Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to correlating functional signal to subcortical structures. “Workshop on mathematical methods in medical image analysis” (organized by Dr. Moo K. Chung). Seoul, South Korea. Sep 27, 2011.

1. Sparse shape representation using the
Laplace-Beltrami eigenfunctions
and its application to correlating functional
signal to subcortical structures
Seung-Goo Kim
BCS @ SNU

2. ACKNOWLEDGEMENT
• Formulation & implementation of LaplaceBeltrami eigenfunction
• Moo K. Chung @ SNU
• “MIDUS II” project: data collection
• Stacey M. Schaefer, Carien van Reekum,
Richard J. Davidson @ U of Wisconsin

4. CONTENTS
• Surface modeling analysis
• Sparse regression on measures
• Effects of normal aging and gender
• + Correlating the anatomical measures
with functional signal

5. MOTIVATION

6. Atlas-based automatic
segmentation using
FreeSurfer
Quadratic decrease
in Hippocampus &
Amygdala
R2
R2
1
2
3
4
5
6
Walhovd et al., 2009, Neurobiol. Aging.
Total n=883

7. Manual segmentation,
84 men: 21-81 yrs
44 women: 20-85 yrs
No significant
aging effects in
Hippocampus
volume,
but significant
decrease in
Amygdala volume.
Sullivan et al., 2005, Neurobiol. Aging.

8. Surface modeling analyses
(Distance from medial axis)
Xu et al., 2008, NeuroImage.
(Normal surface momentum)
Qiu & Miller, 2008, NeuroImage.

9. METHODS

11. Manual segmentations on
Individual MRIs
52 healthy subjects
Age: 38-79 yrs
Gender: 16 M, 36 F

12. Manual segmentations on
Individual MRIs
Template image
Advanced
Normalization Tools (ANTS)
52 healthy subjects
Age: 38-79 yrs
Gender: 16 M, 36 F

13. Manual segmentations on
Individual MRIs
Template image
Advanced
Normalization Tools (ANTS)
Averaged surfaces
52 healthy subjects
Age: 38-79 yrs
Gender: 16 M, 36 F

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

15. Displacement Demo: from template to 37/F

16. Displacement Demo: from template to 37/F

17. Displacement Demo: from template to 73/M

18. Displacement Demo: from template to 73/M

20. Why Smoothness?
The MathWorksTM

21. Why Smoothness?
b) N(0,52); SNR=0.05
1
200
200
400
0.5
600
800
6
20
10
400
0
400
600
200 400 600 800
0
4
200
800
2
0
600
800
200 400 600 800
200 400 600 800
The MathWorksTM
d) FWHM=5; SNR=0.38
20
2
200
400
0
600
800
200
15
400
10
5
600
200 400 600 800
800
0
200 400 600 800

22. Why Smoothness?
• To boost up SNR & statistical power,
• To reduce sampling noise,
• To Random Field Theory to work,
b) N(0,52); SNR=0.05
1
200
200
400
0.5
600
800
6
20
10
400
0
400
600
200 400 600 800
0
4
200
800
2
0
600
800
200 400 600 800
200 400 600 800
d) FWHM=5; SNR=0.38
20
2
200
400
0
200
15
400
10
600
800
800
5
600
200 400 600 800
0
200 400 600 800

23. Parametrization of measurement

24. Parametrization of measurement
Measurement model
Y(p) = θ(p) + (p)
p ∈ M ⊂ R3

25. Parametrization of measurement
Measurement model
Y(p) = θ(p) + (p)
p ∈ M ⊂ R3
Fourier expansion
θ(p) =
k
i=0
β j ψj

26. Parametrization of measurement
Laplcae-Beltrami
Eigenfunctions
Measurement model
Y(p) = θ(p) + (p)
p∈M⊂R
3
Fourier expansion
θ(p) =
k
i=0
β j ψj
∆ψj = λj ψj

27. Parametrization of measurement
Laplcae-Beltrami
Eigenfunctions
Measurement model
Y(p) = θ(p) + (p)
p∈M⊂R
3
Fourier expansion
θ(p) =
k
∆ψj = λj ψj
Cotan discretization*:
Cψ = λAψ
β j ψj
i=0
* Anqi et al.,Smooth functional and structural maps on the neocortex via
orthonormal bases of the Laplace-Beltrami operator, IEEE TMI., 2006.

