Weierstrass Institute for
Applied Analysis and Stochastics
Position Orientation Adaptive Smoothing (POAS) in Diffusion
Weighted Imaging
Jörg Polzehl (joint work with Karsten Tabelow and Saskia Becker)
Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de
Neuroimaging Data Analysis, SAMSI, June 9, 2013
Outline
MR Physics
Data properties and random effects
Modeling
Adaptive smoothing
Examples
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 2 (36)
Magnetic resonance imaging (MRI)
Figure: Kasuga Huang (Wikimedia)
Figure: Franz Wilhelmstötter (Wikimedia)
From O. Friman “Adaptive Analysis of Functional MRI Data”, PhD Thesis, 2003
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Magnetic resonance imaging (MRI)
Many different imaging contrasts
“There is nothing that nuclear spins will not do for you, as
Figure: Kasuga Huang (Wikimedia)
long as you treat them as human beings”
(Erwin L. Hahn 1949)
Figure: Franz Wilhelmstötter (Wikimedia)
From O. Friman “Adaptive Analysis of Functional MRI Data”, PhD Thesis, 2003
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Magnetic resonance imaging (MRI)
T2 -weighted
T1 -weighted
Diffusion-weighted
thanks to: F. Godtliebsen
(University Tromsœ), H.U. Voss
(Weill Cornell Medical College,
NY) and M. Deppe (Uniklinikum
Münster)
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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White matter anatomy
White matter characterized
by fiber bundles (axon
bundles)
diameter
30 − 50µm
length up to
20 − 25cm
highly anisotropic structures
Source: John A Beal, PhD Dep’t. of Cellular Biology &
Anatomy, Louisiana State University Health Sciences
Center Shreveport; Wikipedia Commons
MRI can be used to measure
water diffusion
water diffusion is restricted by
anatomic structure
Beaulieu, C. The basis of anisotropic water diffusion in
the nervous system - a technical review NMR in
Biomedicine, 2002, 15, 435-455
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Diffusion MRI
Figure: Thomas Schultz (Wikimedia)
Notation (Diffusion Propagator)
P (~r, ~r 0 , τ ) - probability density for a particle (spin) to “travel” from position ~r 0 to ~r in time τ
p(~r 0 ) is the initial probability density of particle location
Random walk, diffusion process
V (Ensemble Averaged Propagator, EAP):
Z
~ τ) =
P (R,
P (~r, ~r 0 , τ ) p(~r 0 ) d~r 0 ,
Aggregate over a voxel
~ r −~
~
r 0 ∈V, R=~
r0
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Diffusion MRI signal and diffusion propagator
Diffusion gradients lead to signal
attenuation due to diffusion process - loss
of phase coherence between precessing
spins:
S(~
q , τ ) = S0 hexp(iϕ)i
Fourier relation
Z
S(~
q , τ ) = S0
~
~ τ ) ei~q R dR
~
P (R,
IR3
Measure S(~
q , τ ) at N voxel locations in
3D for 3, . . . , 200 vectors ~
q
3D +
S2
Spatial resolution: 0.6-2 mm
1 - ... different b-values
Magnetic field strength: 3-7 T
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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MR acquisition and noise
complex signal in K-space (one coil):
2
sc (k) ∼ N (xc (k), σK
)
FFT provides complex image
Sc (x) ∼ N (ξc (x), σI2 )
MR image:
S(x) usually obtained as
norm of linear combinations of Sc from
L receiver coils
Notation: Si = |S(xi )|
|S(x)|2 /σI2 distributed as a linear
combination of noncentral χ22 RV, with
spatially varying coefficients.
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Properties of dMRI images
Signal of interest in voxel location
ζi =
sX
xi :
ac (xi )ξc (xi )ξ¯c (xi )
c
ac (x) depend on the reconstruction algorithm
For
ac (xi ) = 1 Si /σ ∼ χ2L;ζi /σ
Signal density Chi_16^2 − Gaussian Approx.
0.6
Signal density Chi_2 − Gaussian Approx.
0.5
NCP 0
NCP 2
NCP 4
Signal attenuation →
decrease in SNR
Bias(NCP,coils)
0
1
2
3
4
x
5
6
Bias/sigma
2
3
1 coil
2 coils
4 coils
8 coils
1
0
0.0
0.0
0.1
0.1
0.2
0.2
dchi
0.3
dchi
0.3
4
0.4
0.4
0.5
NCP 0
NCP 2
NCP 4
0
2
4
