Evgeniya Balmashnova
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The document discusses mathematical techniques for diffusion magnetic resonance imaging (MRI). It introduces diffusion tensor imaging (DTI) and describes how DTI models water diffusion using a tensor. It then presents equations for calculating the probability of water diffusion over time using the diffusion tensor. Finally, it discusses DTI scale space and describes equations for smoothing diffusion tensors over scale.
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Evgeniya Balmashnova
- 1. Mathematical Techniques for Diffusion MRI Evgeniya Balmashnova 10 September 2009
- 4. Diffusion Tensor Imaging 1 2 D −1 P(r, t) = 1 (4πDt)3/2 e− 4t r t Diffusion time D Diffusion coefficient P Probability of travel to point r in time t
- 5. Diffusion Tensor Imaging 1 T −1 P(r, t) = 1 (4π|D|t)3/2 e− 4t r D r Dxx Dxy Dxz D= Dyx Dyy Dyz Dzx Dzy Dzz
- 6. Diffusion Tensor Imaging: Application
- 7. DTI Scale Space 1 −1 −1 −1 x ρ+ ¨ Γρ x µ x ν = 0 µν ˙ ˙ with Γλ = ρν Dλµ ∂ν Dρµ + ∂ρ Dµν − ∂µ Dνρ µν 2 µ
- 8. DTI Scale Space 1 −1 −1 −1 x ρ+ ¨ Γρ x µ x ν = 0 µν ˙ ˙ with Γλ = ρν Dλµ ∂ν Dρµ + ∂ρ Dµν − ∂µ Dνρ µν 2 µ
- 9. DTI Scale Space 1 −1 −1 −1 x ρ+ ¨ Γρ x µ x ν = 0 µν ˙ ˙ with Γλ = ρν Dλµ ∂ν Dρµ + ∂ρ Dµν − ∂µ Dνρ µν 2 µ
- 10. DTI Scale Space 1 −1 −1 −1 x ρ+ ¨ Γρ x µ x ν = 0 µν ˙ ˙ with Γλ = ρν Dλµ ∂ν Dρµ + ∂ρ Dµν − ∂µ Dνρ µν 2 µ
- 11. DTI Scale Space 1 −1 −1 −1 x ρ+ ¨ Γρ x µ x ν = 0 µν ˙ ˙ with Γλ = ρν Dλµ ∂ν Dρµ + ∂ρ Dµν − ∂µ Dνρ µν 2 µ
- 14. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 15. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 16. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 17. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 18. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 19. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 20. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 21. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 22. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 23. DTI Scale Space inv Dσ = exp (ln D ∗ φσ ) − − → Dσ = exp ln D −1 ∗ φσ −− −1 exp exp ln D ∗ φσ ln D −1 ∗ φσ = − ln D ∗ φσ ∗φσ ∗φ σ ln D ln D −1 = − ln D ln ln inv D −−→ −− D −1
- 24. Diffusion Tensor Imaging: Application
- 25. Diffusion Tensor Imaging: Application
- 26. Diffusion Tensor Imaging: Application
- 27. Diffusion Tensor Imaging: Application
- 28. Diffusion Tensor Imaging: Application
- 29. Diffusion Tensor Imaging: Application
- 30. Diffusion Tensor Imaging: Application
- 31. Diffusion Tensor Imaging: Application
- 32. Diffusion Tensor Imaging: Application
- 33. Diffusion Tensor Imaging: Application
- 34. Diffusion Tensor Imaging: Application
- 35. Diffusion Tensor Imaging: Application
- 36. MRI and Diffusion Tensor Imaging
- 37. MRI and Diffusion Tensor Imaging
- 38. MRI and Diffusion Tensor Imaging
- 39. MRI and Diffusion Tensor Imaging
- 40. MRI and Diffusion Tensor Imaging
- 41. High Angular Resolution Diffusion Imaging
- 42. HARDI Stejskal & Tanner Equation S(g) = S0 e−bD(g) (Stejskal & Tanner, 1965)
- 43. HARDI Stejskal & Tanner Equation S(g) = S0 e−bD(g) (Stejskal & Tanner, 1965)
- 44. Diffusion Orientation Distribution Function
- 45. Diffusion Orientation Distribution Function
- 46. Diffusion Orientation Distribution Function
- 47. Diffusion Orientation Distribution Function
- 48. Alternative Decompositions Spherical harmonics N DN (g) = c m Y m (g) =0 m=− High order tensors 3 3 i ...iN DN (g) = ... D1 gi . . . gi 1 N i1 =1 iN =1 Hierarchial tensors N 3 3 i ...i DN (g) = ... D1 gi . . . gi (Florack & Balmashnova, 2008) 1 =0 i1 =1 iN =1