28. Parametrization of measurement
Laplcae-Beltrami
Eigenfunctions
Measurement model
Y(p) = θ(p) + (p)
p∈M⊂R
3
Fourier expansion
θ(p) =
k
i=0
∆ψj = λj ψj
Cotan discretization*:
Cψ = λAψ
β j ψj
Coefficient Estimation
Y = ψβ
* Anqi et al.,Smooth functional and structural maps on the neocortex via
orthonormal bases of the Laplace-Beltrami operator, IEEE TMI., 2006.

29. Coefficient Estimation
Y = ψβ
Least Square estimation
β = (ψ ψ)−1 ψ Y

30. Coefficient Estimation
Y = ψβ
Least Square estimation
l1-penalty*
β = (ψ ψ)−1 ψ Y
min ||Y − ψβ||2 +λ||β||1
2
β

31. Coefficient Estimation
Y = ψβ
Least Square estimation
l1-penalty*
β = (ψ ψ)−1 ψ Y
min ||Y − ψβ||2 +λ||β||1
2
β
5
4
2
LSE
l1 penalty
1.5
3
1
2
0.5
1
0
0
500
1000
0
80
100
120
140
* Implementation: Kim et al., An Interior-Point Method for Large-Scale l1-Regularized
Least Squares. IEEE J. Select.Topics Signal Processing, 2007.

32. LSE vs. l1-minimization

33. RESULTS

34. Volumetric analysis
Volume = β1 + β2 · Brain + β3 · Age + β4 · Gender +
1000
40
50
60
70
age (yr)
Not significant, p=0.25
4000
3000
2000
1000
40
50
60
70
age (yr)
Not significant, p=0.26
2500
2000
1500
1000
male
female
80
Total Amygdala (mm3)
1500
80
Total Hippocampus (mm3)
2000
Not significant, p=0.23
2500
2000
1500
1000
40
50
60
70
age (yr)
Not significant, p=0.53
4000
3000
2000
1000
40
50
60
70
age (yr)
Not significant, p=0.47
2500
2000
1500
1000
male
female
Total Amygdala (mm3)
Right Amygdala (mm3)
80
2500
b)
Left Amygdala (mm3)
80
Right Hippocampus (mm3)
Not significant, p=0.4
Right Amygdala (mm3)
Left Hippocampus (mm3)
Left Amygdala (mm3)
a)
Not significant, p=0.29
5000
male
female
4000
3000
2000
40
50
60
70
age (yr)
Not significant, p=0.34
80
50
80
8000
6000
4000
2000
40
60
70
age (yr)
Not significant, p=0.34
5000
4000
3000
2000
male
female

35. 1000
50
60
70
age (yr)
Not significant, p=0.25
5000
80
2000
1500
1000
40
50
60
70
age (yr)
Not significant, p=0.53
80
4000
male
female
4000
3000
2000
40
50
60
70
age (yr)
Not significant, p=0.34
80
8000
1000
40
50
60
70
age (yr)
Not significant, p=0.26
2500
2000
1500
1000
male
female
4000
3000
2000
1000
male
female
gender
80
2000
1000
40
50
60
70
age (yr)
Not significant, p=0.47
2500
2000
1500
1000
male
female
gender
Not significant, p=0.12
4000
3000
2000
1000
male
female
gender
80
Total Amygdala (mm3)
2000
3000
Total Hippocampus (mm3)
3000
Right Amygdala (mm3)
Volume = β1 + β2 · Brain + β3 · Age + β4 · Gender +
gender
Left Hippocampus (mm3)
Not significant, p=0.29
Volumetric analysis
4000
b)
Left Amygdala (mm3)
40
2500
Total Hippocampus (mm3)
1500
Right Hippocampus (mm3)
2000
Not significant, p=0.23
Total Amygdala (mm3)
2500
Right Amygdala (mm3)
Not significant, p=0.4
Right Hippocampus (mm3)
Left Hippocampus (mm3)
Left Amygdala (mm3)
a)
6000
4000
2000
40
50
60
70
age (yr)
Not significant, p=0.34
5000
4000
3000
2000
male
female
gender
Not significant, p=0.054
8000
6000
4000
2000
male
female
gender
80

