Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI

Estimation and Processing of Ensemble Average
Propagator and Its Features in Diffusion MRI
Jian Cheng
Center for Computational Medicine, LIAMA
Institute of Automation, Chinese Academy of Sciences
Athena Project-Team, INRIA Sophia Antipolis - Méditerranée
PhD Defense, May 30, 2012
Jian Cheng (ccm, athena) Estimation and Processing of EAPs/ODFs May 30, 2012 1 / 81

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Outline
1 Thesis Overview
2 Background and Motivation
Diffusion Tensor Imaging (DTI)
Modeling Beyond DTI: High Angular Resolution Diffusion
Imaging (HARDI)
3 Spherical Polar Fourier Imaging (SPFI)
SPF basis and analytical forms
Analytical Fourier Transform in Spherical Coordinate (AFT-SC)
4 Riemannian Framework for ODF and EAP Computing
Riemannian Framework for general PDF Computing
Applications of the Riemannian framework
5 Summary and conclusion

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Water diffusion in human brain
Water diffusion in human brain is hindered by surrounding tissues.
[Johansen-Berg and Behrens 2009]

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Diffusion Data in 6D space
3D x-space (spatial space) and 3D R-space (diffusion displacement
space)
Ensemble Average Propagator (EAP) P(R)
EAP field: P(x, R); ODF field: Φk
(x, R)
[Descoteaux 2008, Hagmann 2006]
EAP profile P(R0r)
Orientation Distribution
Function (ODF) Φk(r)
Φk(r) def
=
1
Z
∞
0
P(Rr)RkdR

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Diffusion Weighted Imaging (DWI)
The Pulse Gradient Spin-Echo (PGSE)
sequence [Stejskal and Tanner 1965]
Narrow pulse assumption (δ ∆)
E(q)
F3D
⇐⇒ P(R)
[Callaghan, 1991]
q = qu
F3D
⇐⇒ R = Rr
E(q) = S(q)
S(0)
q = γδG/2π
Fourier relation
P(R) = E(q)e−2πiq·Rdq
E(q) = P(R)e−2πiq·RdR

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Research Content in Computational dMRI

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Thesis overview

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Outline
1 Thesis Overview
Imaging (HARDI)

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Free diffusion, Gaussian diffusion assumption [Basser 1994]
P(R) =
1
(4πτ)3|D|
exp −
1
4τ
RTD−1R D =
3
i=1
λi
vi
vT
i
Diffusion tensor model [Basser 1994]
E(q) = S(q)/S(0) = exp(−4π2τqTDq) = exp(−buTDu)
b = 4π2τ q 2
Tensor estimation (>6 DWIs)

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From tensors to Tissue Properties
From tensor valued images to scalar images
tenor ﬁeld FA map MD map RGB map
FA =
√
3||D − 1
3
Trace(D)I||
√
2||D||
=
3
2
(λ1 − ¯
λ)2 + (λ2 − ¯
λ)2 + (λ3 − ¯
λ)2
λ2
1
+ λ2
2
+ λ2
3
MD =
1
3
Trace(D) =
λ1
+ λ2
+ λ3
3

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Riemannian framework for tensors and Gaussian distributions
Metric selection for processing tensor field.
Parametric family PF (a set of PDFs) parametrized by tensor Σ
PF = P(R|Σ) =
1
(2π)3|Σ|
exp −
1
2
RTΣ−1R : Σ ∈ Sym+
3
Fisher metric in information geometry theory: [Rao, 1945, Atkinson 1981,
Skovgaard 1984, Lenglet 2006, Pennec 2006, Fletcher 2004]
gij
=
R3
∂
√
P(R|Σ)
∂Di
∂
√
P(R|Σ)
∂Dj
dR =
1
2
Tr(Σ−1Ei
Σ−1Ej)
Exponential map, Logarithmic map, geodesic
mean of tensors (Gaussian distributions)
interpolation, filtering, atlas estimation
definite positive tensor estimation
revisited later

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Advantages and limitation of tensor model
Advantages and limitations of DTI
>6 DWIs, normally 20-30 DWIs
useful scalar indices, FA, MD, etc.
DTI works well in the area with isotropic diffusion and single
directional diffusion
DTI can not handle non-Gaussian diffusion and complex ﬁber
conﬁgurations
From Gaussian distributions to general non-Gaussian distributions

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Outline
1 Thesis Overview
Imaging (HARDI)

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Modeling Beyond DTI: HARDI
Sampling in DTI, in Cartesian grid, single shell and multiple shells
sampling in DTI
[Basser1994]
Dense Cartesian grid
Diffusion Spectrum Imaging (DSI)
[Wedeen 200, 2005]
Single shell sampling
sHARDI:
Q-Ball Imaging (QBI), exact QBI
[Tuch 2004,Descoteaux 2007,Aganj
2010]
Diffusion Orientation Transform
(DOT)
[Özarslan 2006]
...
...
Multiple shell or arbitrarily sampling
mHARDI:
Diffusion Propagator Imaging (DPI)
[descoteaux 2010]
Spherical Polar Fourier Imaging
(SPFI)
[Assemlal 2009, Cheng 2010]
Simple Harmonic Oscillator based
reconstruction and estimation
(SHORE)
[Özarslan 2009, Cheng 2011a]
...
P(R) = E(q)e−2πiq·Rdq

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Diffusion Spectrum Imaging (DSI)
Measure the signal in a 3D Cartesian grid in q-space
Perform a numerical Fourier transform to obtain EAP [Wedeen 2000, 2005]
Advantages and limitations of DSI
model-free, no Gaussian assumption
too many samples, large acquisition time
numerical Fourier transform, re-griding (interpolation, extrapolation)
Numerical integration for ODFs

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Diffusion Orientation Transform (DOT) for P(R0r)
Goal: simplify the FT based on mono-exponential decay
assumption, [Ozarslan 2006]
. E(qu) = E(q0u)q2/q2
0
3D Fourier transform to obtain EAP proﬁle P(R0r)
Advantages and limitations of DOT
single shell data, analytical solution via spherical harmonic basis
mono-exponential decay assumption is unrealistic.
the estimated EAP is the true EAP convolved by F3D{E(qu)q2/q2
0 E(qu)−1}
it has difﬁculty to handle multi-shell data.

