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Development of optical
tomography methods with
discretized path integral
D136580
Bingzhi Yuan
Tomography devices
MRI Magnetic Resonance ImagingPET Positron Emission tomography
X-ray image is from http://en.wikipedia.org/wiki/X-ray_computed_tomography MRI image is from http://en.wikipedia.org/wiki/MRI PET image is from http://en.wikipedia.org/wiki/Positron_emission_tomography
High
magnetic
field
radioactive
X-ray CT
radioactive
2
Research motivation
Light Source
Camera
Develop an optical tomography method can provide
better and sharper result than DOT
Diffusion Optical Tomography
(DOT)
Blurred
results
3[Chung SC et al. 2015]
Problem and Challenges in the optical
tomography
X-rayInfrared Light
Scattering+Attenuation Attenuation
4
Framework of optical tomography
Forward problem
Modeling the light transport
Light source
Detector
Inverse problem
Estimating the property of the
material
Light source
Detector
Scattering
Attenuation
5
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
Inverse
problem
6
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
7
Related Works
Single ScatteringDiffusion Optical Tomography
Distribution of the scattered
light
Incident light
Distribution of the scattered
light
Incident light
Phase function has a spherical shape
[Florescu et al. 2009]
[Florescu et al. 2010]
[Gonzalez-Rodriguez et al. 1960]
[Arridge et al. 1999]
[Gibson et al. 2005]
Forward Scattering
[Tamaki et al. 2013]
[Ishii et al. 2013]
Multiple Scattering
[Tamaki et al. 2013]
[Ishii et al. 2013]
[Arridge et al. 1999]
[Gibson et al. 2005]
8
Path Integral
Light source
Detector
All the possible paths between the light
source and the detector
should be accumulated.
Path integral is an efficient and promising method
to describe the light transport in computer graphics
[Simon Premože et al. 2003]
9
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
10
Simplify the forward problem
Modeling the light transport
Light source
Detector
2D layered material
N grids
Layer 1
Layer M
……
Layer 2
Our proposed idea
11
2D Layered Material
N grids
Layer 1
Layer M
……
Layer 2
N grids
Layer 1
Layer M
……
Layer 2
Homogeneous in every grid. No back scattering
2.Forward scattering1.Homogenous
Phase function will be introduced in the next section
12
2D Layered Material
N grids
Layer 1
Layer M
……
Layer 2
N grids
Layer 1
Layer M
……
Layer 2
Scatters at the center of a grid,
and points to the center of a grid
Multiple(=NM-2) paths
for a given (i, j)
4.Multiple paths3.Scatter at center
Light source position i
Detector position j
13
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
14
Common phase function in 3D
1 1
Henyey-Greenstein’s phase function
θ is the angle between the scattered light and the incident
light.
p(θ) is the probability of having such an angle.
g describes the scattering property of a media, from back
scattering to isotropic scattering to forward scattering.
g=1
g=0
forward
spherical
Disadvantage: it’s a phase function in 3D space and the back
scattering part can’t be excluded.
g
15
Gaussian as the phase function
Distribution of the Gaussian
σ describe the scattering property, smaller value indicate
the more peaked forward scattering.
