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Algorithmes de traitement dimage pour lestimation des caract eristiques locales de la diffusion. Orateur : D. Tschumperl e Travail en collaboration avec Haz-Edine Assemlal and Luc Brun GREYC-ENSICAEN (CNRS UMR 6072), CAEN, FRANCE

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SLIDE 1

Algorithmes de traitement d’image pour l’estimation des caract´ eristiques locales de la diffusion.

Orateur : D. Tschumperl´ e Travail en collaboration avec Haz-Edine Assemlal and Luc Brun

GREYC-ENSICAEN (CNRS UMR 6072), CAEN, FRANCE

JIRNFI’2009, Septembre 2009

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 1 / 68

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SLIDE 2

Outline

1

Context

2

DTI model : Diffusion Tensor Imaging

3

QBI model : Q-Ball Imaging

4

Measuring the PDF : DSI and Multi Q-Ball

5

Conclusions & Perspectives

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 2 / 68

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SLIDE 3

Diffusion MRI: Brownian Motion

a: Brownian motion of water molecules b: Constraints due to body internal structures c: Signal = Local macroscopic diffusion

Objective

To estimate the probability P of the macroscopic displacement p during a diffusion time dt.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 3 / 68

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SLIDE 4

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 5

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 6

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 7

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 8

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 9

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 10

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 11

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 12

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 13

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 14

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 15

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 16

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 17

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 4 / 68

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SLIDE 18

Q-Space

MRI-based image modality is able to measure water diffusion within tissues. Acquisition of several raw images under different magnetic field magnitudes and orientations (Q-space).

(a) Acquisition in q-space on a sphere of radius ||q||: a baseline image S0 and several other gradients images S(q). (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 5 / 68

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SLIDE 19

1

Context

2

DTI model : Diffusion Tensor Imaging

3

QBI model : Q-Ball Imaging

4

Measuring the PDF : DSI and Multi Q-Ball

5

Conclusions & Perspectives

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 6 / 68

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SLIDE 20

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 7 / 68

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SLIDE 21

2nd-order Diffusion tensors

A volume of Diffusion Tensors can be estimated from these raw images. Diffusion tensors represents Gaussian models of the water diffusion in the voxels, and are 3x3 symmetric and positive-definite matrices. Representation of a DTI reconstructed image with a volume of ellipsoids :

(a) Diffusion tensor field, giving a Gaussian diffusion model on each voxel. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 8 / 68

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SLIDE 22

Diffusion tensors for fibertracking

Diffusion MRI images give structural informations about fiber networks within tissues. Fiber tractography can be performed by tracking the principal diffusion directions. Raise a lot of image processing problems : Estimation of diffusion profiles, Regularization (Raw images are noisy), Visualization.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 9 / 68

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SLIDE 23

Diffusion MRI: fiber-tracking

Fiber-tracking through human tissues (brain white matter) is usually done by following the principal eigen vector of tensors

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 10 / 68

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SLIDE 24

DTI : The Stejskal-Tanner Equation

Consider D(x,y,z) =   a b c b d e c e f  , the diffusion tensor modeling the local Gaussian diffusion at voxel (x, y, z). Measured (raw) images Sk are related to the projection of the tensors D(x,y,z) along a particular gradient orientation qk : ∀(x, y, z), Sk(x,y,z) = S0(x,y,z) e−b qT

k D(x,y,z) qk

At least 7 images are needed to get an estimation of D.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 11 / 68

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DTI Estimation : A Direct Approach

Method proposed by [Westin:2002], using a dual tensor basis.

