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![Page 1: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/1.jpg)
Sparse shape representation using the Laplace-Beltrami eigenfunctions
and its application to correlating functional signal to subcortical structures
Seung-Goo KimBCS @ SNU
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ACKNOWLEDGEMENT• Formulation & implementation of Laplace-
Beltrami eigenfunction
• Moo K. Chung @ SNU
• “MIDUS II” project: data collection
• Stacey M. Schaefer, Carien van Reekum, Richard J. Davidson @ U of Wisconsin
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CONTENTS
• Surface modeling analysis
• Sparse regression on measures
• Effects of normal aging and gender
• + Correlating the anatomical measures with functional signal
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MOTIVATION
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Walhovd et al., 2009, Neurobiol. Aging.
R2
Atlas-based automatic segmentation using FreeSurfer
Quadratic decrease in Hippocampus & Amygdala
1
23
4
56
Total n=883
R2
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Manual segmentation,84 men: 21-81 yrs44 women: 20-85 yrs
No significant aging effects in Hippocampus volume,but significant decrease in Amygdala volume.
Sullivan et al., 2005, Neurobiol. Aging.
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(Distance from medial axis)Xu et al., 2008, NeuroImage.
(Normal surface momentum)Qiu & Miller, 2008, NeuroImage.
Surface modeling analyses
![Page 9: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/9.jpg)
METHODS
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Manual segmentations on Individual MRIs
52 healthy subjectsAge: 38-79 yrs
Gender: 16 M, 36 F
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Manual segmentations on Individual MRIs
Template image
Advanced Normalization Tools (ANTS)
52 healthy subjectsAge: 38-79 yrs
Gender: 16 M, 36 F
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Manual segmentations on Individual MRIs
Template image
Advanced Normalization Tools (ANTS)
Averaged surfaces
52 healthy subjectsAge: 38-79 yrs
Gender: 16 M, 36 F
![Page 14: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/14.jpg)
Displacement field of the LEFT HIPPOCAMPUS of a subject (37/F)
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Displacement Demo: from template to 37/F
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Displacement Demo: from template to 37/F
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Displacement Demo: from template to 73/M
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Displacement Demo: from template to 73/M
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Why Smoothness?
The MathWorksTM
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Why Smoothness?
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The MathWorksTM
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Why Smoothness?
• To boost up SNR & statistical power,
• To reduce sampling noise,
• To Random Field Theory to work,
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Parametrization of measurement
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Parametrization of measurement
p ∈ M ⊂ R3
Y(p) = θ(p) + �(p)Measurement model
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Parametrization of measurement
p ∈ M ⊂ R3
Y(p) = θ(p) + �(p)Measurement model
θ(p) =k�
i=0
βjψj
Fourier expansion
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Parametrization of measurement
p ∈ M ⊂ R3
Y(p) = θ(p) + �(p)Measurement model
θ(p) =k�
i=0
βjψj
Fourier expansion
∆ψj = λjψj
Laplcae-Beltrami Eigenfunctions
![Page 27: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/27.jpg)
Parametrization of measurement
p ∈ M ⊂ R3
Y(p) = θ(p) + �(p)Measurement model
θ(p) =k�
i=0
βjψj
Fourier expansion
∆ψj = λjψj
Laplcae-Beltrami Eigenfunctions
Cψ = λAψCotan discretization*:
* Anqi et al.,Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator, IEEE TMI., 2006.
![Page 28: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/28.jpg)
Parametrization of measurement
p ∈ M ⊂ R3
Y(p) = θ(p) + �(p)Measurement model
θ(p) =k�
i=0
βjψj
Fourier expansion
∆ψj = λjψj
Laplcae-Beltrami Eigenfunctions
Cψ = λAψCotan discretization*:
* Anqi et al.,Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator, IEEE TMI., 2006.
