Oriented Tensor Reconstruction. Tracing Neural Pathways from DT-MRI

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Diffusion tensor DT-MRI tractography. Talk at Vis 2002

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Oriented Tensor Reconstruction. Tracing Neural Pathways from DT-MRI

  1. 1. Oriented Tensor Reconstruction: Tracing Neural Pathways from DT-MRI Leonid Zhukov Alan H. BarrDepartment of Computer ScienceCalifornia Institute of Technology 9/10/11 Computer Graphics Group 1
  2. 2. Talk outline •  Introduction –  Tensor visualization: previous work –  Motivation: brain anatomy –  Diffusion tensor DT-MRI overview •  Algorithm for directional tensor reconstruction: –  Data interpolation and filtering –  Moving Least Squares method –  Fiber tracing algorithm •  Results: –  Extracted anatomical structures: corona radiata, corpus callosum, cingulum bundle, U-shape fibers etc •  Conclusions9/10/11 Computer Graphics Group 2
  3. 3. Previous work•  Tensor visualization –  Tensor fields (stress–strain tensors) (Delmarcelle & Hesselink 92)•  Diffusion tensor based segmentation –  Anisotropy measures ( Basser 96 ) –  Ellipsoid classification (Westin 97)•  Diffusion tensor visualization –  DT-MRI 2D – ellipsoids (Laidlaw 98) –  DT-MRI 3D volume rendering (Kindlmann 99)•  Diffusion tensor based fiber tracing – streamline integration –  Tensorlines, streamtubes (Weinstein 98, Laildlaw 01) –  In vivo fiber tractography (Basser 2000) –  Anatomical brain connectivity (Parker 01) 9/10/11 Computer Graphics Group 3
  4. 4. Brain structure Photo:University of Iowa Virtual Hospital9/10/11 Computer Graphics Group 4
  5. 5. Diffusion tensor y•  Diffusion – random thermal motion(Brownian motion) of water molecules: x •  Diffusion equation: 9/10/11 Computer Graphics Group 5
  6. 6. DT- MRI •  Diffusion tensor data Dxx Dxy Dxz Dyx Dyy Dyz Dzx Dzy DzzData: SCI Institute, University of Utah 9/10/11 Computer Graphics Group 6
  7. 7. Eigenvalues/vectors•  Eigenvalues/eigenvectors basis e3•  In e1,e2,e3 local Cartesian frame - tensor diagonal e2 e1D every voxel•  Interpretation: ellipsoid = D * sphere•  Bilinear form –invariant 9/10/11 Computer Graphics Group 7
  8. 8. Diffusion ellipsoids9/10/11 Computer Graphics Group 8
  9. 9. DT-MRI & fibers9/10/11 Computer Graphics Group 9
  10. 10. DT-MRI & fibers9/10/11 Computer Graphics Group 10
  11. 11. Fiber tracing 1) continues representation 2) local averaging filter “with memory” and look ahead (oriented anisotropic)9/10/11 Computer Graphics Group 11
  12. 12. Method•  Build continues representation (super-sampling) for tensor data –  Static preprocessing –  Component-wise filtering –  Tri-linear interpolation•  Dynamic adaptive local filtering + fibertracing –  Anisotropic local filter, orientation determined by the fiber –  Local least squares approximation to the data (MLS) –  Forward Euler type integration9/10/11 Computer Graphics Group 12
  13. 13. Super-sampling Continues tensor field – component-wise tri-linear interpolation Kindlmann, Weinstein, 20009/10/11 Computer Graphics Group 13
  14. 14. Moving filter Local filter – moving oriented least squares (MLS) filter for tensors9/10/11 Computer Graphics Group 14
  15. 15. Moving Least Squares•  Find best approximation in LS sense - minimizing functional: scalar scalar tensor tensor •  Polynomial approximation: tensor tensor•  Minimization: scalar tensor tensor (every tensor component separately!) 9/10/11 Computer Graphics Group 15
  16. 16. Moving Least Squares•  Polynomial approximation : tensor tensor•  Approximated tensor : tensor tensor•  Approximated tensor-zero-order polynomial : tensor scalar tensor 9/10/11 Computer Graphics Group 16
  17. 17. Integration Streamline integration: vector vector Forward Euler (RG) integration (diverging) : vector vector vector Inverse Euler –implicit scheme integration (converging): vector vector vector9/10/11 Computer Graphics Group 17
  18. 18. Diffusion ellipsoids9/10/11 Computer Graphics Group 18
  19. 19. Anisotropy measures C Westin, 979/10/11 Computer Graphics Group 19
  20. 20. Anisotropy DT MRI Anisotropy Cl9/10/11 Computer Graphics Group 20
  21. 21. Tracing algorithmTracing Procedure: trace = fiber_trace(P,e) { trace->add(P);for (every starting point P) { do { Tp = filter(T,P,sphere); Pn = integrate_forward(P,e1,dt); cl = anisotropy(Tp); Tp = filter(T,Pn,ellipsoid,e1); if (cl > eps) { cl = anisotropy(Tp) e1 = direction(Tp); if ( c1 > eps ) { trace->add(Pn); trace1 = fiber_trace(P, e1); P = Pn; trace2 = fiber_trace(P,-e1); e1 = direction(Tp); trace = trace1 + trace2; } } } while (cl >eps) } return(trace); }9/10/11 Computer Graphics Group 21
  22. 22. Tracing algorithm9/10/11 Computer Graphics Group 22
  23. 23. Results9/10/11 Computer Graphics Group 24
  24. 24. MLS effect9/10/11 Computer Graphics Group 25
  25. 25. Results9/10/11 Computer Graphics Group 26
  26. 26. Results9/10/11 Computer Graphics Group 28
  27. 27. Results9/10/11 Computer Graphics Group 30
  28. 28. Results9/10/11 Computer Graphics Group 31
  29. 29. Invariant volumes DT MRIDiffusivity I Anisotropy Cl9/10/11 Computer Graphics Group 33
  30. 30. Conclusions•  Contributions: –  New method for non-linear tensor filtering –  Smooth reconstruction of anatomically recognizable brain structures •  Future work: –  additional analytic developments –  needs a good validation9/10/11 Computer Graphics Group 36
  31. 31. Acknowledgements•  Gordon Kindlmann and SCI institute for brain dataset•  Yarden Livnat and David Breen•  Supported by NSF grants•  Human Brain Project9/10/11 Computer Graphics Group 37

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