6
x
8
0
2
4
6
8
NCP
Strong Bias E S/σ − ζ/σ for small SN R. Models thermal noise, other sources of noise ?
ac (xi ) 6= 1 ? Varying ac (xi ) lead to heteroskedasticity ..., varying effective number of coils.
Noise estimation –> Aja Fernandez (2009, 2011)
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 8 (36)
10
The diffusion tensor model
Gaussian diffusion: (Homogeneity within a voxel, no effect of fiber structure)
1
~
uT D−1 ~
u
~ τ ) = P (r~
exp −r2
.
P (R,
u, τ ) = p
4τ
det D(4πτ )3
Diffusion Tensor Model:
E(~
q , τ ) = E(q~
u, τ ) = e−b~u
Fully characterized by the Diffusion Tensor D (or R :
Estimation by nonlinear regression:
R(Si ; θ, R)
θ̂i
R̂i
T
D~
u
D = RR> )
Si = {Si (~0), Si (~
q1 ), . . . , Si (~
qm )}
=
m
X
(Si (~
qj ) − θ exp(−bj (~
u>
uj R)> ))2
j R)(~
2
σ
j=0
=
argmin R(Si ; θ, R)
!
θ,R
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Characteristics in the diffusion tensor model
Mean diffusivity
T r(D) = µ1 + µ2 + µ3 = 3µ̄
Fractional anisotropy (FA)
FA =
3 (µ1 − µ̄)2 + (µ2 − µ̄)2 + (µ3 − µ̄)2
2
µ21 + µ22 + µ23
1/2
Geodesic anisotropy (GA) (Fletcher (2004), Corouge
(2006))
GA =
3
X
(log(µi ) − log(µ))2
i=1
3
1X
log(µ) =
log(µi )
3 i=1
!1/2
Color coded FA / GA maps
FA / GA coded as image
intensity
Principal eigenvector coded
in RGB
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Models addressing shortcomings of DTI
Diffusion tensor model (DTI) Assumptions:
Assumes anisotropic Gaussian diffusion
does not reflect effects of fiber geometry.
homogeneous fiber structure within a voxel
Reality: high percentage of voxel with fiber crossings or bifurcations
Consequences:
Uninformative tensor estimates
Reduction in FA, Biased or non-existent directional information
Need a better description of the data!!
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 11 (36)
Models addressing shortcomings of DTI
Diffusion tensor model (DTI) Assumptions:
Assumes anisotropic Gaussian diffusion
does not reflect effects of fiber geometry.
homogeneous fiber structure within a voxel
Reality: high percentage of voxel with fiber crossings or bifurcations
Consequences:
Uninformative tensor estimates
Reduction in FA, Biased or non-existent directional information
More detailed models:
Orientation distribution function (Tuch 2004, Wedeen 2005, Aganji 2010, Özarslan 2006)
Positive definite EAP and ODF estimation (Cheng 2012)
Tensor Mixture Models (Behrens 2003/2007, Hosey 2005, Tabelow 2012, Jian 2007,
Leow 2009)
Kurtosis Imaging (Özarslan 2003, Liu 2005, Hui 2008, Jensen 2010, Tabesh 2011)
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Clinical Applications
Mean diffusivity / FA
MD/FA based diagnostics:
decreased Mean Diffusivity –> Stroke
changes of FA over time
differences in FA between groups (control/patient)
Surgery planning
Color coded FA
Color coded FA / MD
Experimental questions:
increased resolution
reduction of recording time
Both lead to significant loss in Signal to Noise Ratio (SNR).
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Modeling example. Data: H. Voss, Weill Cornell Medical College
Tensor ODF
ODF (Qball)
Tensor Mixture ODF
3.0 Tesla GE Signa Excite MRI Scanner
8 Channel receive only head coil
10S0 images and 140 gradient
directions
T E = 73.2ms, T R = 14s
66 slices
Acquisition matrix size: 128 × 128 zero
filled to 256 × 256
CCFA
Kurtosis Tensor
Voxelsize; 0.9 × 0.9 × 1.8mm3
b-value 1000
s
mm2
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 13 (36)
Research problems, high resolution
interest in higher resolution (leads to loss in SNR)
interest in stronger signals (higher field strength 7T)
high resolution Magnetom 7T Siemens. Data: R. Heidemann (MPI Leipzig/Siemens)
s
1000 mm
2,
FOV 143 × 147mm,
0.8mm isotropic.
b-value
left: 60-gradients
7 S0 images
recording time 15 min.
right: 240-gradients
28 S0 images
recording time 65 min.
Need for image enhancement / noise reduction methods preserving the structure