σ = 0.2
σ = 0.4
Advantage: works in 2D space, and easy to
implement
16
Discretize the Gaussian
for the 2D layered material
N grids
Layer 1
Layer M
……
Layer 2
17
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
18
Scattering
N grids
Layer m
Layer M
……
Layer m+1 k
Contribution for a path
Measure at the scattering point
scattering coefficient
Gaussian for every scattering point19
Attenuation
Light source
Detector
Model by the integral of the
extinction coefficient along
the path
: extinction coefficient at grid I
: Intensity of the light source : Intensity observed by the detector 20
Discretized path integral in layered material
I0
I1
N grids
Layer 1
Layer M
……
Layer 2
1.Homogenous
2.Forward scattering
3.Scatter at center
4.Multiple paths
Model by the inner product of
the distance vector and
extinction coefficient vector
: vector contains distances at every grid
: vector contains extinction coefficient at every grid 21
Scattering & Attenuation
I0
Iijk
N grids
Layer 1
Layer M
……
Layer 2
1.Homogenous
2.Forward scattering
3.Scatter at center
4.Multiple paths
Light source position i
Detector position j
AttenuationContribution of this path
: Light source position
: Detector position
: index for light path with (i, j)
: Intensity for path ijk
: Intensity for light source
Intensity for path ijk
22
Discretized path integral for forward problem
I0
Iij
N grids
Layer 1
Layer M
……
Layer 2
1.Homogenous
2.Forward scattering
3.Scatter at center
4.Multiple paths
Light source position i
Detector position j
Multiple(𝑁 =NM-2) paths for a given (i, j)
: Light source position
: Detector position
: index for light path with (i, j)
: Intensity for path ijk
: Intensity for light source
: Observed Intensity at (i, j)
Sum all paths
23
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
24
Inverse problem-Construct an inverse problem
Calculate from the model of the forward problem
I0
N grids
Layer 1
Layer M
……
Layer 2
Light source position i
Detector position j
: Light source position
: Detector position
: index for light path with (i, j)
25
Inverse problem
Difference between observed and calculated intensityObserved light intensity
I0
N grids
Layer 1
Layer M
……
Layer 2
Light source position i
Detector position j
: Light source position
: Detector position
: index for light path with (i, j)
26
Inverse problem
N grids
Layer 1
Layer M
……
Layer 2
Light source position i
Detector position j
: Light source position
: Detector position
: index for light path with (i, j)
27
More Observations by changing
configuration
T2B case
28
More Observations by changing
configuration
T2B case R2L case B2T case L2R case
29
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
30
Log-Barrier Interior-point approach to the
inverse problem
constrained optimization problem
Combine the constraints and the cost function
into one equation with log-barrier term
31Possible area
1
2
σ∗
Newton & Quasi Newton for optimization
Newton
Second order derivative
(Hessian) is calculated
Quasi-Newton
Second order derivative is
approximated
Solver for optimization
Accuracy
Computation Cost High Low
LowHigh
32
Numerical simulation
Tested with 5 different 2D layered materials
Material size is 24 by 24
I0
N grids
Layer 1
Layer M
……
Layer 2
Light source position i
Detector position j
33
Given Target
the observed light intensity
the light source
the contribution of have a light path ijk
the distance between each point in the light
path ijk
the extinction coefficient
Given
Target
Comparison
with DOT
Provided by EIDORS
Material size is 24 by 24
solver: Gauss-Newton(GN)
Primal-Dual(PD)
34
EIDOR is from http://eidors3d.sourceforge.net/
Goals in next step:
Improve efficiency
Maintain accuracy
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
35
Primal-Dual interior point approach
General form
Reform as
Lagrangian L
KKT system
KKT condition
Update the result
36
Thesis Structure
• Chapter 1: Introduction
• Chapter 2: Related Works
• Chapter 3: Forward Problem
• Chapter 4: Log-Barrier Interior Point
Approach
• Chapter 5: Primal-Dual Interior Point
Approach
• Chapter 6: Conclusion
• Related work
• 2D Layered material
• Our phase function
• Discretized path integral formulation
• Constraint optimization problem
• Log-Barrier Interior Point Approach
• Primal-Dual Interior Point Approach
• Efficient formulation
Presentation outline
37
Numerical results
LB:
PD:
New:
Old:
Newton:
Quasi-Newton:
Log-Barrier approach
Primal-Dual approach
New formulations were used
Old formulations were used
Newton is the solver for optimization
Quasi-Newton is the solver for
optimization
In total, there are 8 combination.