Direct estimation :

D =

6

  • k=1

< D, gkgT

k > gkgT k = 6

  • k=1

−1 b ln Sk S0

  • gkgT

k

⊕ At least six non-colinear diffusion gradient directions, i.e 7 images are then theoretically sufficient. ⊕ Real-time. ⊖ Positive definiteness of the tensors is not insured. ⊖ Not robust to noise.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 12 / 68

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SLIDE 26

DTI Estimation : Minimizing Least Square Error

When n > 7 raw images are available, D can be retrieved by a least square minimization method : min

D n

  • k=1
  • −1

b ln Sk S0

  • − gT

k Dgk

2 ⇒ minX AX − b2 ˆ X = (ATA)−1ATb ⊕ Least square methods are more robust to noise, by using all the raw image informations. ⊖ Positive definiteness of the tensors is not insured. ⇒ Reprojection of the tensors into the positive tensor space is needed by both estimation methods.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 13 / 68

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DTI Estimation : Variational Approach

Minimizing the following criterion, in the constrained positive tensor space : min

D∈P(3)

  • Ω

n

  • k=1

ψ

  • ln

S0 Sk

  • − gT

k Dgk

  • + α φ(∇D) dΩ

The corresponding gradient descent that respect the positive-definite property of the tensors is (Euler-Lagrange) :    T(t=0) = Id

∂T ∂t = (G + GT)T2 + T2(G + GT)

where G corresponds to the unconstrained velocity matrix defined as : Gi,j = n

k=1 ψ

′(|vk|)sign(vk)

  • gkgT

k

  • i,j + αdiv
  • φ

′(∇T)

∇T

∇Ti,j

  • ,

with vk = ln

  • S0

S

  • − gT

k Tgk.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 14 / 68

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SLIDE 28

DTI Estimation : Variational Approach

Comparison results with synthetic datasets :

(a) Partial set of noisy raw images Sk (in 6 different gradient directions) (b) True tensor field (c) Direct estimation (d) LS estimation (e) Variational method (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 15 / 68

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SLIDE 29

Effect on fiber-tracking

(a) Mean diffusivity (left) and Fractional Anisotropy (right) (b) LS estimation + fibers (c) Variational estimation + fibers (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 16 / 68

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Fiber scale space

Tensor (left) & Fibers (right) Variational estimation with α = 0.1 Variational estimation with α = 0.3. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 17 / 68

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SLIDE 31

Fiber scale space

Tensor (left) & Fibers (right) Variational estimation with α = 0.1 Variational estimation with α = 1. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 18 / 68

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SLIDE 32

Limits of Tensor Models

When n >> 7 images are available, there is an average of diffusion orientations for each voxel. Gaussian diffusion models cannot handle the crossing problem.

(a) DTI are sufficient when fibers are mainly monodirectional (b) DTI fails at modeling fiber crossing (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 19 / 68

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SLIDE 33

Limits of Tensor Models

In case of crossing, the “best” case happens when the directions are

  • rthogonals, so the fibertracking algorithm may stop.

When crossing directions are more close, fibertracking directions just go wrong.

(b) DTI fails at modeling fiber crossing (from Descoteaux-etal:05) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 20 / 68

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SLIDE 34

1

Context

2

DTI model : Diffusion Tensor Imaging

3

QBI model : Q-Ball Imaging

4

Measuring the PDF : DSI and Multi Q-Ball

5

Conclusions & Perspectives

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 21 / 68

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SLIDE 35

Signal acquisition

Figure: Diffusion MRI acquisition steps.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 22 / 68

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SLIDE 36

DTI versus higher order models

Fibers Raw data DTI 2nd-order tensor [Basser94] higher orders approaches Numerous approaches exist: GDTI [Liu03], PASMRI [Jansons03], HODT / DOT [¨ Ozarslan03,06], Spherical Deconvolution [Tournier04], ODF [Tuch04, Descoteaux-etal05], etc.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 23 / 68

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SLIDE 37

Spherical Harmonics: Definition

Spherical harmonics are a basis for complex functions on the unit sphere. We use a modified basis constrained to be real and symmetric (imaginary and non-symmetric parts = noise) ∀(θq, φq) ∈ ΩS = [0, π] × [0, 2π) , S : ΩS → R S(θq, φq) = N

j=0 cjYj(θq, φq) = ˜

BCj(θq, φq), with ˜ B =    Y1(θ1, φ1) . . . YN(θ1, φ1) . . . ... . . . Y1(θns, φns) . . . YN(θns, φns)    (1)