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Exact QBI for ODFs, mono-exponential decay assumption
Two types of ODFs: [Tuch 2004, Wedeen 2005]
Φt(r) = 1
Z
∞
0
P(Rr)dR Φw(r) = ∞
0
P(Rr)R2dR
Projection slice theorem: [Aganj 2010, Canales-Rodriguez 2009, Tristan-Vega 2010]
Φt(r) = 1
Z Πr
E(qu)qdqdu Φw(r) = 1
4π
− 1
8π2 Πr
∆bE(qu)
q
dqdu
Πr
= {qu : uTr = 0}
Goal: simplify the plane integral based on mono-exponential
decay assumption, E(qu) = E(q0u)q2/q2
0 .
Φt(r) = − q2
0
2Z S2
1
ln(E(q0u))
δ(rTu)du
Φw(r) = 1
4π
+ 1
16π2 S2
∆b ln(− ln(E(qu)))δ(rTu)du
Funk-Radon Transform (FRT): FRT{f(u)}(r) =
S2
f(u)δ(rTu)du
SH is the eigenfunction of FRT and Laplace-Beltrami operator
S2
Ym
l
(u)δ(rTu)du = 2πPl(0)Ym
l
(r) ∆bYm
l
(u) = −l(l + 1)Ym
l
(u)

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Exact QBI for ODFs, mono-exponential decay assumption
Advantages and limitations of exact QBI
single shell data, analytical solution via spherical harmonic basis
mono-exponential decay assumption is unrealistic.

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Q-ball Imaging for Φt(r), delta function decay assumption
Goal: simplify the plane integral based on delta function
decay assumption, E(qu) = E(q0u)δ(q − q0).
Q-ball Imaging (QBI) for Φt(r), Funk-Radon Transform [Tuch 2004, Descoteaux
2007, Hess 2006]
E(q0u) =
lm
clmYm
l
(u)
Φt(r) ≈
1
Z
FRT{E(q0u)}(r) =
1
Z
lm
2πPl(0)clmYm
l
(r)

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Q-ball Imaging for Φt(r), delta function decay assumption
Advantages and limitations of QBI
single shell data, analytical solution is fast
delta function decay assumption is unrealistic.
the estimated Φt(r) is too smooth, spherical deconvolution

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The state-of-the-art in local estimation
sampling Φt(r) = 1
Z
∞
0
0
P(Rr)R2dR P(R)
DTI

exp(−4π2τqTDq)
DSI
[Wedeen 2000, 2005]
original QBI [Tuch 2004] [Tristán-Vega 2009]
many negative values
δ(q − q0) [Descoteaux 2007] not converge
exact QBI
[Canales-Rodriguez 2009]
[Aganj 2010] DOT
exp(−4π2τD(u)) [Tristán-Vega 2010] [Özarslan 2006]
SPFI [Assemlal 2009, Cheng 2010a] [Cheng 2010b]
{Gn(q)Ym
l
(u)} proposition proposition proposition
SHORE proposition proposition
{Gnl(q)Ym
l
(u)} [Özarslan 2009, Cheng 2011a]
DPI
{qlYm
l
(u), q−l−1Ym
l
(u)} [Descoteaux 2010]

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Outline
1 Thesis Overview
Imaging (HARDI)

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numerical Spherical Polar Fourier Imaging (nSPFI)
Spherical Polar Fourier expression of signals [Assemlal 2009]
E(q) =
N
n=0
L
l=0
l
m=−l
anlmGn(q|ζ)Ym
l
(u) BSPF
nlm
(q) = Gn(q|ζ)Ym
l
(u)
Gaussian Laguerre polynomial
Gn(q) = κn(ζ) exp −q2
2ζ
L1/2
n
(q2
ζ
), κn(ζ) = 2
ζ3/2
n!
Γ(n+3/2)
1/2

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Analytical Spherical Polar Fourier Imaging (SPFI)
P(R) is analytically obtained from Fourier dual SPF (dSPF) basis
[Cheng 2010a]
P(Rr) =
N
n=0
L
l=0
l
m=−l
anlmFnl(R)Ym
l
(r) BdSPF
nlm
(R) = Fnl(R)Ym
l
(r)
Fourier dual SPF (dSPF) basis, orthonormal basis in R-space
BdSPF
nlm
(R) =
R3
BSPF
nlm
(q)e−2πiqTRdq = Fnl(R)Ym
l
(r)
Fnl(R) = 4(−1)l/2
ζ0.5l+1.5πl+1.5Rl
0
Γ(l + 1.5)
κn(ζ)
n
i=0
(−1)i n + 0.5
n − i
1
i!
20.5l+i−0.5Γ(0.5l + i + 1.5)1F1(
2i + l + 3
2
; l +
3
2
; −2π2R2ζ)
A linear transform from {anlm} to EAP proﬁle represented by SH
basis
For a given R0
, EAP proﬁle
P(R0r) =
L
l=0
l
m=−l
cP
lm
(R0)Ym
l
(r), cP
lm
(R0) =
N
n=0
anlmFnl(R0)

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Analytical Spherical Polar Fourier Imaging (SPFI)
Linear transform for ODF by Tuch Φt(r) def
= 1
Z
∞
0
P(Rr)dR [Cheng 2010b]
Φt(r) =
L
l=0
l
m=−l
cΦt
lm
Ym
l
(r)
cΦt
lm
=
2πζ
Z
N
n=0
n
i=0
κn(ζ)
i − 0.5
i
(−1)n−iPl(0)anlm
Linear transform for ODF by Wedeen Φw(r) def
= ∞
0
P(Rr)R2dR [Cheng 2010b]
Φw(r) =
L
l=0
l
m=−l
cΦw
lm
Ym
l
(r)
cΦw
lm
=
1
√
4π
δ(l)δ(m) −
1
8π
N
n=1
n
i=1
(−1)iκn(ζ)
n + 0.5
n − i
2i
i
Pl(0)(−l)(l + 1)anlm
Analytical transform avoids the numerical error
{E(qi)}→{anlm
}→{cP
lm
(R0)}, {cΦt
lm
}, {cΦw
lm
}

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Scalar measurements in SPFI
Return to origin probability (RTO): Po def
= P(0)
Po =
R3
E(q)dq = 4
√
πζ 3
4
N
n=0
(−1)n
Γ(n + 1.5)
n!
an00
Generalized Fractional Anisotropy (GFA) for EAPs:
GFA def
= 1 − S2
P(Rr)Y0
0
(r)dr Y0
0
(r)
2
P(R) 2
= 1 −
N
n=0
a2
n00
N
n=0
L
l=0
l
m=−l
a2
nlm
Mean-Squared Displacement (MSD): MSD def
=
R3
P(R)R2dR
MSD
3
8π2.5
N
n=0
an00
κn(ζ)
ζ
(2L3/2
n−1
(0) + L1/2
n
(0))

