1. CIS 4930/6930-902 S CIENTIFIC V ISUALIZATION TENSOR FIELD VISUALIZATION Paul Rosen Assistant Professor University of South Florida Slide credit X. Tricoche

2. O UTLINE Tensor basics Tensor glyphs Hyperstreamlines DTI visualization

3. T ENSORS p-ranked tensor in n-space: linear transformation between vector spaces Special cases: 0th order (rank): scalars 1st order: vectors 2nd order: matrices In Visualization “tensors” are mostly 2nd order tensors

4. T ENSORS 2nd order tensors map vectors to vectors Symmetric / antisymmetric Tt = ±T with Represented* by matrices in cartesian basis (*) tensors exist independently of any matrix representation

5. T ENSORS Eigenvalues, eigenvectors Real symmetric tensors: eigenvalues are real and eigenvectors are orthogonal. Sorted eigenvalues. Invariants: quantities (function of the tensor value) that do not change with the reference frame Eigenvalues and all functions of the eigenvalues Trace (sum), determinant (product), FA, mode, …

6. E XAMPLES Forces stress: cause of deformation strain: deformation description Derivative Jacobian: 1st-order derivative of a vector field Hessian: 2nd-order derivative of a scalar field Diffusion tensor field

7. T ENSORS Anisotropy characterizes tensor shape Example: ink diffusion KleenexNewspaper

8. T ENSORS Eigenvectors: non-oriented directional info. Have no intrinsic norm Have no intrinsic orientation Eigenvectors ≠ vectors! Tensor visualization requires combined visualization of eigenvectors and eigenvalues

9. S YMMETRIC T ENSOR G LYPHS A 2nd order symmetric 3D tensor is fully characterized by its 3 real eigenvalues (shape) and associated orthogonal eigenvectors (orientation)

10. S YMMETRIC T ENSOR G LYPHS

13. Shortcomings

14. S YMMETRIC T ENSOR G LYPHS Shortcomings

15. S UPERQUADRICS A. B ARR, S UPERQUADRICS AND ANGLE - PRESERVING TRANSFORMATIONS, IEEE C OMPUTER G RAPHICS AND A PPLICATIONS 18(1), 1981

16. with S UPERQUADRIC T ENSOR G LYPHS Parameters and are a function of the tensor’s anisotropy measures: G. K INDLMANN, S UPERQUADRIC T ENSOR G LYPHS, J OINT E UROGRAPHICS /IEEE VGTC S YMPOSIUM ON V ISUALIZATION 2004

17. S UPERQUADRIC T ENSOR G LYPHS Superquadric glyphs

18. S UPERQUADRIC T ENSOR G LYPHS G. K INDLMANN, S UPERQUADRIC T ENSOR G LYPHS, J OINT E UROGRAPHICS /IEEE VGTC S YMPOSIUM ON V ISUALIZATION 2004

19. S UPERQUADRIC T ENSOR G LYPHS G. K INDLMANN, S UPERQUADRIC T ENSOR G LYPHS, J OINT E UROGRAPHICS /IEEE VGTC S YMPOSIUM ON V ISUALIZATION 2004

20. C OMPARISON G. K INDLMANN, S UPERQUADRIC T ENSOR G LYPHS, J OINT E UROGRAPHICS /IEEE VGTC S YMPOSIUM ON V ISUALIZATION 2004

21. C OMPARISON G. K INDLMANN, S UPERQUADRIC T ENSOR G LYPHS, J OINT E UROGRAPHICS /IEEE VGTC S YMPOSIUM ON V ISUALIZATION 2004

22. C OMPARISON G. K INDLMANN, S UPERQUADRIC T ENSOR G LYPHS, J OINT E UROGRAPHICS /IEEE VGTC S YMPOSIUM ON V ISUALIZATION 2004

23. S YMMETRIC T ENSOR G LYPHS Color-coding can be used to facilitate the interpretation of the orientation e.g., e max mapped to R=|x|, G=|y|, B=|z|

24. C OMPARISON

25. S YMMETRIC T ENSOR G LYPHS

26. +λ2+λ2 ( 1, 1, 1 \√3 √3 √3+λ3( 1, 1, 1 \√3 √3 √3+λ3 +λ1+λ1 +λ1+λ1 +λ2−λ3+λ2−λ3 λ 2 = 0= 0 λ 1 = 0= 0 λ 2 = − λ 3 λ 1 = − λ 2 λ 1 = − λ 3 λ 1 = λ= λ 2 λ 2 = λ= λ 3 λ 1 = λ= λ 2 λ 2 = λ= λ 3 λ 2 > 0> 0 λ 2 < 0< 0 λ 1 λ 2 = λ 2 λ 3 − λ 1 λ 2 = λ 2 λ 3 − (1, 0, 0) ( 1,0,( 1,0, √ 2 −√ 2 1 \1 \ ( 1, 1,( 1, 1, √3 √3 −√3√3 √3 −√3 1 \1 \ ( 1, 1,0\√2 √2( 1, 1,0\√2 √2 ( 1, 1, 1 \ √ 3 −√ 3 −√ 3 (0, 0, 1) − (0,− √2 − √2(0,− √2 − √2 1,1, 1 \1 \ λ 3 = 0= 0 λ 3 > 0> 0 λ 3 < 0< 0 λ 1 > 0> 0 λ 1 < 0< 0 indeinite ( 1, 1, 1 \ − √ 3 − √ 3 − √ 3 negative deinite positive deinite S YMMETRIC T ENSOR G LYPH Glyphs for general symmetric tensors? Eigenvalues can be positive or negative