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 24 / 68

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SLIDE 38

Angular Atoms

ym

l

=    √ 2 R(Y m

l )

if 0 < m ≤ l Y 0

l ,

if m = 0 √ 2 ℑ(Y |m|

l

) if −l ≤ m < 0 with l ∈ 2N (2)

Figure: Real and symmetric spherical harmonics: first orders l = 0, 2, 4, 6. Blue indicates a negative value, whereas indicates red a positive value.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 25 / 68

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SLIDE 39

Angular profile using Spherical Harmonics

(a)

Figure: Square sampled along a 5-subdivided icosahedron.

(a) 6 coefficients

Figure: Angular reconstruction along with increasing truncation order L.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68

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SLIDE 40

Angular profile using Spherical Harmonics

(a)

Figure: Square sampled along a 5-subdivided icosahedron.

(a) 15 coefficients

Figure: Angular reconstruction along with increasing truncation order L.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68

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SLIDE 41

Angular profile using Spherical Harmonics

(a)

Figure: Square sampled along a 5-subdivided icosahedron.

(a) 28 coefficients

Figure: Angular reconstruction along with increasing truncation order L.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68

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SLIDE 42

Angular profile using Spherical Harmonics

(a)

Figure: Square sampled along a 5-subdivided icosahedron.

(a) 45 coefficients

Figure: Angular reconstruction along with increasing truncation order L.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68

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SLIDE 43

Angular profile using Spherical Harmonics

(a)

Figure: Square sampled along a 5-subdivided icosahedron.

(a) 66 coefficients

Figure: Angular reconstruction along with increasing truncation order L.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68

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SLIDE 44

Angular profile using Spherical Harmonics

(a)

Figure: Square sampled along a 5-subdivided icosahedron.

(a) 91 coefficients

Figure: Angular reconstruction along with increasing truncation order L.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68

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SLIDE 45

Angular profile using Spherical Harmonics

(a)

Figure: Square sampled along a 5-subdivided icosahedron.

(a) 120 coefficients

Figure: Angular reconstruction along with increasing truncation order L.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68

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SLIDE 46

Angular profile using Spherical Harmonics

(a)

Figure: Square sampled along a 5-subdivided icosahedron.

(a) 153 coefficients

Figure: Angular reconstruction along with increasing truncation order L.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68

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SLIDE 47

Orientation Density Function (ODF)

The ODF Ψ at direction u is defined as the radial projection of the diffusion PDF Ψ(u) = ∞

  • P(αu)dα

=

  • P(r, θ, z)δ(θ, z)rdrdθdz

(3) ODF Problem: How to get the PDF ?

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 27 / 68

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SLIDE 48

Funk-Radon Transform (FRT)

The FRT (4) is a smoothed estimation of the true ODF (3) [Tuch04] Gq′(u) = 2πq′

  • P(r, θ, z)J0(2πq′r)rdrdθdz

(4) FRT Besides, the ODF can be directly expressed from diffusion signal in spherical harmonics by a Least Square minimization [Descoteaux06] ODF ≈ Gq′ = ˜ P ˜ BC =

  • j

2πPlj(0)cjYj (5)

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 28 / 68

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SLIDE 49

Least Square ODF estimation

Least Square ODF estimation : Enables resolution of any specific local structure (crossing fibers) Model-free method: no assumption on macroscopic diffusion Light matrix computations But... Least Square method not adapted to Rice noise model [Sijbers98] No guaranty on spatial coherence of the ODF field