Material size is 24 by 24
Same method in last simulation
38
39
40
41
cv
42
cv
43
cv
44
45
46
47
48
49
50
51
Conclusion
• Build a simplified mathematical model for the forward
problem of optical tomography
• 2D-layered Material
• Phase function approximated by Gaussian
• Propose 2 different approach to solve the inverse problem of
optical tomography
• Log-barrier Interior Point approach
• Primal-Dual Interior Point approach
• Efficient formulation
52
Additional Slides
53
Efficient formulations-Jacobian
Old formulation New formulation
TotalTotal
54
Efficient formulations-Hessian
Total
Total
55
)
Old formulation New formulation

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A Presentation on Artificial Intelligence

Development of optical tomography methods with discretized path integral

  • 1. Development of optical tomography methods with discretized path integral D136580 Bingzhi Yuan
  • 2. Tomography devices MRI Magnetic Resonance ImagingPET Positron Emission tomography X-ray image is from http://en.wikipedia.org/wiki/X-ray_computed_tomography MRI image is from http://en.wikipedia.org/wiki/MRI PET image is from http://en.wikipedia.org/wiki/Positron_emission_tomography High magnetic field radioactive X-ray CT radioactive 2
  • 3. Research motivation Light Source Camera Develop an optical tomography method can provide better and sharper result than DOT Diffusion Optical Tomography (DOT) Blurred results 3[Chung SC et al. 2015]
  • 4. Problem and Challenges in the optical tomography X-rayInfrared Light Scattering+Attenuation Attenuation 4
  • 5. Framework of optical tomography Forward problem Modeling the light transport Light source Detector Inverse problem Estimating the property of the material Light source Detector Scattering Attenuation 5
  • 6. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline Inverse problem 6
  • 7. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline 7
  • 8. Related Works Single ScatteringDiffusion Optical Tomography Distribution of the scattered light Incident light Distribution of the scattered light Incident light Phase function has a spherical shape [Florescu et al. 2009] [Florescu et al. 2010] [Gonzalez-Rodriguez et al. 1960] [Arridge et al. 1999] [Gibson et al. 2005] Forward Scattering [Tamaki et al. 2013] [Ishii et al. 2013] Multiple Scattering [Tamaki et al. 2013] [Ishii et al. 2013] [Arridge et al. 1999] [Gibson et al. 2005] 8
  • 9. Path Integral Light source Detector All the possible paths between the light source and the detector should be accumulated. Path integral is an efficient and promising method to describe the light transport in computer graphics [Simon Premože et al. 2003] 9
  • 10. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline 10
  • 11. Simplify the forward problem Modeling the light transport Light source Detector 2D layered material N grids Layer 1 Layer M …… Layer 2 Our proposed idea 11
  • 12. 2D Layered Material N grids Layer 1 Layer M …… Layer 2 N grids Layer 1 Layer M …… Layer 2 Homogeneous in every grid. No back scattering 2.Forward scattering1.Homogenous Phase function will be introduced in the next section 12
  • 13. 2D Layered Material N grids Layer 1 Layer M …… Layer 2 N grids Layer 1 Layer M …… Layer 2 Scatters at the center of a grid, and points to the center of a grid Multiple(=NM-2) paths for a given (i, j) 4.Multiple paths3.Scatter at center Light source position i Detector position j 13
  • 14. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline 14
  • 15. Common phase function in 3D 1 1 Henyey-Greenstein’s phase function θ is the angle between the scattered light and the incident light. p(θ) is the probability of having such an angle. g describes the scattering property of a media, from back scattering to isotropic scattering to forward scattering. g=1 g=0 forward spherical Disadvantage: it’s a phase function in 3D space and the back scattering part can’t be excluded. g 15
  • 16. Gaussian as the phase function Distribution of the Gaussian σ describe the scattering property, smaller value indicate the more peaked forward scattering. σ = 0.2 σ = 0.4 Advantage: works in 2D space, and easy to implement 16
  • 17. Discretize the Gaussian for the 2D layered material N grids Layer 1 Layer M …… Layer 2 17
  • 18. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline 18
  • 19. Scattering N grids Layer m Layer M …… Layer m+1 k Contribution for a path Measure at the scattering point scattering coefficient Gaussian for every scattering point19
  • 20. Attenuation Light source Detector Model by the integral of the extinction coefficient along the path : extinction coefficient at grid I : Intensity of the light source : Intensity observed by the detector 20
