Diese Präsentation wurde erfolgreich gemeldet.
Wir verwenden Ihre LinkedIn Profilangaben und Informationen zu Ihren Aktivitäten, um Anzeigen zu personalisieren und Ihnen relevantere Inhalte anzuzeigen. Sie können Ihre Anzeigeneinstellungen jederzeit ändern.

Development of optical tomography methods with discretized path integral

111 Aufrufe

Veröffentlicht am

This is the presentation for the phd defense of Bingzhi Yuan

Veröffentlicht in: Technologie
  • Loggen Sie sich ein, um Kommentare anzuzeigen.

Development of optical tomography methods with discretized path integral

  1. 1. Development of optical tomography methods with discretized path integral D136580 Bingzhi Yuan
  2. 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. 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. 4. Problem and Challenges in the optical tomography X-rayInfrared Light Scattering+Attenuation Attenuation 4
  5. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 17. Discretize the Gaussian for the 2D layered material N grids Layer 1 Layer M …… Layer 2 17
  18. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 28. More Observations by changing configuration T2B case 28
  29. 29. More Observations by changing configuration T2B case R2L case B2T case L2R case 29
  30. 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. 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. 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. 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. 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. 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. 36. Primal-Dual interior point approach General form Reform as Lagrangian L KKT system KKT condition Update the result 36
  37. 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. 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
  39. 39. 39
  40. 40. 40
  41. 41. 41 cv
  42. 42. 42 cv
  43. 43. 43 cv
  44. 44. 44
  45. 45. 45
  46. 46. 46
  47. 47. 47
  48. 48. 48
  49. 49. 49
  50. 50. 50
  51. 51. 51
  52. 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
  53. 53. Additional Slides 53
  54. 54. Efficient formulations-Jacobian Old formulation New formulation TotalTotal 54
  55. 55. Efficient formulations-Hessian Total Total 55 ) Old formulation New formulation

×