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Scale Estimation
based on typical value: ζ = 1
8π2τD0
, D0
= 0.7 × 10−3mm2/s
BSPF
000
∝ exp(−4π2τq2D0) [Cheng 2010a, 2010b]
optimization on both A and ζ (slow, local optimal)
minA,ζ E − MSPFA 2
based on generalized High Order Tensor (GHOT) model fitting
Theorem (Set the scale in SPF basis via GHOT model)
When using SPF basis to fit the diffusion signal attenuation represented by
GHOT model as E(q) = exp − ∞
n=1
∞
l=0
l
m=−l
bnlm
q2
ζ1
n
Ym
l
(u) , if b100
> 0, the
optimal scale parameter ζ of SPF basis for any given L and large enough N is
ζ =
ζ1
√
π
b100
=
1
8π2τDp
Pseudo-ADC in GHOT model, coefficient of 4π2τq2. Dp
= b100
8π5/2τζ
Spherical Polar Non-Polynomial (SPNP) basis: BSPNP
nlm
(q) = q2
ζ1
n
Ym
l
(u)

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Consider the prior E(0) = 1 in estimation
Two ways to consider the prior E(0) =
R3
P(R)dR = 1 in estimation.
Consider artiﬁcial shell E(0u) = 1 [Cheng 2010a, 2010b]
min
A
E − MSPFA 2 + w2 E(0) − M(0)
SPF
A 2 + ATΛA
E(0) = 1 =⇒ N
n=0
anlmGn(0) =
√
4πδ0
l
, 0 ≤ l ≤ L, −l ≤ m ≤ l
a0lm
=
1
G0(0)








√
4πδ0
l
−
N
n=1
anlmGn(0)








, 0 ≤ l ≤ L, −l ≤ m ≤ l
N
n=1
L
l=0
l
m=−l
anlmYm
l
(u) Gn(q) −
Gn(0)
G0(0)
G0(q) = E(q) −
G0(q)
G0(0)
A = (M T
SPF
M
SPF
+ Λ )−1M T
SPF
E

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Whole estimation process
1. Scale ζ Estimation based on typical value or GHOT model:
a) based on typical value: ζ = 1
8π2τD0
, D0
= 0.7 × 10−3mm2/s
b) M: basis matrix generated from basis q2
0.5q2
max
n
Ym
l
(u)
b = (MTM)−1MT ln E, b = (b100
, · · · , bN L L )T is the coefﬁcient vector
ζ =
0.5q2
max
√
π
b100
2. Least Square Estimation of coefﬁcient vector A with E(0) = 1 consideration:
M
SPF
= [Gn(q) − Gn(0)
G0(0)
G0(q)]: Ns x NA
matrix from SPF basis, NA
= N(L + 1)(L + 2)/2
E =














E(q1) − G0(q1)
G0(0)
.
.
.
E(qNS
) − G0(qNs
)
G0(0)














: Ns x 1 dimensional data vector
Λ =












Λ
100
· · · 0
.
.
.
...
.
.
.
0 · · · Λ
NLL












:
NA
x NA
regularization matrix
Λ
nlm
= λll2(l + 1)2 + λnn2(n + 1)2
A = (M T
SPF
M
SPF
+ Λ )−1M T
SPF
E A0
= (a000
, · · · , a0LL)T, a0lm
=
1
G0(0)








√
4πδ0
l
−
N
n=1
anlmGn(0)








A =
A0
A
3. Analytical Calculation of EAP and its features from A:
P(Rr) =
N
n=0
L
l=0
l
m=−l
anlmFnl(R)Ym
l
(r), P(R0r) =
L
l=0
l
m=−l
cP
lm
(R0)Ym
l
(r), cP
lm
(R0) =
N
n=0
anlmFnl(R0)
Φt(r) =
L
l=0
l
m=−l
cΦt
lm
Ym
l
(r), Φw(r) =
L
l=0
l
m=−l
cΦw
lm
Ym
l
(r), Po, GFA, MSD

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Invariance of diffusion time τ
We do not know τ, only know b values and gradients {ui}
b = 4π2τq, q = qu
Theorem (Invariance for different diffusion time τ)
For a given data set with known b values and gradients {uj}Ns
j=1
and
unknown diffusion time τ, if ζ is set by ﬁtting GHOT model, then the
estimated Φt(r), Φt(r, C), Φw(r), Φw(r, C), RTO, MSD, GFA are invariant
for different diffusion time τ. If we denote the EAP estimated from τ by
P(Rr, τ), then
P(Rr, τ2) =
τ1
τ2
3/2
P(
τ1
τ2
Rr, τ1)
The theorem is independent of the estimation (L1, L2, etc) of A. It
is also for typical scale setting.
N(R|2τ2D) =
τ1
τ2
3/2
N(
τ1
τ2
R|2τ1D)

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Outline
1 Thesis Overview
Imaging (HARDI)

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The state-of-the-art in local estimation
sampling Φt(r) = 1
Z
∞
0
0
P(Rr)R2dR P(R)
DTI

exp(−4π2τqTDq)
DSI
[Wedeen 2000, 2005]
original QBI [Tuch 2004] [Tristán-Vega 2009]
many negative values
δ(q − q0) [Descoteaux 2007] not converge
exact QBI
[Canales-Rodriguez 2009]
[Aganj 2010] DOT
exp(−4π2τD(u)) [Tristán-Vega 2010] [Özarslan 2006]
SPFI [Assemlal 2009, Cheng 2010a] [Cheng 2010b]
{Gn(q)Ym
l
(u)} proposition proposition proposition
SHORE proposition proposition
{Gnl(q)Ym
l
(u)} [Özarslan 2009, Cheng 2011a]
DPI
{qlYm
l
(u), q−l−1Ym
l
(u)} [Descoteaux 2010]

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E(q) is represented by {Bk(qu)}, where Bk(qu) = Rk(q)Qk(u).
E(q) =
K
k=0
ck
Bk(qu) =
K
k=1
ck
Rk(q)Qk(u)
Theorem (Plane Wave Expansion)
e±2πiq·R = 4π
∞
l=0
l
m=−l
(±i)ljl(2πqR)Ym
l
(u)Ym
l
(r)
Analytical Fourier Transform
P(R) =
K
k=0
ck
Dk(R) =
K
k=0
∞
l=0
l
m=−l
ck
Fkl(R)Tklm(r)
Radial Integration: Fkl(R) = 4π(−i)l ∞
0
Rk(q)jl(2πqR)q2dq
Spherical Integration: Tklm(r) =
S2
Qk(u)Ym
l
(u)du Ym
l
(r)
If Qk(u) = Ym
l
(u), then Tklm(r) = Ym
l
(r)

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QBI in AFT-SC framework
Delta function decay assumption:
E(q) =
L
l=0
l
m=−l
alm
δ(q − q0)Ym
l
(u)
P(R) =
L
l=0
l
m=−l
alm4π(−1)l/2jl(2πq0R)q2
0
Ym
l
(r)
Φt(r) =
1
Z
∞
0
P(Rr)dR =
q0
Z
L
l=0
l
m=−l
alm2πPl(0)Ym
l
(r)
Φw(r) =
∞
0
P(Rr)R2dR =
1
2π2q0
L
l=0
l
m=−l
alm(−1)l/2Ym
l
(r)
∞
0
jl(x)x2dx
P(R) has many negative values when b is big. Φw(r) does not exist.
Addition theorem, Funk-Hecke theorem, plane wave expansion theorem.

