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 20
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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.
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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, 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