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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

Brain white matter connections

Short and long association fibers in the right hemisphere

([Williams-etal97])

Cerebral Anatomy

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

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

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

- 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]

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

- 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)

- 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

Q-Ball Imaging (QBI) [Tuch; MRM04][Tuch; MRM04]

z = 1000

J0(2z)

[Tuch; MRM04]

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

Funk-Radon ~= ODF- Funk-Radon Transform
- True ODF

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]

Sketch of the approach

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 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*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

- 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)-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

Funk-Hecke Theorem

[Funk 1916, Hecke 1918]

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)

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

Synthetic Data Experiment

- Multi-Gaussian model for input signal
- Exact ODF

Strong Agreement

Multi-Gaussian model with SNR 35

Average difference

between exact ODF

and estimated ODF

b-value

Tuch reconstruction vsAnalytic reconstruction

Analytic ODFs

Tuch ODFs

Difference: 0.0356 +- 0.0145

Percentage difference: 3.60% +- 1.44%

[INRIA-McGill]

Human Brain

Analytic ODFs

Tuch ODFs

Difference: 0.0319 +- 0.0104

Percentage difference: 3.19% +- 1.04%

[INRIA-McGill]

Summary

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

- 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 diffusionODF?
- 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

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]

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