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SHORE in AFT-SC framework
SHO-3D basis: BSHO3
nlm
(q|ζ) = Gnl(q|ζ)Ym
l
(u) [Özarslan 2009, Cheng 2011a]
Gnl
(q|ζ) = κnl
(ζ)(
q2
ζ
)l/2 exp −
q2
2ζ
Ll+1/2
n−l/2
(
q2
ζ
) κnl
(ζ) =
2
ζ3/2
(n − l/2)!
Γ(n + l/2 + 3/2)
1/2
E(q) =
N
n=0
n
l=0
l
m=−l
anlm
Gnl
(q|ζ)Ym
l
(u) ⇐⇒ P(R) =
N
n=0
n
l=0
l
m=−l
anlm
(−1)nGnl
(R|
1
4π2ζ
)Ym
l
(r)
Theorem (Fourier transform of SHO-3D basis)
F3D{BSHO3
nlm
(q|ζ)} = (−1)nBSHO3
nlm
(R| 1
4π2ζ
)
Φk
(r) =
1
Z
2N
l=0
l
m=−l
clm
Ym
l
(r)
clm
= (4π2ζ)− k+1
2
N
n=l/2
n−l/2
j=0
anlm
(−1)n+j
j!
n + l/2 + 1/2
n − l/2 − j
2k+l−1
2
+jΓ(
k + l − 1
2
+ j)
MSD, RTO, GFA

View Slide

SHORE in AFT-SC framework
E(0) = 1 in estimation
E(0) = 1 =⇒
N
n=0
an00Gn0(0) =
√
4π
a000
=
1
G00(0)








√
4π −
N
n=1
an00Gn0(0)








N
n=1
2n
l=0
l
m=−l
anlm







Gnl(q) −
Gn0(0)δ0
l
G00(0)
G00(q)







Ym
l
(u) = E(q) −
G00(q)
G00(0)

View Slide

DPI in AFT-SC framework
DPI0
: solution of Laplace’s equation [Descoteaux, 2010]
E(q) =
1
n=0
L
l=0
l
m=−l
cnlm
Rnl
(q)Ym
l
(u) Fkl
(R) = 4π(−i)l
qmax
0
Rk
(q)jl
(2πqR)q2dq
R0l
(q) = (
q
√
ζ
)l R1l
(q) = (
q
√
ζ
)−l−1
P(R0
r) ≈
1
Z
qmax
0 S2
E(qu)q2 exp −2πiqR0
uTr dqdu
Φt
(r) ≈
1
Z
Rmax
0
P(Rr)dR, Φw
(r) ≈
1
Z
Rmax
0
P(Rr)R2dR
Advantages and limitations of DPI
analytical solution is fast, generalization of QBI
model-based, ∆E(q) = 0
large modeling error when q is small and q is large, E(0) does not exist
approximation based qmax
and Rmax

View Slide

DPI in AFT-SC framework
DPI0
: solution of Laplace’s equation [Descoteaux, 2010]
E(q) =
1
n=0
L
l=0
l
m=−l
cnlm
Rnl
(q)Ym
l
(u) Fkl
(R) = 4π(−i)l
qmax
0
Rk
(q)jl
(2πqR)q2dq
DPI1
: a speciﬁc case of SHO-3D basis
(
q
√
ζ
)lYm
l
(u), (
q
√
ζ
)−l−1Ym
l
(u) =⇒ exp −
q2
2ζ
(
q
√
ζ
)lYm
l
(u) = Gnl
(q)Ym
l
(u)|n=l/2

View Slide

DOT in AFT-SC framework
DOT0
: E(q) = exp(−4π2q2D(u)), mono-exponential decay [Özarslan 2009]
Il
(R, u) = 4π(−1)l/2
∞
0
E(q)jl
(2πqR)q2dq =
Γ(0.5l + 1.5)1
F1
(0.5l + 1.5, l + 1.5, − R2
4τD(u)
)
R−l(−1)l/22l+1π0.5(D(u)τ)0.5l+1.5Γ(l + 1.5)
P(Rr) =
S2
Il(R, u)Ym
l
(u)du Ym
l
(r)
DOT1
: E(q) = nlm
anlm(q2
ζ
)nYm
l
(u)
D(u) = lm
bm
l
Ym
l
(u)
E(q) = ∞
n=1
(−4π2q2)n
n!
( lm
bm
l
Ym
l
(u))n = nlm
anlmq2nYm
l
(u)
DOT2
: E(q) = nlm
anlm exp −q2
2ζ
q2
ζ
n
Ym
l
(u) (SPNP basis)
E(q) = exp(−4π2τb0
0
Y0
0
q2) exp







lm,l 0
4π2τq2bm
l
Ym
l
(u)








= exp −
q2
2ζ








∞
n=0
(−4π2τq2)n
n!
(
lm,l 0
bm
l
Ym
l
(u))n








= exp −
q2
2ζ
nlm
anlm
q2
ζ
n
Ym
l
(u)

