Tuch reconstruction vsAnalytic reconstruction

Tuch ODFs Analytic ODFs

Difference: 0.0356 +- 0.0145Percentage difference: 3.60% +- 1.44%

[INRIA-McGill]

Human Brain

Tuch ODFs Analytic ODFs

Difference: 0.0319 +- 0.0104Percentage difference: 3.19% +- 1.04%

[INRIA-McGill]

Genu of the corpus callosum - frontal gyrus fibers

FA map + diffusion tensors ODFs

Corpus callosum - corona radiata - superior longitudinal

FA map + diffusion tensors ODFs

Corona radiata diverging fibers - longitudinal fasciculus

FA map + diffusion tensors ODFs

Discussion & Conclusion

SummaryS in Q-space

Spherical harmonic

description of S

ODF

Physically meaningfulspherical harmonicbasis

Analytic solution usingFunk-Hecke formula

Fiber directions

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

• 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?

• Tracking fibers!

• Can it be done properly from the diffusion ODF?

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

Thank you!

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

=>

1)

2)

3)

Nice trick!

=>

Spherical Harmonics

• SH

• SH PDE

• Real• Modified basis

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 DTIrank-2 tensor

HARDIODF reconstruction