adaptive smoothing for functional data in
adaptive smoothing for image data in
R3 (Tabelow et al (2009))
S 2 n R3 (Becker et al (2012, 2013))
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 14 (36)
Research problems, high resolution
interest in higher resolution (leads to loss in SNR)
interest in stronger signals (higher field strength 7T)
really bad DTI images 3T GE, 40-gradients, 3 S0 images, b-value 1000s/mm2
128x128 (1.72x1.72x0.9)
256x256 (0.86x0.86x0.9)
dito 4 averages
Need for image enhancement / noise reduction methods preserving the structure
adaptive smoothing for functional data in
adaptive smoothing for image data in
R3 (Tabelow et al (2009))
2
S n R3 (Becker et al (2012, 2013))
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 14 (36)
Structural adaptive smoothing: Setup/General idea
Design:
x1 , . . . , xn ∈ X ⊆ IRp
Yi ∼ Pθ(xi )
θ : IRp → Θ (i.i.d.)
SM
Structural assumption: ∃ Partitioning X = m=1 Xm such that
Observations:
(Polzehl & Spokoiny 2000,2006)
Y1 , . . . , Yn ∈ Y ⊂ IRq
θ(x) = θ(xi ) ⇔ ∃m : x ∈ Xm ∧ xi ∈ Xm
i.e. θ constant on each Xm – local homogeneity structure
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Structural adaptive smoothing: Setup/General idea
Design:
x1 , . . . , xn ∈ X ⊆ IRp
Yi ∼ Pθ(xi )
θ : IRp → Θ (i.i.d.)
SM
Structural assumption: ∃ Partitioning X = m=1 Xm such that
Observations:
(Polzehl & Spokoiny 2000,2006)
Y1 , . . . , Yn ∈ Y ⊂ IRq
θ(x) = θ(xi ) ⇔ ∃m : x ∈ Xm ∧ xi ∈ Xm
i.e. θ constant on each Xm – local homogeneity structure
General idea:
Determine structure and estimates in an iterative procedure
local partitioning described by a weighting scheme
“Learn” about partitioning from estimates
“Learn” about
(k)
(k)
W (k) (xi ) = wi1 , . . . , win ∀i
θ̂(xi ) by local comparisons
θ(xi ) using the weighting scheme W (k) (xi ) in weighted log-likelihood
estimation.
(
ideally for large
k:
(k)
wij
≈
1
0
if ∃m : xi , xj ∈ Xm
else.
go from small to large scales with iterations
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Structural adaptive smoothing: Algorithm
Propagation-Separation algorithm
(0)
(0)
k = 0, Wi such that wij = δij , θ̂(xi ) as weighted likelihood or
least squares estimate, h(0) = 1.
Initialization: Step
Adaptation:
∀i, j define
(k)
(k)
wij = Kloc (∆(xi , xj )/h(k) )Ks (sij /λ)
Estimation:
∀i define
θ̂(k) (xi ) = arg max l(Y, Wi ; θ)
θ
Iterate: Stop if
∗
( or
k ≥ k , else consider next scale h
(k+1)
arg min R(Y, Wi ; θ))
θ
= ch h(k) , set k := k + 1 and
continue with adaptation.
Statistical penalty:
(k−1)
sij = Ni
KL(Pθ̂(k−1) (xj ) , Pθ̂(k−1) (xi ) ),
(k)
Ni
(k−1)
= max(Ni
,
n
X
j=1
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 16 (36)
(k)
wij )
Illustration
Univariate local constant regression with additive Gaussian errors
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Illustration
Univariate local constant regression with additive Gaussian errors
Weighting schemes
Location weights
Statistical penalty
Combined weights
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 18 (36)
Parameters
Parameters
k∗ determines the maximum bandwidth
ch = 1.251/p provides exponential growth of sum of location weights
Kernels
Kloc (z) = (1 − z 2 )+ and Ks (z) = min(1, 2(1 − z))+
Propagation condition (Becker & Mathé, 2013)
Let the probability law
Let
θ(xi ) ≡ θ and
Pθ and the metric of the design space X be specified
(k)
N̄i
be the sum of non-adaptive weights (= without Ks ()) in step k
(k)
(k)
Zλ (k, p, θ) = inf{z > 0 : P (N̄i K(θ̂i (λ), θ) > z) ≤ p}
λ is set to be chosen according to the Propagation Condition at level if
Zλ (., p, θ) is non-increasing ∀p > A value
λ can be chosen by simulation using functions from our R-packages aws or dti
the value of
λ does not depend on the data at hand
Theoretical results for exponential family models in Becker & Mathé, 2013
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Propagation condition