  • 21. Discretized path integral in layered material I0 I1 N grids Layer 1 Layer M …… Layer 2 1.Homogenous 2.Forward scattering 3.Scatter at center 4.Multiple paths Model by the inner product of the distance vector and extinction coefficient vector : vector contains distances at every grid : vector contains extinction coefficient at every grid 21
  • 22. Scattering & Attenuation I0 Iijk N grids Layer 1 Layer M …… Layer 2 1.Homogenous 2.Forward scattering 3.Scatter at center 4.Multiple paths Light source position i Detector position j AttenuationContribution of this path : Light source position : Detector position : index for light path with (i, j) : Intensity for path ijk : Intensity for light source Intensity for path ijk 22
  • 23. Discretized path integral for forward problem I0 Iij N grids Layer 1 Layer M …… Layer 2 1.Homogenous 2.Forward scattering 3.Scatter at center 4.Multiple paths Light source position i Detector position j Multiple(𝑁 =NM-2) paths for a given (i, j) : Light source position : Detector position : index for light path with (i, j) : Intensity for path ijk : Intensity for light source : Observed Intensity at (i, j) Sum all paths 23
  • 24. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline 24
  • 25. Inverse problem-Construct an inverse problem Calculate from the model of the forward problem I0 N grids Layer 1 Layer M …… Layer 2 Light source position i Detector position j : Light source position : Detector position : index for light path with (i, j) 25
  • 26. Inverse problem Difference between observed and calculated intensityObserved light intensity I0 N grids Layer 1 Layer M …… Layer 2 Light source position i Detector position j : Light source position : Detector position : index for light path with (i, j) 26
  • 27. Inverse problem N grids Layer 1 Layer M …… Layer 2 Light source position i Detector position j : Light source position : Detector position : index for light path with (i, j) 27
  • 28. More Observations by changing configuration T2B case 28
  • 29. More Observations by changing configuration T2B case R2L case B2T case L2R case 29
  • 30. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline 30
  • 31. Log-Barrier Interior-point approach to the inverse problem constrained optimization problem Combine the constraints and the cost function into one equation with log-barrier term 31Possible area 1 2 σ∗
  • 32. Newton & Quasi Newton for optimization Newton Second order derivative (Hessian) is calculated Quasi-Newton Second order derivative is approximated Solver for optimization Accuracy Computation Cost High Low LowHigh 32
  • 33. Numerical simulation Tested with 5 different 2D layered materials Material size is 24 by 24 I0 N grids Layer 1 Layer M …… Layer 2 Light source position i Detector position j 33 Given Target the observed light intensity the light source the contribution of have a light path ijk the distance between each point in the light path ijk the extinction coefficient Given Target
  • 34. Comparison with DOT Provided by EIDORS Material size is 24 by 24 solver: Gauss-Newton(GN) Primal-Dual(PD) 34 EIDOR is from http://eidors3d.sourceforge.net/ Goals in next step: Improve efficiency Maintain accuracy
  • 35. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline 35
  • 36. Primal-Dual interior point approach General form Reform as Lagrangian L KKT system KKT condition Update the result 36
  • 37. Thesis Structure • Chapter 1: Introduction • Chapter 2: Related Works • Chapter 3: Forward Problem • Chapter 4: Log-Barrier Interior Point Approach • Chapter 5: Primal-Dual Interior Point Approach • Chapter 6: Conclusion • Related work • 2D Layered material • Our phase function • Discretized path integral formulation • Constraint optimization problem • Log-Barrier Interior Point Approach • Primal-Dual Interior Point Approach • Efficient formulation Presentation outline 37
  • 38. Numerical results LB: PD: New: Old: Newton: Quasi-Newton: Log-Barrier approach Primal-Dual approach New formulations were used Old formulations were used Newton is the solver for optimization Quasi-Newton is the solver for optimization In total, there are 8 combination. Material size is 24 by 24 Same method in last simulation 38
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  • 52. Conclusion • Build a simplified mathematical model for the forward problem of optical tomography • 2D-layered Material • Phase function approximated by Gaussian • Propose 2 different approach to solve the inverse problem of optical tomography • Log-barrier Interior Point approach • Primal-Dual Interior Point approach • Efficient formulation 52
  • 54. Efficient formulations-Jacobian Old formulation New formulation TotalTotal 54