Coefficient EstimationY = ψβ
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Coefficient Estimation
Least Square estimation�β = (ψ �ψ)−1ψ �Y
Y = ψβ
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Coefficient Estimation
Least Square estimation�β = (ψ �ψ)−1ψ �Y
l1-penalty*minβ
||Y −ψβ||22+λ||β||1
Y = ψβ
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Coefficient Estimation
Least Square estimation�β = (ψ �ψ)−1ψ �Y
l1-penalty*minβ
||Y −ψβ||22+λ||β||1
0 500 10000
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LSEl1 penalty
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* Implementation: Kim et al., An Interior-Point Method for Large-Scale l1-Regularized Least Squares. IEEE J. Select. Topics Signal Processing, 2007.
Y = ψβ
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LSE vs. l1-minimization
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RESULTS
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40 50 60 70 801000
1500
2000
2500
age (yr)
Left
Amyg
dala
(mm
3 ) Not significant, p=0.4
40 50 60 70 801000
1500
2000
2500
age (yr)
Rig
ht A
myg
dala
(mm
3 ) Not significant, p=0.23
40 50 60 70 802000
3000
4000
5000
age (yr)
Tota
l Am
ygda
la (m
m3 ) Not significant, p=0.29
40 50 60 70 801000
2000
3000
4000
age (yr)
Left
Hip
poca
mpu
s (m
m3 ) Not significant, p=0.25
40 50 60 70 801000
2000
3000
4000
age (yr)Rig
ht H
ippo
cam
pus
(mm
3 )
Not significant, p=0.53
40 50 60 70 802000
4000
6000
8000
age (yr)Tota
l Hip
poca
mpu
s (m
m3 ) Not significant, p=0.34
malefemale
male female1000
1500
2000
2500
gender
Left
Amyg
dala
(mm
3 ) Not significant, p=0.26
male female1000
1500
2000
2500
gender
Rig
ht A
myg
dala
(mm
3 ) Not significant, p=0.47
male female2000
3000
4000
5000
gender
Tota
l Am
ygda
la (m
m3 ) Not significant, p=0.34
male female1000
2000
3000
4000
gender
Left
Hip
poca
mpu
s (m
m3 )
male female1000
2000
3000
4000
genderRig
ht H
ippo
cam
pus
(mm
3 )
Not significant, p=0.12
male female2000
4000
6000
8000
genderTota
l Hip
poca
mpu
s (m
m3 ) Not significant, p=0.054
a)
b)
40 50 60 70 801000
1500
2000
2500
age (yr)
Left
Amyg
dala
(mm
3 ) Not significant, p=0.4
40 50 60 70 801000
1500
2000
2500
age (yr)
Rig
ht A
myg
dala
(mm
3 ) Not significant, p=0.23
40 50 60 70 802000
3000
4000
5000
age (yr)
Tota
l Am
ygda
la (m
m3 ) Not significant, p=0.29
40 50 60 70 801000
2000
3000
4000
age (yr)
Left
Hip
poca
mpu
s (m
m3 ) Not significant, p=0.25
40 50 60 70 801000
2000
3000
4000
age (yr)Rig
ht H
ippo
cam
pus
(mm
3 )
Not significant, p=0.53
40 50 60 70 802000
4000
6000
8000
age (yr)Tota
l Hip
poca
mpu
s (m
m3 ) Not significant, p=0.34
malefemale
male female1000
1500
2000
2500
gender
Left
Amyg
dala
(mm
3 ) Not significant, p=0.26
male female1000
1500
2000
2500
gender
Rig
ht A
myg
dala
(mm
3 ) Not significant, p=0.47
male female2000
3000
4000
5000
gender
Tota
l Am
ygda
la (m
m3 ) Not significant, p=0.34
male female1000
2000
3000
4000
gender
Left
Hip
poca
mpu
s (m
m3 )
male female1000
2000
3000
4000
genderRig
ht H
ippo
cam
pus
(mm
3 )
Not significant, p=0.12
male female2000
4000
6000
8000
genderTota
l Hip
poca
mpu
s (m
m3 ) Not significant, p=0.054
a)
b)
Volumetric analysisVolume = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender+ �
![Page 35: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/35.jpg)
40 50 60 70 801000
1500
2000
2500
age (yr)
Left
Amyg
dala
(mm
3 ) Not significant, p=0.4
40 50 60 70 801000
1500
2000
2500
age (yr)
Rig
ht A
myg
dala
(mm
3 ) Not significant, p=0.23
40 50 60 70 802000
3000
4000
5000
age (yr)
Tota
l Am
ygda
la (m
m3 ) Not significant, p=0.29
40 50 60 70 801000