Propagation condition for Poisson variates on a 3D grid.
0
5
1.08
10
1.47
15
step
20
2.24 3.12
bandwidth
25
4.49
30
6.51
35
9.44
Poisson 3 −dim. ladj= 1 Exceed. Prob.
0
5
10
15
20
25
z
Level: = 0.001,
Red lines correspond to the non-adaptive estimates.
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 20 (36)
Smoothing of Diffusion Weighted Data
Two different views:
DWI as functional data:
consider observations related to a voxel as sample from a function on one (multiple)
spheres
S : R3 → Rn
describe this function by a model
define the statistical penalty in terms of a distance between models or their
parameters
DWI as data in orientation space
consider observations related to a voxel as sample from a function on one (multiple)
spheres S : R3 n S2 → RB+1
define the statistical penalty in Kullback-Leibler-divergences between noncentral
χ-distributions.
first considered in Franken (2008) and Duits and Franken (2011) using nonlinear
diffusion methods
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Functional approach based on Tensor Model
Tabelow et al 2008
Smoothing algorithm
(0)
(0)
Initialization: k = 1, h(1) = ch . Set ζ̂b,i = Sb,i , D̂i
(0)
(0)
, θ̂0,i Ni
= 1.
Adaptation: For every pair i, j compute
(k−1)
Ni
(k)
sij =
λCi
(g, h(k−1) )
(k)
(k−1)
wij = Kloc ∆(i, j, D̃i
(k−1)
R(ζ̂.,i
(k−1)
, θ̂0,j
(k)
(k)
=
Pn
j=1
(k−1)
) − R(ζ̂.,i
(k−1)
, θ̂0,i
θ̂0,i
(k)
D̂i
(k−1)
, D̂i
(k) )/h(k) Kst sij /σj2 ,
Estimation of diffusion weighted images:
Ni
(k−1)
, D̂j
!
(k)
= arg minθ,D R(ζ̂.,i , θ, D), Set
(k)
wij .
Stopping or Iterate: Stop if k = k ∗ , otherwise h(k+1) = ch h(k) , k := k + 1
The algorithm may incorporate a Bias correction for ζ̂ .
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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)
Comparison
Adaptive smoothing provides more stable estimates without loss of structure
enables to reduce recording time
sensitive only to contrasts reflected by the model
A: unsmoothed
B: non-adaptive
C: adaptive
Data: M. Deppe, Univ. Münster. Color scheme codes FA intensity only.
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 23 (36)
Position orientation adaptive smoothing (POAS)
Becker et al. (2012)
Position orientation adaptive smoothing (POAS) Algorithm ( model=“chi” in dwi.smooth)
Choose a sequences of bandwidths
Initialization (k
= 0):
Adaptation (Step
(0)
sg1 g2
q , k)}k=0,··· ,k∗
{h(~
q , k)}k=0,··· ,k∗ and {κ(~
= 0, h(~
q , 0) = 1
k): ∀g1 , g2 ∈ R3 o S2 define adaptive weights
!
(k)
∆κ (g1 , g2 )2
sg1 g2
(k)
wg1 g2 =Kloc
Ks
h2k
λ
X (k)
(k)
(k−1)
(k−1)
sg1 g2 =
wg1 g2 · KL Ŝg1 /σ̂, Ŝg2 /σ̂
g2
Estimation (Step
k): ∀g1 ∈ R3 o S2 define weighted mean as
X
X
Ŝg(k)
:=
wg(k)
S /
wg(k)
1
1 g2 g2
1 g2
g2 ∈R3 oS2
Iterate: Stop if
g2 ∈R3 oS2
k ≥ k∗ , else k := k + 1 and continue.
(k−1)
The KL-divergence between noncentral χ2L -distributions KL Ŝg1
(k−1)
/σ̂, Ŝg2
/σ̂ needs
to be approximated. Distributions are parametrized by their expected values.
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Discrepancy function on R3 × S2
Embedding of R3 × S2 into SE(3)
Notations
S2 := {~
q ∈ R3 : k~
q k = 1} = 2-sphere
SE(3) := R3 o SO(3) = 3-dimensional Euclidean motion group,
where SO(3) := {R ∈ R3o3 : RT = R−1 , det(R) = 1}
stab(~ez ) := {R ∈ SO(3) : R = rotation around the z -axis}
Then it holds
S2 ∼
= SO(3)/stab(~ez )
and
R3 × S2 ∼
= SE(3)/(0 o stab(~ez )).
F : SO(3) → R, which satisfies F (R(α,β,γ) ) = F (R(0,β,γ) ) for all
α ∈ [0, 2π), can be identified one-to-one with a function f : S2 → R.
Any function
Parametrization:
~
e
Parametrization of
SO(3): R(α,β,γ) := R~eγz Rβy R~eαz ∈ SO(3) for β ∈
/ {0, π}
Parametrization of
S2 :
~
q(β,γ) := (cos γ sin β, sin γ sin β, cos β)T ∈ S2
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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From geodesics to a discrepancy function on R3 o S2
Riemannian 2-norm k.kR on SE(3) - Approximation
kgkR ≈ inf