View Slide

Summary of Analytical EAP Estimation Methods
It is very easy to propose an analytical EAP estimation methods
E(q) =
K
k=0
ck
Bk(qu) =
K
k=1
ck
Rk(q)Qk(u)
P(R) =
K
k=0
ck
Dk(R) =
K
k=0
∞
l=0
l
m=−l
ck
Fkl(R)Tklm(r)
method Rk(q) Qk(u) Fkl(R) = 4π(−i)l ∞
0
Rk(q)jl(2πqR)q2dq Tklm(r)
QBI R(q) = δ(q − q0) Ym
l
(u) Fl(R) = 4π(−1)l/2jl(2πq0R)q2
0
Ym
l
(r)
SHORE Rnl(q) = Gnl(q|ζ) Ym
l
(u) Fnl(R|ζ) = (−1)nGnl(R| 1
4π2ζ
) Ym
l
(r)
SPFI Rn(q) = Gn(q|ζ) Ym
l
(u) Fnl(R) = ∗ ∗ ∗ Ym
l
(r)
DPI0
R0l(q) = ( q
√
ζ
)l
Ym
l
(u)
F0l
= (−1)l/2ql+1.5
max
ζ−0.5lR−1.5Jl+1.5
(2πqmax
R)
Ym
l
(r)
R1l(q) = ( q
√
ζ
)−l−1 F1l
= (−1)l/2R−1.5ζ0.5l+0.5((πR)l−0.5
Γ(l+0.5)
− Jl−0.5(2πqmaxR)
ql−0.5
max
)
DPI1 Rl(q) = (q2
ζ
)l/2 exp(−q2
2ζ
) Ym
l
(u) Fl(R) = 2l+1.5ζ0.5l+1.5πl+1.5Rl exp(−2ζπ2R2) Ym
l
(r)
DOT1 Rn(q) = ( q
√
ζ
)2n Ym
l
(u) q2n+l+3
max
πl+1.5RlΓ(1.5+0.5l+n)1F2(1.5+0.5l+n;1.5+l,2.5+0.5l+n;−π2qmaxR)
(−1)l/2ζnΓ(1.5+l)Γ(2.5+0.5l+n)
Ym
l
(r)
DOT2
Rn(q) = (q2
ζ
)n exp(−q2
2ζ
) Ym
l
(u) 2n+0.5l+1.5ζ0.5l+1.5πl+1.5RlΓ(n+0.5l+1.5)1F1(n+0.5l+1.5,l+1.5,−2ζπ2R2)
(−1)l/2Γ(l+1.5)
Ym
l
(r)
(SPNP)

View Slide

Theoretical Analysis and Comparisons
Completeness.
{Bk(q)} is complete if it can represent any square integrable E(q) in
3D space.
More samples =⇒ less MSE. No modeling error if all samples are
known.
Representability.
P(R) is globally affected by E(q) in R3. Fitting the given {E(qi)} is
not enough for a good EAP estimation.
Prior 1 (P1): E(0) = 1 because
R3
P(R)dR = 1
Prior 2 (P2): E(q) tends to 0 as q increases
Prior 3 (P3): E(q) radially decays like (but NOT) a Gaussian
Orthogonality.
Numerical stability
Coefﬁcients under orthonormal basis are independent with the
basis order chosen in Least Square approximation if all samples
are given.

View Slide

Theoretical Analysis and Comparisons
mHARDI methods for single shell data.
Unstable, Rn(q0)Ym
l
(u)
Rn
(q0)Ym
l
(u)
is a constant independent with u
Considering E(0) = 1, SPF basis with N = 1 is stable
Separation information between spherical and radial parts
method Rk(q) Qk(u)Completeness P1 P2 P3 orthogonal single shell Separation
QBI R(q) = δ(q − q0) Ym
l
(u) in S2 No No No in S2 Yes Yes
SHORE Rnl(q) = Gnl(q|ζ) Ym
l
(u) Yes YesYesYes Yes No No
SPFI Rn(q) = Gn(q|ζ) Ym
l
(u) Yes YesYesYes Yes Yes, if N = 1 Yes
DPI0
R0l(q) = ( q
√
ζ
)l
Ym
l
(u) No No No No No No No
R1l(q) = ( q
√
ζ
)−l−1
DPI1 Rl(q) = (q2
ζ
)l/2 exp(−q2
2ζ
)Ym
l
(u) No YesYesYes No Yes No
DOT1 Rn(q) = (q2
ζ
)n Ym
l
(u) in a ball Yes No No No No Yes
DOT2
Rn(q) = (q2
ζ
)n exp(−q2
2ζ
) Ym
l
(u) Yes YesYesYes No No Yes
(SPNP)
DOT0
— — in S2 YesYes No in S2 Yes Yes
Prior 1 (P1): E(0) = 1 because
R3
P(R)dR = 1
Prior 2 (P2): E(q) tends to 0 as q increases
Prior 3 (P3): E(q) radially decays like (but NOT) a Gaussian

View Slide

More comparison between SHO-3D basis and SPF basis
Spherical Polar Polynomial (SPP) basis:
BSPP
nlm
(q|ζ) = exp −
q2
2ζ
q2
ζ
n
Ym
l
(u), n ≥ l/2
Spherical Polar Non-Polynomial (SPNP) basis:
BSPNP
nlm
(q|ζ) = exp −
q2
2ζ
q2
ζ
n
Ym
l
(u)
Theorem (Equivalent bases form the same function space)
{BSPP
nlm
(q|ζ)}n≤N ⇐⇒ {BSHO3
nlm
(q|ζ)}n≤N
{BSPNP
nlm
(q|ζ)}n≤N,l≤L ⇐⇒ {BSPF
nlm
(q|ζ)}n≤N,l≤L
Theorem (Linear span of SHO-3D basis and SPF basis)
Span{BSHO3
nlm
(q|ζ)}n≤N ⊂ Span{BSPF
nlm
(q|ζ)}n≤N,l≤2N
Span{BSHO3
nlm
(q|ζ)}n≤N Span{BSPF
nlm
(q|ζ)}n≤N,l≤2N

View Slide

Experimental comparisons: noise free synthetic data
Cylinder model [Söderman 1995]
mHARDO (SPFI, DPI, SHORE), b=500/1500/3000 s/mm2, 60 × 3
samples, without or with little regularization
sHARDI (DOT, QBI, exact QBI), b=1500 s/mm2, 60 samples

View Slide

Experimental comparisons: EAPs from noisy data
mHARDI, b=500/1500/3000 s/mm2, 60 × 3 samples, SNR=10,
λl
= λn
= 10−8
sHARDI, b=1500 s/mm2, 60 samples, λ = 0.006
EAP results:
(A), (B): mHARDI
methods with three
shell data
(C), (D): sHARDI and
mHARDI methods with
single shell data

View Slide

λl
= λn
= 10−8
ODF results:
(A), (B): mHARDI
methods with three
shell data
(C), (D): sHARDI and
mHARDI methods with
single shell data

View Slide

λl
= λn
= 10−8
Performance summary
mHARDI methods with multiple shell data have better results than
with single shell data
EAP results are generally better than ODF results.
EAPs (SPFI, 3 shells) > EAPs (SPFI, SHORE, 1 shell) > EAPs
(DOT, 1 shell)

