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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

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

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])

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

- 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]

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

- 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

High Angular Resolution Diffusion Imaging

Q-Space Imaging

Q-Ball Imaging

…

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

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

Diffusion profile

Diffusion Orientation

Distribution Function (ODF)

Fiber distribution

- 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

ODF can be computed directly from the HARDI signal over a single ball

Integral over the perpendicular equator

Funk-Radon Transform

[Tuch; MRM04]

FRT

->

ODF

Diffusion Signal

z = 1

z = 1000

J0(2z)

[Tuch; MRM04]

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

- Funk-Radon Transform
- True ODF

- 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

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

- 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]

- 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]

- Squared error between spherical function F and its smooth version on the sphere ∆bF
- SH obey the PDE
- We have,

- 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

- For any ()

S = [d1, d2, …, dN]

For l = 6,

C = [c1, c2 , …, c28]

Funk-Hecke Theorem

Solve the Funk-Radon integral

Delta sequence

[Funk 1916, Hecke 1918]

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

- 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

Delta sequence

=>

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

- 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

- Tuch ODF reconstruction:
- N = 90, k = 48-> rat data set
= 100, k = 51-> human brain

= 321, k = 90-> cat data set

- N = 90, k = 48-> rat 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

- Multi-Gaussian model for input signal
- Exact ODF

Multi-Gaussian model with SNR 35

Average difference

between exact ODF

and estimated ODF

b-value

55 crossing

b = 3000

b = 1500

SNR 15

order 6

90 crossing

Real Data Experiment

Biological phantom

Human Brain

[Campbell et al.

NeuroImage 05]

T1-weigthed

Diffusion tensors

Analytic ODFs

Tuch ODFs

Difference:0.0356 +- 0.0145

Percentage difference:3.60% +- 1.44%

[INRIA-McGill]

Analytic ODFs

Tuch ODFs

Difference:0.0319 +- 0.0104

Percentage difference:3.19% +- 1.04%

[INRIA-McGill]

FA map + diffusion tensors

ODFs

FA map + diffusion tensors

ODFs

FA map + diffusion tensors

ODFs

Discussion & Conclusion

S in Q-space

Physically meaningful

spherical harmonic

basis

Spherical harmonic

description of S

Analytic solution using

Funk-Hecke formula

ODF

Fiber directions

- 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

- 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

- 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

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

1)

2)

=>

3)

=>

- SH
- SH PDE
- Real
- Modified basis

- 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]

Classical DTI

rank-2 tensor

HARDI

ODF reconstruction