36. Deformation-based shape analysis
Length = β1 + β2 · Brain + β3 · Age + β4 · Gender +

37. Deformation-based shape analysis
Length = β1 + β2 · Brain + β3 · Age + β4 · Gender +

38. LSE vs. l1-minimization

39. LSE vs. l1-minimization

40. t-statistic maps

41. t-statistic maps

42. + Correlating with
functional measures

43. Preliminary:
use of EMG as an
emotional response

44. Defensive behaviors as objective
measure of emotionality
www.somewhre.com

45. Defensive behaviors as objective
measure of emotionality
• Startle Reflex is known to subject to the
presence of threats in animals.
www.somewhre.com

46. 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

47. 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 measure
of emotionality in laboratory.
www.somewhre.com

48. Electromyography (EMG)
for eye blink reflex
!
Lang et al., 1990, Psychol. Rev.

49. Eye Blink Reflex Emotionality
Bradley et al., 2001, Emotion.

50. Eye Blink Reflex Emotionality
Bradley et al., 2001, Emotion.

51. Eye Blink Reflex Emotionality
Bradley et al., 2001, Emotion.

52. Experiment procedure

53. International Affective
Picture System (IAPS)

54. International Affective
Picture System (IAPS)

55. International Affective
Picture System (IAPS)

56. (c) ICPSR

57. probe A
probe B
probe C
(c) ICPSR

58. 3 picture-conditions
x 3 probe-timings
= 9 types of trials
probe A
probe B
probe C
(c) ICPSR

59. 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
BLUMENTHAL et al., 2005, Psyphysiol.

60. 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
BLUMENTHAL et al., 2005, Psyphysiol.

61. 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
BLUMENTHAL et al., 2005, Psyphysiol.

62. 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
BLUMENTHAL et al., 2005, Psyphysiol.

63. Age interaction with EMG
Length = β1 + β2 · Brain + β3 · Age + β4 · Gender
+ β5 · EMG +
• EMG effect: β
5
was not significant (p’s0.33)

64. Age interaction with EMG
Length = β1 + β2 · Brain + β3 · Age + β4 · Gender
+ β5 · EMG +
• EMG effect: β
5
was not significant (p’s0.33)
Length = β1 + β2 · Brain + β3 · Age + β4 · Gender
+ β5 · EMG + β6 · Age · EMG +
• But found significant AGE x EMG interactions (β )
• Positive picture @ Probe C (1.9 s after offset)
• Neutral picture @ Probe A (2.9 s after onset)
6

65. Positive, Probe C
Residual = Length − (β1 + β2 · Brain + β3 · Age
+ β4 · Gender + β5 · EMG)

66. Positive, Probe C
Residual = Length − (β1 + β2 · Brain + β3 · Age
+ β4 · Gender + β5 · EMG)

67. Neutral, Probe A

68. Neutral, Probe A

69. Neutral, Probe A

70. Conclusions

71. Conclusions
•
Surface modeling analysis gives more sensitivity
than volumetric analysis.

72. Conclusions
•
Surface modeling analysis gives more sensitivity
than volumetric analysis.
•
l1-minimization gives sparse solution of β
constructing more smooth data than LSE.

73. 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.

74. 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.

75. Thank you for your attention!