sparse shape representation using the laplace-beltrami eigenfunctions and its application to...

Sparse shape representation using the Laplace-Beltrami eigenfunctions

and its application to correlating functional signal to subcortical structures

Seung-Goo KimBCS @ SNU

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

CONTENTS

• Surface modeling analysis

• Sparse regression on measures

• Effects of normal aging and gender

• + Correlating the anatomical measures with functional signal

MOTIVATION

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

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.

(Distance from medial axis)Xu et al., 2008, NeuroImage.

(Normal surface momentum)Qiu & Miller, 2008, NeuroImage.

Surface modeling analyses

METHODS

Manual segmentations on Individual MRIs

52 healthy subjectsAge: 38-79 yrs

Gender: 16 M, 36 F


Template image

Advanced Normalization Tools (ANTS)


Gender: 16 M, 36 F


Template image

Advanced Normalization Tools (ANTS)

Averaged surfaces


Gender: 16 M, 36 F

Displacement field of the LEFT HIPPOCAMPUS of a subject (37/F)

Displacement Demo: from template to 37/F

Displacement Demo: from template to 73/M

Why Smoothness?

The MathWorksTM

Why Smoothness?

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b) N(0,52); SNR=0.05

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d) FWHM=5; SNR=0.38

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The MathWorksTM

Why Smoothness?

• To boost up SNR & statistical power,

• To reduce sampling noise,

• To Random Field Theory to work,

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d) FWHM=5; SNR=0.38

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Parametrization of measurement


p ∈ M ⊂ R3

Y(p) = θ(p) + �(p)Measurement model


p ∈ M ⊂ R3


θ(p) =k�

i=0

βjψj

Fourier expansion


p ∈ M ⊂ R3


θ(p) =k�

i=0

βjψj

Fourier expansion

∆ψj = λjψj

Laplcae-Beltrami Eigenfunctions


p ∈ M ⊂ R3


θ(p) =k�

i=0

βjψj

Fourier expansion

∆ψj = λjψj


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.


p ∈ M ⊂ R3


θ(p) =k�

i=0

βjψj

Fourier expansion

∆ψj = λjψj


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

Coefficient Estimation

Least Square estimation�β = (ψ �ψ)−1ψ �Y

Y = ψβ



l1-penalty*minβ

||Y −ψβ||22+λ||β||1

Y = ψβ



l1-penalty*minβ

||Y −ψβ||22+λ||β||1

0 500 10000

1

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LSEl1 penalty

80 100 120 1400

0.5

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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

LSE vs. l1-minimization

RESULTS

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


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


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


malefemale

male female1000

1500

2000

2500

gender

Left

Amyg

dala

(mm


male female1000

1500

2000

2500

gender

Rig

ht A

myg

dala

(mm


male female2000

3000

4000

5000

gender

Tota

l Am

ygda

la (m


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 )


male female2000

4000

6000

8000

genderTota

l Hip

poca

mpu

s (m


a)

b)

40 50 60 70 801000

1500

2000

2500

age (yr)

Left

Amyg

dala

(mm


40 50 60 70 801000

1500

2000

2500

age (yr)

Rig

ht A

myg

dala

(mm


40 50 60 70 802000

3000

4000

5000

age (yr)

Tota

l Am

ygda

la (m


40 50 60 70 801000

2000

3000

4000

age (yr)

Left

Hip

poca

mpu

s (m


40 50 60 70 801000

2000

3000

4000

age (yr)Rig

ht H

ippo

cam

pus

(mm

3 )


40 50 60 70 802000

4000

6000

8000

age (yr)Tota

l Hip

poca

mpu

s (m


malefemale

male female1000

1500

2000

2500

gender

Left

Amyg

dala

(mm


male female1000

1500

2000

2500

gender

Rig

ht A

myg

dala

(mm


male female2000

3000

4000

5000

gender

Tota

l Am

ygda

la (m


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 )


male female2000

4000

6000

8000

genderTota

l Hip

poca

mpu

s (m


a)

b)

Volumetric analysisVolume = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender+ �

Deformation-based shape analysisLength = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender+ �

LSE vs. l1-minimization

t-statistic maps

+ Correlating with functional measures

Preliminary: use of EMG as an

emotional response

Defensive behaviors as objective measure of emotionality

www.somewhre.com

Defensive behaviors as objective measure of emotionality• Startle Reflex is known to subject to the

presence of threats in animals.

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• Also in human, startling reflex as eye blink can reflect the inner state affected by threats (Lang et al., 1997).

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• 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.

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Electromyography (EMG) for eye blink reflex

!

Lang et al., 1990, Psychol. Rev.

Eye Blink Reflex & Emotionality

Bradley et al., 2001, Emotion.

Experiment procedure

International Affective Picture System (IAPS)

(c) ICPSR

(c) ICPSR

probe Aprobe B

probe C

(c) ICPSR

3 picture-conditionsx 3 probe-timings= 9 types of trials

probe Aprobe B

probe C

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

Age interaction with EMG

• EMG effect: β5 was not significant (p’s>0.33)

Length = β1 + β2 · Brain+ β3 ·Age+ β4 ·Gender

+ β5 · EMG+ �

Age interaction with EMG

• EMG effect: β5 was not significant (p’s>0.33)


+ β5 · EMG+ �


+ 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)

Positive, Probe C

Residual = Length− (�β1 + �β2 · Brain+ �β3 ·Age

+ �β4 ·Gender+ �β5 · EMG)

Neutral, Probe A

Conclusions

Conclusions

• Surface modeling analysis gives more sensitivity than volumetric analysis.

Conclusions


• l1-minimization gives sparse solution of β constructing more smooth data than LSE.

Conclusions



• Large displacements on the hippocampal tails are associated with aging.

Conclusions



• Large displacements on the hippocampal tails are associated with aging.

• Some eye blink reflex measures interact with the age on amygdalar and hippocampal structures.

Thank you for your attention!

sparse shape representation using the laplace-beltrami eigenfunctions and its application to...

Technology

age yr

p pmr3fourier expansionp

j j coefcient estimationy

j j cotan discretization

amygdala volume

hippocampus amygdala

hippocampus volume

yrs gender