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 29 / 68

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SLIDE 50

Variational estimation of ODF

minC:ΩC →RN

  • E(C) =
  • ΩS [ns

k ψ(|Dk|)] + αϕ(||∇C||)dΩS

  • ,

with Dk(p) = Sk(p) −

j ˜

Bk,j ˜ P−1

j

Cj(p) (6) The best fitting coefficients are computed with a gradient descent coming from the Euler-Lagrange derivation of the energy E. This leads to a set of multi-valued partial derivate equation.      Ct=0 = U0

∂Cj ∂t

=

q ψ′(|Dq|) sign(Dq)P−1 j

˜ Bq,j + α div(ϕ

′(||∇C||)

||∇C||

∇C) (7)

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 30 / 68

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SLIDE 51

Advantages: regularity

Ensure a global regularity of the ODF field: ϕ(||∇C(p)||) Spherical harmonics coefficients characterize anistropy [Frank02] l = 0 Isotropic diffusion l = 2 Single fibers

  • rientation

l = 4 Several fibers

  • rientation

Example of possible regularization function ϕ

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 31 / 68

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SLIDE 52

Advantages: adaptive to noise distribution

ψ-likelihood function adapted to MRI noise law:

The best ψ function is the one specific to MR scanners, ie. Rice distribution We seek Sr which maximizes a posteriori (MAP) the log-posterior probability log p(Sr|S) = log p(S|Sr) + log p(Sr) − log p(S) (8) Consequently the pointwise likelihood is ψ(Sr) = log p(S|Sr, σ) = log S σ2 − (S2 + S2

r )

2σ2 + log I0 S · Sr σ2

  • (9)

[Basu,2006]

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 32 / 68

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SLIDE 53

Simulation: influence of Rice model

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 33 / 68

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SLIDE 54

Results on synthetical data

DTI field ODF field

Results are good on perfect datasets, what about MRI acquisition noise ?

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 34 / 68

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SLIDE 55

Results on human brain hardi data

GFA DTI field

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 35 / 68

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SLIDE 56

Results on human brain hardi data

GFA ODF field

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 36 / 68

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SLIDE 57

Simulation: energy minimization

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 37 / 68

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SLIDE 58

Simulation: regularization

without with GFA zoom

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 38 / 68

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SLIDE 59

GFA DTI

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 39 / 68

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SLIDE 60

Consequences on fiber-tracking

GFA LS

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 40 / 68

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SLIDE 61

Consequences on fiber-tracking

GFA LS

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 41 / 68

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SLIDE 62

Consequences on fiber-tracking

GFA PDE

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 42 / 68

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SLIDE 63

Fibertracking: regularization

As for DTI models, ODF fibertracking is very sensitive to noise.

without regularization with regularization

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 43 / 68

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SLIDE 64

1

Context

2

DTI model : Diffusion Tensor Imaging

3

QBI model : Q-Ball Imaging

4

Measuring the PDF : DSI and Multi Q-Ball

5

Conclusions & Perspectives

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 44 / 68

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SLIDE 65

Signal acquisition

Figure: Diffusion MRI acquisition steps.

Why measuring the PDF ? The PDF brings new important radial information.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 45 / 68

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SLIDE 66

Interest of Radial part of the PDF

Information on cells micro-structure that composed the organic tissue. Ex: axon diameter, number of compartments.

Spinal cord [Cohen02]

May increase detection of anomalies such as demyelinization, a symptom of multiple sclerosis.

Figure: Myelination of an axone [www.jdaross.cwc.net]

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 46 / 68

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SLIDE 67

Diffusion in a bi-homogeneous environment

Figure: Experimental graph: human erythocytes rate for decreasing values of

  • hematocrites. [Kuchel97]

Empirical approximation of signal by a bi-exponential function (compartiments: intra/extra diffusion).