View Slide

Phantom data
Public phantom data with 3 shells (b=500/1500/2000 s/mm2).
It is more isotropic than normal data, thus the results needs
min-max normalization.
mHARDI, 3 shells, 64 × 3 samples, without or with little
regularization
sHARDI, b=1500 s/mm2, 64 samples

View Slide

EAP proﬁles in phantom data at 25µm
SPFI SHORE DPI SPFI SHORE DPI
three shell data, b=650,1500,3000 s/mm2 single shell data, b=1500 s/mm2
SPFI SHORE DPI DOT DOT
single shell data, b=2000 s/mm2 b=1500 s/mm2 b=2000 s/mm2

View Slide

Monkey data: scale selection
3 shell, b=500/1500/3000, 30 × 3 samples.
scale selection (N , L ) based on pseudo-ADC map.
pseudo-ADC, (N , L ) = (1, 4) GFA, (N, L) = (1, 4) MSD, (N, L) = (1, 4)
pseudo-ADC, (N , L ) = (2, 4) GFA, (N, L) = (2, 4) RTO, (N, L) = (1, 4)

View Slide

Monkey data: EAP proﬁle esimation
scale selection (N, L) based on GFA map (without regularization)
SPFI SHORE DPI
3 shell data, b = 500, 1500, 3000s/mm2
SPFI SHORE DPI DOT
single shell data, b = 1500s/mm2

View Slide

Monkey data: EAP proﬁle esimation
scale selection (N, L) based on GFA map (without regularization)
SPFI SHORE DPI
3 shell data, b = 500, 1500, 3000s/mm2
SPFI SHORE DPI DOT
single shell data, b = 3000s/mm2

View Slide

Monkey data: ODF esimation
No min-max normalization.
SPFI SHORE
ODF by Wedeen Φw(r), 3 shell data
SPFI SHORE exact QBI
ODF by Wedeen Φw(r), b=1500 s/mm2

View Slide

Monkey data: ODF esimation
No min-max normalization.
SPFI SHORE
ODF by Tuch Φt(r), 3 shell data
SPFI SHORE exact QBI
ODF by Tuch Φt(r), b=1500 s/mm2

View Slide

Outline
1 Thesis Overview
Imaging (HARDI)

View Slide

Multinomial distribution family
Multinomial distribution family
PF = p(x|p) = K
i=1
pi
δ(x = i) : p ∈ RK, pi
≥ 0, K
i=1
pi
= 1,
original parametrization
PS = p = (p1
, p2
, · · · , pK)T ∈ RK : K
i=1
pi
= 1, pi
≥ 0
square root parametrization, gij
= 4δij
PS = c = (c1
, c2
, · · · , cK)T ∈ RK : K
i=1
c2
i
= 1, ci
≥ 0

View Slide

Riemannian framework for tensors and Gaussian distributions
Parametric family PF (a set of PDFs) parametrized by tensor Σ
PF = P(R|Σ)= 1
√
(2π)3|Σ|
exp − 1
2
RTΣ−1R : Σ is positive deﬁnite matrix
Fisher metric: [Rao, 1945, Atkinson 1981, Skovgaard 1984, Lenglet 2006, Pennec 2006, Fletcher 2004]
gij
=
R3
∂
√
P(R|Σ)
∂Di
∂
√
P(R|Σ)
∂Dj
dR = 1
2
Tr(Σ−1Ei
Σ−1Ej)
Exponential map:
ExpΣ(W) = Σ1
2 exp(Σ−1
2 WΣ− 1
2 )Σ1
2
Logarithmic map:
LogΣ(M) = Σ1
2 log(Σ−1
2 MΣ−1
2 )Σ1
2
Geometric Anisotropy:
GA(Σ) = d(Σ, u), where u is the nearest isotropic tensor [¯
λ, ¯
λ, ¯
λ]
Log-Euclidean framework:
F(Σ) = Logu(Σ), where u is the identity matrix [1, 1, 1]
Afﬁne-invariant metric:
d(Σ, M) = d(AΣAT, AMAT),
d(P(R|Σ), P(R|M)) = d(P(R|AΣAT), P(R|AMAT))

View Slide

PDF family
PDF family PFK represented by orthonormal basis {Bi(x)} [Cheng 2009]
PFK
= p(x|c)= K
i=1
ciBi(x)
2
:
χ
p(x|c)dx= K
i=1
c2
i
=1, K
i=1
ciBi(x)≥0
Wavefunction (from Quantum Mechanics)
ψ(x|c) =
√
P(x|c) = K
i=1
ciBi(x)
χ can be any ﬁeld, {Bi(x)} can be any orthonormal basis
For ODF, χ is S2 and {Bi} is chosen as Spherical Harmonics {Ym
l
}
For EAP, χ is R3 and {Bi} could be chosen as SPF basis {BSPF
nlm
}

View Slide

Riemannian Framework
Parameter space {c}: a closed convex subset of SK−1, contained in
a convex cone with 90o
Fisher Information metric: gij
= 4δij
Riemannian distance:
dgij
(p(·|c), p(·|c )) = dδij
(c, c ) = arccos(cTc )
Exponential map:
Expc(vc) = c = c cos ϕ + vc
vc
sin ϕ, where ϕ = vc
Logarithmic map:
Logc(c ) = vc
= c −c cos ϕ
c −c cos ϕ
ϕ, where ϕ = arccos(cTc )

View Slide

Diffeomorphism Invariance
Theorem (Diffeomorphism Invariance)
Fisher information metric is diffeomorphism invariant by deﬁnition.
Mixture of Gaussian model, P(R) = w1
P(R|D1) + w2
P(R|D2)
Transformed model, (A ◦ P)(R) = w1
P(R|ATD1
A) + w2
P(R|ATD2
A)

View Slide

Weighted Riemannian Mean/Median & Geometric Anisotropy
Weighted mean: µw
= arg minc∈PSK
N
i=1
wid(c, c(i))2
[Cheng 2009]
Weighted median: mw
= arg minc∈PSK
N
i=1
wid(c, c(i)) [Cheng 2010c]
Theorem (Existance and Uniqueness)
Weighted Riemannian mean and median uniquely exist in PS. They
can be estimated via a simple gradient descent method.