2000
3000
4000
age (yr)
Left
Hip
poca
mpu
s (m
m3 ) Not significant, p=0.25
40 50 60 70 801000
2000
3000
4000
age (yr)Rig
ht H
ippo
cam
pus
(mm
3 )
Not significant, p=0.53
40 50 60 70 802000
4000
6000
8000
age (yr)Tota
l Hip
poca
mpu
s (m
m3 ) Not significant, p=0.34
malefemale
male female1000
1500
2000
2500
gender
Left
Amyg
dala
(mm
3 ) Not significant, p=0.26
male female1000
1500
2000
2500
gender
Rig
ht A
myg
dala
(mm
3 ) Not significant, p=0.47
male female2000
3000
4000
5000
gender
Tota
l Am
ygda
la (m
m3 ) Not significant, p=0.34
male female1000
2000
3000
4000
gender
Left
Hip
poca
mpu
s (m
m3 )
male female1000
2000
3000
4000
genderRig
ht H
ippo
cam
pus
(mm
3 )
Not significant, p=0.12
male female2000
4000
6000
8000
genderTota
l Hip
poca
mpu
s (m
m3 ) Not significant, p=0.054
a)
b)
40 50 60 70 801000
1500
2000
2500
age (yr)
Left
Amyg
dala
(mm
3 ) Not significant, p=0.4
40 50 60 70 801000
1500
2000
2500
age (yr)
Rig
ht A
myg
dala
(mm
3 ) Not significant, p=0.23
40 50 60 70 802000
3000
4000
5000
age (yr)
Tota
l Am
ygda
la (m
m3 ) Not significant, p=0.29
40 50 60 70 801000
2000
3000
4000
age (yr)
Left
Hip
poca
mpu
s (m
m3 ) Not significant, p=0.25
40 50 60 70 801000
2000
3000
4000
age (yr)Rig
ht H
ippo
cam
pus
(mm
3 )
Not significant, p=0.53
40 50 60 70 802000
4000
6000
8000
age (yr)Tota
l Hip
poca
mpu
s (m
m3 ) Not significant, p=0.34
malefemale
male female1000
1500
2000
2500
gender
Left
Amyg
dala
(mm
3 ) Not significant, p=0.26
male female1000
1500
2000
2500
gender
Rig
ht A
myg
dala
(mm
3 ) Not significant, p=0.47
male female2000
3000
4000
5000
gender
Tota
l Am
ygda
la (m
m3 ) Not significant, p=0.34
male female1000
2000
3000
4000
gender
Left
Hip
poca
mpu
s (m
m3 )
male female1000
2000
3000
4000
genderRig
ht H
ippo
cam
pus
(mm
3 )
Not significant, p=0.12
male female2000
4000
6000
8000
genderTota
l Hip
poca
mpu
s (m
m3 ) Not significant, p=0.054
a)
b)
Volumetric analysisVolume = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender+ �
![Page 36: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/36.jpg)
Deformation-based shape analysisLength = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender+ �
![Page 37: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/37.jpg)
Deformation-based shape analysisLength = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender+ �
![Page 38: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/38.jpg)
LSE vs. l1-minimization
![Page 39: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/39.jpg)
LSE vs. l1-minimization
![Page 40: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/40.jpg)
t-statistic maps
![Page 41: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/41.jpg)
t-statistic maps
![Page 42: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/42.jpg)
+ Correlating with functional measures
![Page 43: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/43.jpg)
Preliminary: use of EMG as an
emotional response
![Page 44: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/44.jpg)
Defensive behaviors as objective measure of emotionality
www.somewhre.com
![Page 45: Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures](https://reader033.fdocuments.us/reader033/viewer/2022052900/5563be41d8b42ac70d8b56ad/html5/thumbnails/45.jpg)
Defensive behaviors as objective measure of emotionality• Startle Reflex is known to subject to the
presence of threats in animals.
www.somewhre.com
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Defensive behaviors as objective measure of emotionality• Startle Reflex is known to subject to the
presence of threats in animals.