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 47 / 68

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SLIDE 68

Diffusion in a complex environment

Figure: Simulation plot: fibers set of various diameters. [Cohen02]

Observations

Important information are found in the radial diffusion profile.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 48 / 68

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SLIDE 69

High Angular Resolution Diffusion Imaging

The Fourier Transform

PDF(p) =

  • q E(q) exp(−2πiqTp)dq [Cory90,Callaghan91]

DSI: Fourier transform [Wedeen00]

Very long acquisition time Needs high gradients ⇒ Magnetic field distortion

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 49 / 68

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SLIDE 70

High Angular Resolution Diffusion Imaging

The Fourier Transform

PDF(p) =

  • q E(q) exp(−2πiqTp)dq [Cory90,Callaghan91]

DSI: Fourier transform [Wedeen00]

Very long acquisition time Needs high gradients ⇒ Magnetic field distortion

Problem

The DSI is not clinical-compliant.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 49 / 68

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SLIDE 71

High Angular Resolution Diffusion Imaging

The Funk-Radon Transform

ODF(k) =

  • u⊥k E(u)du

HARDI: High Angular Diffusion Imaging [Tuch02]

Reduced acquisition time Lack of radial information

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 49 / 68

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SLIDE 72

High Angular Resolution Diffusion Imaging

The Funk-Radon Transform

ODF(k) =

  • u⊥k E(u)du

HARDI: High Angular Diffusion Imaging [Tuch02]

Reduced acquisition time Lack of radial information

Problem

The ODF does not give any radial information.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 49 / 68

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SLIDE 73

HARDI Extension: multi-sphere imaging

Figure: Example of HARDI extension [Assaf05, ¨ Ozarslan06, Wu07, Khachaturian07, Assemlal-et.al08, Assemlal-et.al09].

Better distribution of samples on the Q-Space.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 50 / 68

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SLIDE 74

HARDI Extension: multi-sphere imaging

Figure: Example of HARDI extension [Assaf05, ¨ Ozarslan06, Wu07, Khachaturian07, Assemlal-et.al08, Assemlal-et.al09].

Better distribution of samples on the Q-Space.

Problem

Still insufficient number of samples for a Fourier transform. Which mathematical tool for the signal estimation ?

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 50 / 68

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SLIDE 75

Continuous representation of the signal

Continuous representation of the MR signal E in the following basis (Spherical Polar Fourier SPF): E(q) =

∞

  • n=0

∞

  • l=0

l

  • m=−l

anlmRn(||q||)ym

l

q ||q||

  • (10)

where anlm expansion coefficients, Rn and ym

l

are respectively are radial and angular atoms. The basis is orthonormal in spherical coordinates:

  • q∈R3
  • Rn(||q||)ym

l

q ||q||

  • ·
  • Rn′(||q||)ym′

l′

q ||q||

  • dq = δnn′ll′mm′

(11)

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 51 / 68

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SLIDE 76

Radial Atoms

Rn (||q||) = 2 γ3/2 n! Γ (n + 3/2) 1/2 exp

  • −||q||2

2γ

  • L1/2

n

||q||2 γ

  • (12)

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 52 / 68

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SLIDE 77

Radial Atoms

Rn (||q||) = 2 γ3/2 n! Γ (n + 3/2) 1/2 exp

  • −||q||2

2γ

  • L1/2

n

||q||2 γ

  • (12)

Figure: Some radial atoms Rn, γ = 100 Figure: Experimental plot [Regan06]

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 52 / 68

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SLIDE 78

Radial Atoms

(a) Signal de diffusion

Figure: Radial reconstruction along with increasing truncation order N.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68

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SLIDE 79

Radial Atoms

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 5 10 15 20 25 30 35 40 MR attenuation q Truth Sample Reconstruction

(a) 1 Coefficient

Figure: Radial reconstruction along with increasing truncation order N.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68

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SLIDE 80

Radial Atoms

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 5 10 15 20 25 30 35 40 MR attenuation q Truth Sample Reconstruction

(a) 2 Coefficients

Figure: Radial reconstruction along with increasing truncation order N.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68

slide-81
SLIDE 81

Radial Atoms

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 5 10 15 20 25 30 35 40 MR attenuation q Truth Sample Reconstruction

(a) 3 Coefficients

Figure: Radial reconstruction along with increasing truncation order N.