View Slide

Weighted Riemannian Mean/Median & Geometric Anisotropy
Weighted mean: µw
= arg minc∈PSK
N
i=1
wid(c, c(i))2
[Cheng 2009]
Weighted median: mw
= arg minc∈PSK
N
i=1
wid(c, c(i)) [Cheng 2010c]
Theorem (Reasonable Riemannian mean and median of PDFs)
Weighted Riemannian mean p(x|µw
) and median p(x|mw) satisﬁes
p(x|µw
) ≥ min{p(x|c(i))}N
i=1
, p(x|mw) ≥ min{p(x|c(i))}N
i=1
, ∀x ∈ χ.

View Slide

Isotropic Distribution, Geodesic Anisotropy (GA)
Isotropic ODF is unique, ψ(r) = 1
4π
, ∀r ∈ S2
Isotropic tensor is not unique, D = [λ, λ, λ]
Nearest isotropic tensor is [¯
λ, ¯
λ, ¯
λ]
Isotropic EAP is not unique, P(Rr) = F(R)
Theorem (The nearest isotropic EAP [Cheng 2011b]
)
For any EAP P(R) = ψ(R)2, its nearest isotropic EAP is Piso(R) = (ψSiso
(R))2,
where ψSiso
(R) is the normalized version of the isotropic part of ψ(R), i.e.
ψSiso
(R) = ψLiso
(R)
ψLiso
(R)
, ψLiso
(R) =
R3
ψ(R)Y0
0
(r)dr Y0
0
(r)
With SPF basis representation, ψ(R) = nlm
cnlm
BSPF
nlm
(R), the nearest isotropic
EAP is ψSiso
(R) = nlm
sc,nlm
BSPF
nlm
(R), sc,nlm
= cnlm
δ00
lm
√
N
n=0
c2
n00

View Slide

Isotropic Distribution, Geodesic Anisotropy (GA)
Geodesic Anisotropy (GA): Riemannian distance from the
nearest isotropic EAP/ODF
GA(p(x|c)) = d(c, sc) = arccos(cTsc)

View Slide

Log-Euclidean Framework & Afﬁne-Euclidean Framework
project all EAPs/ODFs onto the tangent space of one typical
isotropic EAP/ODF with coordinate s = (1, 0, 0, ..., 0)T
Projection diffeomorphism:
F(c) = Logs(c) (Log-Euc)
F(c) = c
sTc
− s (Afﬁne-Euc)
Closed form of mean
µw
= F−1 N
i=1
wiF(c(i))

View Slide

Comparison between Euclidean metric and Riemannian metric
Different results if the PDFs are far from the uniform PDF.
The uniform ODF is the unique isotropic ODF. Similar results.
Uniform EAP does not exist. Isotropic EAP is not a unform PDF.
Different results.

View Slide

Outline
1 Thesis Overview
Imaging (HARDI)

View Slide

Wavefunction estimation
Represent E(q) using wavefunction ψ(R) = nlm
cnlmBnlm(R)
E(q|c) =
R3
N
n=0
L
l=0
l
m=−l
cnlmBnlm(R)
2
e−i2πqTRdR
Theorem (Plane Wave Expansion)
e±2πiq·R = 4π
∞
l=0
l
m=−l
(±i)ljl(2πqR)Ym
l
(u)Ym
l
(r)
E(q|c) =
nlm n l m αβ
4π(−1)α
2 cnlmcn l m Inn α(q)Qmm β
ll α
Yβ
α(u) = cTK(q|ζ)c
Kn l m
nlm
(q|ζ) =
2L
α=0
α
β=−α
4π(−1)α
2 Inn α(q)Qmm β
ll α
Yβ
α(u)
Inn α(q) =
∞
0
Rn(R)Rn (R)jα(2πqR)R2dR Qmm β
ll α
=
S2
Ym
l
(r)Ym
l
(r)Yβ
α(r)dr

View Slide

Wavefunction estimation
least square cost function with radial and spherical regularizations
c = arg min
c =1
M(c), M(c) =
1
2
Ns
i=1
cTK(qi|ζ)c − Ei
2
+
1
2
cTΛc
gradient in tangent space
∇M(c) = ∂M(c)
∂c
− cT ∂M(c)
∂c
c
∂M(c)
∂c
= Ns
i=1
2 cTK(qi
|ζ)c − Ei K(qi
|ζ)c + Λc
gradient descent method
c(k+1) = Expc(k)
−dt
∇M(c(k))
∇M(c(k))
, where Expc(v) = c cos v +
v
v
sin v
initialization: c(0) = (1, 0, ..., 0)T, typical isotropic Gaussian EAP

View Slide

Nonnegative deﬁnite ODF/EAP estimation
Square Root Parameterized Estimation (SRPE): [Cheng 2012]
nonnegative deﬁnite ODF/EAP in whole domain
the integration of ODF/EAP is 1
From wavefunction to P(R0r)
P(R0
r) = (ψ(R))2 =
2L
α=0
α
β=−α






nlm n l m
cnlm
cn l m
Gn
(R0
)Gn
(R0
)Qmm β
nn α







Yβ
α
(r)
From wavefunction to Φw(r)
Φw
(r) =
∞
0
(ψ(R))2 R2dR =
nlm n l m
∞
0
Gn
(R)Gn
(R)R2dR cnlm
cn l m
Ym
l
(r)Ym
l
(r)
=
nlm l m
cnlm
cnl m
Ym
l
(r)Ym
l
(r) =
2L
α=0
α
β=−α nlm l m
cnlm
cnl m
Qmm β
ll α
Yβ
α
(r)
From wavefunction to Φt(r)
Φt
(r) =
1
Z
∞
0
(ψ(R))2 dR =
1
Z
nlm n l m
∞
0
Gn
(R)Gn
(R)dR cnlm
cn l m
Ym
l
(r)Ym
l
(r)

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Comparison between SRPE and SPFI
[Cheng MICCAI 2012]

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Interpolation: swelling effect, reasonable mean value interpolation
Mean value interpolation (tensors, ODFs, EAPs)
swelling effect:
Tensor interpolation with Euclidean metric. Larger determinant.
|Σ| > max{Σi
} =⇒ N(0|Σ) < min{N(0|Σi)}
Deﬁnition: Reasonable Mean Value Interpolation
For a PDF valued function p(s, x), a mean value interpolation based on Ns
samples {ps(i)
(x)}Ns
i=1
at position {s(i)}Ns
i=1
is reasonable if the interpolated PDF
ps(x) at position s satisﬁes
ps(x) ≥ min{ps(i)
(x)}N
i=1
, ∀x ∈ χ
Theorem (Reasonable Interpolations)
Riemannian and Euclidean mean value interpolations on general EAPs/ODFs
are both reasonable. Riemannian mean value interpolation on tensors is
reasonable, and more precisely
max{Ns(i)
(R|Σi)}Ns
i=1
≥ Ns(R|Σ) ≥ min{Ns(i)
(R|Σi)}Ns
i=1
, ∀R ∈ R3