6
 X

!1/2
ki2
:
i=1
6
Y
i=1


exp(ki Ai ) = Mg ≡ g ∈ SE(3)

using
g ≡ Mg :=
R(α,β,γ)
0
0
0
~v T
1
!
and the left invariant basis matrices {Ai }6i=1 .
Discrepancy on R3 o S2 We set for g1 , g2 ∈ R3 × S2 with gj = (~
vj , q~j )
∆κ (g1 , g2 ) ≈ inf

3
 X

where ĝ :=
!1/2
ki2 + κ−2 (k42 + k52 + |k6 |)
i=1
:
6
Y
exp(ki Ai ) = Mĝ
i=1
−1
−1
2
2
Rq~ (~
v1 − ~v2 ), Rq~ Rq~1


,

and κ balances between distances on S2 and in R3 .
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Multishell POAS,
Becker et al 2013
Multi-shell data
More sophisticated models for dMRI require measurements for different b-values:
multi-shell data
Sb : V × Gb 3 (~v , ~g ) 7→ Sb (~v , ~g ) ∈ R,
b ∈ B0 := B ∪ {0}
Consider a vector form
S : V × G 3 (~v , ~g ) 7→ (S0 (~v ), Sb1 (~v , ~g ), ..., SbB (~v , ~g ))T ∈ RB+1
some
Sb (~v , ~g ) may be unobserved
Improvements:
Incorporates information from different shells including S0
Improved and faster approximation for KL-divergence
Simplified metric on
R3 o S 2
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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msPOAS
Multi-shell POAS algorithm (iteration only)
b ∈ B and m := (~vm , ~gm ) ∈ V × Gb fill the missing values
(k)
(m) and Ñm,b , b0 ∈ B \ {b} by spherical interpolation
Iteration: For each
(k−1)
of S̃b0
Compute the statistical penalty
(k−1)
(k)
smn
=
X
(k−1)
Ñm,b KL
S̃b
σ̂
b∈B0
(k−1)
(m) S̃b
,
(n)
!
σ̂
,
n ∈ V × Gb ,
and the adaptive weights
(k,b)
(k)
w̃mn
= Kloc δκ (m, n)/h(k) · Kad smn
/λ ,
n ∈ V × Gb .
Compute the adaptive estimator
X
(k)
S̃b (m) =
(k)
(k,b)
w̃mn
Sb (n)/Ñm,b
n∈V ×Gb
and the corresponding sum of weights