- 49. Alternative Decompositions Spherical harmonics N DN (g) = c m Y m (g) =0 m=− High order tensors 3 3 i ...iN DN (g) = ... D1 gi . . . gi 1 N i1 =1 iN =1 Hierarchial tensors N 3 3 i ...i DN (g) = ... D1 gi . . . gi (Florack & Balmashnova, 2008) 1 =0 i1 =1 iN =1
- 50. Alternative Decompositions Spherical harmonics N DN (g) = c m Y m (g) =0 m=− High order tensors 3 3 i ...iN DN (g) = ... D1 gi . . . gi 1 N i1 =1 iN =1 Hierarchial tensors N 3 3 i ...i DN (g) = ... D1 gi . . . gi (Florack & Balmashnova, 2008) 1 =0 i1 =1 iN =1
- 51. Alternative Decompositions Spherical harmonics N DN (g) = c m Y m (g) =0 m=− High order tensors 3 3 i ...iN DN (g) = ... D1 gi . . . gi 1 N i1 =1 iN =1 Hierarchial tensors N 3 3 i ...i DN (g) = ... D1 gi . . . gi (Florack & Balmashnova, 2008) 1 =0 i1 =1 iN =1
- 52. Spherical Harmonics High Order Tensors Regularization No straightforward regularization Simple formula for ODF No straightforward ODF formulas Requires bookkeeping Simple bookkeeping No maxima detection Maxima detection algorithms
- 53. Spherical Harmonics High Order Tensors Regularization No straightforward regularization Simple formula for ODF No straightforward ODF formulas Requires bookkeeping Simple bookkeeping No maxima detection Maxima detection algorithms
- 54. Spherical Harmonics High Order Tensors Regularization No straightforward regularization Simple formula for ODF No straightforward ODF formulas Requires bookkeeping Simple bookkeeping No maxima detection Maxima detection algorithms
- 55. Spherical Harmonics High Order Tensors Regularization No straightforward regularization Simple formula for ODF No straightforward ODF formulas Requires bookkeeping Simple bookkeeping No maxima detection Maxima detection algorithms
- 56. Orientation Distribution Function N 3 3 DN (g)= =0 i1 =1 ... iN =1 D i1 ...i gi1 ...gi N 3 3 ODF (g)= =0 i1 =1 ... iN =1 2πPl (0)D i1 ...i gi1 ...gi
- 57. Regularization N 3 3 DN (g)= =0 i1 =1 ... iN =1 D i1 ...i gi1 ...gi N 3 3 Dt (g)=et g D(g)= =0 i1 =1 ... iN =1 D i1 ...i (t) gi1 ...gi D i1 ...i (t) = e−tl(l+1) D i1 ...i
- 58. Regularization N 3 3 DN (g)= =0 i1 =1 ... iN =1 D i1 ...i gi1 ...gi N 3 3 Dt (g)=et g D(g)= =0 i1 =1 ... iN =1 D i1 ...i (t) gi1 ...gi D i1 ...i (t) = e−tl(l+1) D i1 ...i
- 59. Regularization N 3 3 DN (g)= =0 i1 =1 ... iN =1 D i1 ...i gi1 ...gi N 3 3 Dt (g)=et g D(g)= =0 i1 =1 ... iN =1 D i1 ...i (t) gi1 ...gi D i1 ...i (t) = e−tl(l+1) D i1 ...i
- 60. Regularization N 3 3 DN (g)= =0 i1 =1 ... iN =1 D i1 ...i gi1 ...gi N 3 3 Dt (g)=et g D(g)= =0 i1 =1 ... iN =1 D i1 ...i (t) gi1 ...gi D i1 ...i (t) = e−tl(l+1) D i1 ...i
- 61. MRI and Diffusion Tensor Imaging
- 62. MRI and Diffusion Tensor Imaging
- 63. MRI and Diffusion Tensor Imaging
- 64. MRI and Diffusion Tensor Imaging
- 65. MRI and Diffusion Tensor Imaging
- 66. MRI and Diffusion Tensor Imaging
- 67. MRI and Diffusion Tensor Imaging
- 68. MRI and Diffusion Tensor Imaging
- 69. MRI and Diffusion Tensor Imaging
- 70. MRI and Diffusion Tensor Imaging
- 71. MRI and Diffusion Tensor Imaging
- 72. MRI and Diffusion Tensor Imaging
- 73. MRI and Diffusion Tensor Imaging
- 74. MRI and Diffusion Tensor Imaging
- 75. Tracking Riemannian metric ⇒ Finsler metric.
- 76. Finsler Metric Riemannian metric ⇒ Finsler metric.
- 77. Finsler Metric
- 78. DTI streamlines
- 80. Open Questions Voxel classification Scales selection Reality check
- 81. Open Questions Voxel classification Scales selection Reality check
- 82. Open Questions Voxel classification Scales selection Reality check