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SLIDE 82

Radial Atoms

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 5 10 15 20 25 30 35 40 MR attenuation q Truth Sample Reconstruction

(a) 4 Coefficients

Figure: Radial reconstruction along with increasing truncation order N.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68

slide-83
SLIDE 83

Radial Atoms

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 5 10 15 20 25 30 35 40 MR attenuation q Truth Sample Reconstruction

(a) 5 Coefficients

Figure: Radial reconstruction along with increasing truncation order N.

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68

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SLIDE 84

Fitting the data

How to fit the data

From the diffusion samples, how do we retrieve the SPF coefficients ?

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SLIDE 85

Linear signal estimation

The coefficient estimation is computed by the linear damped least square method: A = arg min

A

||E − MA||2 + λl||L||2 + λn||N||2 (13) = (MTM + λlLTL + λnNTN)−1MTE (14) where M is the basis matrix, E is the MR signal vector and A is the coefficient vector: M =      R0(||q1||)y0

  • q1

||q1||

  • . . .

RN(||q1||)yL

L

  • q1

||q1||

  • .

. . ... . . . R0(||qns||)y0

  • qns

||qns ||

  • . . .

RN(||qns||)yL

L

  • qns

||qns ||

    , (15) E = (E(q1), . . . , E(qns))T (16) A = (a000, . . . , aNLL)T (17)

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SLIDE 86

Simulation: linear least square reconstruction

Figure: N = 0, L = 4, γ = 100,

1 sphere – 42 directions, PSNR: 33.337902, 30 Coefficients

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SLIDE 87

Simulation: linear least square reconstruction

Figure: N = 3, L = 4, γ = 70,

3 spheres – 42 directions, PSNR: 45.172752, 45 Coefficients

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SLIDE 88

Simulation: linear least square reconstruction

Figure: N = 5, L = 6, γ = 50,

10 spheres – 162 directions, PSNR: 50.255381, 168 Coefficients

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SLIDE 89

Features of the PDF

Now that we have a continuous reconstruction of the diffusion signal E, how do we compute the PDF ?

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SLIDE 90

Features of the PDF

Now that we have a continuous reconstruction of the diffusion signal E, how do we compute the PDF ? We don’t. This would require a lot of computation. Besides, the PDF is cumbersome to display.

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SLIDE 91

Features of the PDF

Now that we have a continuous reconstruction of the diffusion signal E, how do we compute the PDF ? We don’t. This would require a lot of computation. Besides, the PDF is cumbersome to display. We are interesting in a data reduction suitable to display: features of the PDF. G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(18)

(GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 57 / 68

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SLIDE 92

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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slide-93
SLIDE 93

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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SLIDE 94

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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slide-95
SLIDE 95

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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SLIDE 96

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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slide-97
SLIDE 97

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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slide-98
SLIDE 98

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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slide-99
SLIDE 99

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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slide-100
SLIDE 100

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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slide-101
SLIDE 101

Features of the PDF: projection

Figure: Example: ODF feature.

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(19)

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slide-102
SLIDE 102

Features of the PDF: projection

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(20)

Hk PDF G(k) Projection

Figure: Overview of the algorithm for the fast computation of a PDF feature G at point k

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SLIDE 103

Features of the PDF: projection

G(k) =

  • p∈R3 PDF(p) Hk(p)dp

(20)

Hk PDF G(k) Projection

Figure: Overview of the algorithm for the fast computation of a PDF feature G at point k

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SLIDE 104

Features of the PDF: projection

G(k) =

  • q∈R3 E(q) hk(q)dq

(20)

Hk PDF hk E G(k) Projection iFFT iFFT

Figure: Overview of the algorithm for the fast computation of a PDF feature G at point k

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slide-105
SLIDE 105

Features of the PDF: projection

G(k) =

∞

  • n,l,m

ak

nlmbnlm

(20)