(k)
Ñm,b
Iterate after increasing

= max
0
k ≤k

X
(k0 ) 
w̃mn
.
n∈V ×Gb
h and adjusting κ.
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 28 (36)
Examples: Really bad SNR
Color Coded FA maps from DTI
really bad DTI images
3T GE, 40-gradients,
3 S0 images,
b-value 1000s/mm2
Reconstructions by POAS, 16 Steps
128x128 (1.72x1.72x0.9)
256x256 (0.86x0.86x0.9)
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
June 9, 2013 · Page 29 (36)
POAS
Repeated acquisition
Acquisition time: 65 min
vs.
Acquisition time: 15 min!
Acquisition time: 15 min
Example: High resolution single-shell data
Data: R. Heidemann, MPI for Cognitive Neuroscience Leipzig
MAGNETOM 7T (Siemens), b-value of 1000s/mm2 , 60(240) directions. FOV: 143 × 147mm2 , 91 slices, isotropic resolution
of800µm.
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Example: High resolution multi-shell data
Observed images from, same gradients but b-values 800 and 2000
smoothed images L = 4, σ = 30
Data: S. Mohammadi, Wellcome Trust Centre for Neuroimaging, UCL Institute of Neurology, London.
3T MAGNETOM Trio scanner (Siemens), reduced FoV-technique (Heidemann2010), FoV: 161 × 58mm around the motor cortex, isotropic
in-plane resolution of 1.2mm. 34 slices of 1.3mm slice thickness. b-values: b = 800s/mm2 and b = 2000s/mm2 each with
100 gradient directions. 21 S0 images.
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Variability-Reduction by Smoothing
Mean absolute deviation of direction from Behrens 1-stick-1-ball model. b) MAD Improvements
vrs. FA. d) sampled directions describing the posterior d. e) MAD orig. vs. smoothed data.
Position Orientation Adaptive Smoothing (POAS) in Diffusion Weighted Imaging · Neuroimaging Data Analysis, SAMSI,
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Effect of smoothing: Deterministic streamline fiber tracks
Original data
Smoothed
Smoothed (roiy = 91)
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Collaborations
Cooperation:
Citigroup Biomedical Imaging Center, Weill Medical College, Cornell University, NY, U.S.A.
University of Münster
BNIC, Charité, Berlin
Max-Plank Institute for Human Cognitive and Brain Sciences, Leipzig
Wellcome Trust Centre for Neuroimaging at UCL, ION UCL, London, UK
R-Community:
CRAN Task View: Medical Image Analysis
Special volume 44 on Magnetic Resonance Imaging in R of Journal of Statistical Software
Acknowledgments:
We thank the Weill Medical College, Cornell University, the Max Planck Institute for Human Cognitive
and Brain Sciences, the Wellcome Trust Centre for Neuroimaging at UCL, the University of Münster
and the NIH/NCRR Center for Integrative Biomedical Computing (P41-RR12553) for providing
functional and diffusion-weighted MR datasets.
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Software
R-Package dti: Modeling, Smoothing, Fiber tracking, Visualization of DWI data
relies on Packages oro.DICOM, oro.nifti, gsl, adimpro, rgl
other related packages: fmri and aws
The examples in this talk have been done using R-Package dti
POAS will also be part of the ACID toolbox for SPM (S. Mohammadi)
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Bibliography (Selection)
J. Polzehl, V. Spokoiny (2000).
Adaptive Weights Smoothing with applications to image restoration,
J. R. Stat. Soc. Ser. B Stat. Methodol., 62: 335–354.
J. Polzehl, V. Spokoiny (2006).
Local Likelihood Modeling by Adaptive Weights Smoothing.
Probability Theory and Related Fields, 135: 335–362.
S. Becker, P. Mathe (2013).
A new perspective on the Propagation-Separation approach: Taking advantage of the propagation condition
WIAS Preprint #1766
K. Tabelow, J. Polzehl, V. Spokoiny, H.U. Voss (2008).
Diffusion tensor imaging: Structural adaptive smoothing.
Neuroimage, 39(4): 1763–1773.
K. Tabelow, H.U. Voss, J. Polzehl (2012).
Modeling the orientation distribution function by mixtures of angular central Gaussian distributions.
Journal of Neuroscience Methods, 203, 200-211.
J. Polzehl, K. Tabelow (2011).
Beyond the Gaussian Model in Diffusion-Weighted Imaging: The Package dti
Journal of Statistical Software, 44(12), 1–26.
S. Becker, K. Tabelow, H.U. Voss, A. Anwander, R. Heidemann, J. Polzehl (2012).
Position-orientation adaptive smoothing of diffusion weighted magnetic resonance data (POAS)
Medical Image Analysis, 16, 1142–1155.
S. Becker, K. Tabelow, S. Mohammadi, N. Weiskopf, J. Polzehl (2013).
Adaptive smoothing of multi-shell diffusion-weighted magnetic resonance data by msPOAS
WIAS Preprint # 17xx
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