• Also in human, startling reflex as eye blink can reflect the inner state affected by threats (Lang et al., 1997).
www.somewhre.com
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Defensive behaviors as objective measure of emotionality• Startle Reflex is known to subject to the
presence of threats in animals.
• Also in human, startling reflex as eye blink can reflect the inner state affected by threats (Lang et al., 1997).
• Thus eye blink can be used as an objective measureof emotionality in laboratory.
www.somewhre.com
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Electromyography (EMG) for eye blink reflex
!
Lang et al., 1990, Psychol. Rev.
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Eye Blink Reflex & Emotionality
Bradley et al., 2001, Emotion.
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Eye Blink Reflex & Emotionality
Bradley et al., 2001, Emotion.
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Eye Blink Reflex & Emotionality
Bradley et al., 2001, Emotion.
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Experiment procedure
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International Affective Picture System (IAPS)
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International Affective Picture System (IAPS)
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International Affective Picture System (IAPS)
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(c) ICPSR
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(c) ICPSR
probe Aprobe B
probe C
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(c) ICPSR
3 picture-conditionsx 3 probe-timings= 9 types of trials
probe Aprobe B
probe C
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BLUMENTHAL et al., 2005, Psyphysiol.
EMG signal process
• Artifacts rejection, rectification, low-pass filtering (smoothing)
• EBR = Peak - Reflex Onset
• Peak: max(EMG) [20,120] ms after probe onset
• Logarithm, then z-score transformation
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BLUMENTHAL et al., 2005, Psyphysiol.
EMG signal process
• Artifacts rejection, rectification, low-pass filtering (smoothing)
• EBR = Peak - Reflex Onset
• Peak: max(EMG) [20,120] ms after probe onset
• Logarithm, then z-score transformation
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BLUMENTHAL et al., 2005, Psyphysiol.
EMG signal process
• Artifacts rejection, rectification, low-pass filtering (smoothing)
• EBR = Peak - Reflex Onset
• Peak: max(EMG) [20,120] ms after probe onset
• Logarithm, then z-score transformation
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BLUMENTHAL et al., 2005, Psyphysiol.
EMG signal process
• Artifacts rejection, rectification, low-pass filtering (smoothing)
• EBR = Peak - Reflex Onset
• Peak: max(EMG) [20,120] ms after probe onset
• Logarithm, then z-score transformation
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Age interaction with EMG
• EMG effect: β5 was not significant (p’s>0.33)
Length = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender
+ β5 · EMG+ �
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Age interaction with EMG
• EMG effect: β5 was not significant (p’s>0.33)
Length = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender
+ β5 · EMG+ �
Length = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender
+ β5 · EMG+ β6 ·Age · EMG+ �
• But found significant AGE x EMG interactions (β6)
• Positive picture @ Probe C (1.9 s after offset)
• Neutral picture @ Probe A (2.9 s after onset)
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Positive, Probe C
Residual = Length− (�β1 + �β2 · Brain+ �β3 ·Age
+ �β4 ·Gender+ �β5 · EMG)
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Positive, Probe C
Residual = Length− (�β1 + �β2 · Brain+ �β3 ·Age
+ �β4 ·Gender+ �β5 · EMG)
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Neutral, Probe A
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Neutral, Probe A
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Neutral, Probe A
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Conclusions
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Conclusions
• Surface modeling analysis gives more sensitivity than volumetric analysis.
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Conclusions
• Surface modeling analysis gives more sensitivity than volumetric analysis.
• l1-minimization gives sparse solution of β constructing more smooth data than LSE.
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Conclusions
• Surface modeling analysis gives more sensitivity than volumetric analysis.
• l1-minimization gives sparse solution of β constructing more smooth data than LSE.
• Large displacements on the hippocampal tails are associated with aging.
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Conclusions
• Surface modeling analysis gives more sensitivity than volumetric analysis.
• l1-minimization gives sparse solution of β constructing more smooth data than LSE.
• Large displacements on the hippocampal tails are associated with aging.
• Some eye blink reflex measures interact with the age on amygdalar and hippocampal structures.
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Thank you for your attention!