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Interpolation: swelling effect
Two ways in the interpolation of Gaussian EAPs:
1. interpolate tensors =⇒ from tensors to EAPs
2. interpolate EAPs

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Interpolation: 1D

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Interpolation in 2D space and PGA

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Filtering

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Robustness in atlas estimation

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Monkey data experiments
EAP ﬁeld at 15 µm ODF ﬁeld GA of EAPs

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Monkey data: EAP and ODF estimation
EAPs at 15 µm EAPs at 25 µm ODFs
proposed SRPE
EAPs at 15 µm EAPs at 25 µm ODFs
SPFI, scale is set by typical ADC

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Monkey data: EAP atlas estimation
RMM median RMM mean EUM mean
original EAP atlas from ﬁve subjects
re-estimated EAP atlas with one noisy subject

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Summary
Local estimation (SPFI, SRPE)
AFT-SC framework, comparison of methods
Riemannian framework for ODFs/EAPs

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Summary
Mean Shift Functional Detection, Riemannian Framework on SK,
2007-2008 [Cheng 2009a]
Riemannian framework for ODFs, Spherical Harmonics in S2,
2008-2009 [Cheng 2009b, Cheng 2010c]
Similar results for ODFs. Probably different results for EAPs. 2009
3D orthonormal basis for EAPs. 2009-2010
MICCAI 2009, Dr. Assemlal, 2009-2010
SPFI, 2009-2010 [Cheng 2010a, 2010b]
Riemannian framework for EAPs, 2010 [Cheng 2011b]
L1-SPFI, 2010-2011
comparisons between EAP estimation methods, 2010-2011 [Cheng
2011a]
nonnegative deﬁnite ODF/EAP estimation 2011-2012 [Cheng 2012]

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Publications
Jian Cheng, Aurobrata Ghosh, Rachid Deriche, Tianzi Jiang,
An Intrinsic Diffeomorphism Invariant Riemannian Framework for Probability Density Function Computing in diffusion
MRI, IEEE Transaction on Medical Imaging (in revision)
Jian Cheng, Aurobrata Ghosh, Tianzi Jiang, Rachid Deriche,
Spherical Polar Fourier Imaging, IEEE transaction on Medical Imaging (to be submitted)
Jian Cheng, Tianzi Jiang, Rachid Deriche,
Nonnegative Definite EAP and ODF Estimation via a Unified Multi-Shell HARDI Reconstruction, MICCAI 2012
Jian Cheng, Aurobrata Ghosh, Tianzi Jiang, and Rachid Deriche,
Diffeomorphism Invariant Riemannian Framework for Ensemble Average Propagator Computing, MICCAI 2011
Jian Cheng, Tianzi Jiang, and Rachid Deriche,
Theoretical Analysis and Practical Insights on EAP Estimation via a Unified HARDI Framework, CDMRI 2011
Jian Cheng, Sylvain Merlet, Aurobrata Ghosh, Emmanuel Caruyer, Tianzi Jiang, and Rachid Deriche,
Compressive Sensing Ensemble Average Propagator Estimation via L1 Spherical Polar Fourier Imaging, CDMRI 2011
Model-free and Analytical EAP Reconstruction via Spherical Polar Fourier Diffusion MRI, MICCAI 2010
Jian Cheng, Aurobrata Ghosh, Rachid Driche, Tianzi Jiang,
Model-Free, Regularized, Fast, and Robust Analytical Orientation Distribution Function Estimation, MICCAI 2010
A Riemannian Framework for Orientation Distribution Function Computing, MICCAI 2009
Jian Cheng, Feng Shi, Kun Wang, Ming Song, Jiefeng Jiang, Lijuan Xu, Tianzi Jiang,
Nonparametric Mean Shift Functional Detection on Functional Space for Task and Resting-state fMRI, MICCAI 2009
Workshop
Riemannian Median and Its Applications for Orientation Distribution Function Computing, ISMRM 2010
Nianming Zuo, Jian Cheng, Tianzi Jiang, Diffusion Magnetic Resonance Imaging for Brainnetome: A Critical Review,
Neuroscience Bulletin (accepted)
Emmanuel Caruyer, Jian Cheng, Christophe Lenglet, Guillermo Sapiro, Tianzi Jiang, Rachid Deriche,
Optimal Design of Multiple Q-shells experiments for Diffusion MRI, CDMRI 2011
Sylvain Merlet, Jian Cheng, Aurobrata Ghosh, Rachid Deriche,
Spherical Polar Fourier EAP and ODF Reconstruction via Compressed Sensing in Diffusion MRI, ISBI 2011

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Perspective
Compressive sensing EAP estimation. [Cheng 2011]
Sampling scheme.
Registration with Riemannian framework.
Better Riemannian framework and SRPE

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Role in Athena and CCM
Christophe Lenglet: Riemannian framework for tensors
from tensors to ODFs and EAPs
Maxime Descoteaux: QBI with single shell data, DPI with
multiple shell data
from single shell to multiple shell, from ODF to EAP, theoretical
comparison, compressive sensing)
Aurobrata Ghosh: High Order Tensor, Cartesian Coordinate,
positive deﬁnite ADC estimation
Spherical Coordinate, coding, etc.
Emmanuel Caruyer: Sampling
Sampling in multiple shell, optimal sampling
Sylvain Merlet: Compressive Sensing estimation
SPF dual basis, L1-SPFI
Yonghui Li, Nianming Zuo, Songma Xie, etc.: diffusion
networks
ODF/EAP estimation ⇒ ﬁber tracking ⇒ network analysis

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Acknowledgements
Rachid Deriche (Athena, INRIA)
Tianzi Jiang (CCM, LIAMA, CASIA)
Aurobrata Ghosh (Athena, INRIA)
Emmanuel Caruyer (Athena, INRIA)
Yonghui Li (CCM, LIAMA, CASIA)
Chuishui Yu (Tianjin Medical University General Hospital)
Wen Qin (Xuanwu Hospital)
Maxime Descoteaux (Sherbrooke University)
Gaolang Gong (Beijing Normal University)
Théodore Papadopoulo (Athena, INRIA)
Hongtu Zhu (UNC-CH)
Rong Xue (Institute of Biophysics, CAS)
Shan Yu (INRIA)

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Thank you for your interests!
Questions?

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Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI

Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI

Jian Cheng

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