Hk PDF hk E ak

nlm

bnlm G(k) Projection iFFT iFFT SPF SPF

Figure: Overview of the algorithm for the fast computation of a PDF feature G at point k

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SLIDE 106

Features of the PDF

G ODF FRT ISO P(0) Hk(p) δ(1 −

|p·k| |p||k| )

2πq′J0(2πq′|p|(1 −

|p·k| |p||k| ))

δ(|p − k|) + δ(|p + k|) δ(p) hk(q) δ(

q·k |q||k| )

δ(|q| − |q′|)δ(

q·k |q||k| )

cos(2πq · k)

1 Z3

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SLIDE 107

Simulation: features of the PDF (ODF)

Truth QBI FRT FRT ODF ODF ODF Low Resolution

  • Med. Res.

High Res.

Figure: ODF comparisons. 1st line: single fiber. 2nd, 3rd lines: crossing fibers (face and profile view)

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SLIDE 108

Variational framework

Robustly estimate and regularize the SPF coefficients by minimizing the functional energy: min

A

  • ΩE

ns

  • k

ψ(ˆ Ek)

  • + αrϕ(||∇A||)dΩE
  • , with ˆ

E = MA (21) The best fitting coefficients A are computed with a gradient descent coming from the Euler-Lagrange derivation of the energy. This leads to a set of multi-valued partial derivate equation.      At=0 = U0

∂Aj ∂t

= ns

k Mk,jψ′(ˆ

Ek) + αr div(ϕ

′(||∇A||)

||∇A||

∇A) (22) iteration 0 iteration 1 ... iteration n

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SLIDE 109

Simulation: Rician vs Gaussian likelihood function

a) Truth b) Noisy c) Gaussian d) Rician |Gauss − Truth| |Rice − Truth|

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SLIDE 110

Simulation: validation on synthetical data (likelihood)

Figure: Synthetic phantom of networks of crossing fibers. Performances of likelihood functions on increasing levels of noise.

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SLIDE 111

Simulation: regularization vs no regularization

a) Phantom b) Original data PSNR: ∞ c) Without regul. PSNR: 12.8 d) With regul. PSNR: 16.6

Figure: Effects of spatial regularization on the GFA. Isotropic area are black, anisotropic area are white. PSNR(noisy,original)=18.5.

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SLIDE 112

1

Context

2

DTI model : Diffusion Tensor Imaging

3

QBI model : Q-Ball Imaging

4

Measuring the PDF : DSI and Multi Q-Ball

5

Conclusions & Perspectives

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SLIDE 113

Conclusions & Perspectives

Diversity of estimation algorithms

Before estimation : Sensitivity to image registration. During estimation : Algorithms with few parameters miss often important signal features (rice noise model, regularity constraints). After estimation : There are at least as far as many fibertracking algorithms as estimation techniques.

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slide-114
SLIDE 114

Conclusions & Perspectives

Diversity of estimation algorithms

Before estimation : Sensitivity to image registration. During estimation : Algorithms with few parameters miss often important signal features (rice noise model, regularity constraints). After estimation : There are at least as far as many fibertracking algorithms as estimation techniques.

Do not trust your algorithms !

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slide-115
SLIDE 115

Conclusions & Perspectives

Modern estimation methods are trying to reconstruct the PDF

Estimation of the PDF or PDF characteristics from a minimal set of images, sampled in the Q-space. Generic approach : The same method is able to compute different diffusion characteristics. Flexible approach : The more number of samples you get, the more precise characteristic you compute.

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slide-116
SLIDE 116

Conclusions & Perspectives

Modern estimation methods are trying to reconstruct the PDF

Estimation of the PDF or PDF characteristics from a minimal set of images, sampled in the Q-space. Generic approach : The same method is able to compute different diffusion characteristics. Flexible approach : The more number of samples you get, the more precise characteristic you compute. Thank you for your attention.

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