27. (d) ( α, β ) base glyph tensor glyph regular superquadric (a, � ) = (0,4) hybrid superquadric (a, �, � W) = (0,4,2) α 1.0 (1,1) (0,2) (1,2) (1,0) (1,4) (1,2) (, )(, ) 4.0 β (0,4,2) 0.0 1.0 2.0 v u (1,2) (, )(, ) (1,4) (1,2) S YMMETRIC T ENSOR G LYPH T. S CHULTZ, G. K INDLMANN, S UPERQUADRIC G LYPHS FOR S YMMETRIC S ECOND -O RDER T ENSORS, IEEE TVCG 16 (6) (IEEE V ISUALIZATION 2010)

28. (a) Glyphs on vertical cutting plane (b) Superquadric tensor glyphs; s ( ∥ D ∥ ) ∝ ∥ D ∥ (c) Superquadric tensor glyphs; s ( ∥ D ∥ ) ∝ ∥ D ∥ 1 / 2 R ESULTS T. S CHULTZ, G. K INDLMANN, S UPERQUADRIC G LYPHS FOR S YMMETRIC S ECOND -O RDER T ENSORS, IEEE TVCG 16 (6) (IEEE V ISUALIZATION 2010)

29. G LYPH P ACKING Distribute (discrete) glyphs over continuous domain in data-driven way Reveal underlying continuous structures Remove artifacts caused by sampling bias G. K INDLMANN AND C.-F. W ESTIN, D IFFUSION T ENSOR V ISUALIZATION WITH G LYPH P ACKING, IEEE V ISUALIZATION 2006

30. Regular gridGlyph packing G LYPH P ACKING G. K INDLMANN AND C.-F. W ESTIN, D IFFUSION T ENSOR V ISUALIZATION WITH G LYPH P ACKING, IEEE V ISUALIZATION 2006

31. Regular gridGlyph packing G LYPH P ACKING G. K INDLMANN AND C.-F. W ESTIN, D IFFUSION T ENSOR V ISUALIZATION WITH G LYPH P ACKING, IEEE V ISUALIZATION 2006

32. H YPERSTREAMLINES Method for symmetric 2nd order tensor fields in 3D Identify eigenvector fields w.r.t. associated eigenvalues

33. H YPERSTREAMLINES Tensor field lines (2D/3D): curve everywhere tangential to a given eigenvector field R. R. D ICKINSON, A U NIFIED A PPROACH TO THE D ESIGN OF V ISUALIZATION S OFTWARE FOR THE A NALYSIS OF F IELD P ROBLEMS, SPIE P ROCEEDINGS V OL. 1083, 1989

34. H YPERSTREAMLINES Remark: numerical integration using e.g. Runge-Kutta is faced with the problem of maintaining orientation consistency R. R. D ICKINSON, A U NIFIED A PPROACH TO THE D ESIGN OF V ISUALIZATION S OFTWARE FOR THE A NALYSIS OF F IELD P ROBLEMS, SPIE P ROCEEDINGS,V OL. 1083, 1989

35. H YPERSTREAMLINES Method Compute tensor field line along major eigenvector. Sweep geometric primitive representing two other eigenvalues and eigenvectors Ellipse stretched along eigenvectors by eigenvalues Cross depicting eigenvectors + eigenvalues Color coding on geometric primitive determined by. T. D ELMARCELLE, L. H ESSELINK, V ISUALIZATION OF S ECOND O RDER T ENSOR F IELDS AND M ATRIX D ATA, IEEE V ISUALIZATION 1992

36. H YPERSTREAMLINES : R EMARKS Eigenvectors are orthogonal: cross section always orthogonal to tensor field line Eigenvalues mapped to length of edges in cross section: problems with negative eigenvalues T. D ELMARCELLE, L. H ESSELINK, V ISUALIZATION OF S ECOND O RDER T ENSOR F IELDS AND M ATRIX D ATA, IEEE V ISUALIZATION 1992

37. H YPERSTREAMLINES T. D ELMARCELLE, L. H ESSELINK, V ISUALIZATION OF S ECOND O RDER T ENSOR F IELDS AND M ATRIX D ATA, IEEE V ISUALIZATION 1992

38. H YPERSTREAMLINES

40. Extension of Hultquist’s stream surfaces to eigenvector fields T. D ELMARCELLE, L. H ESSELINK, V ISUALIZATION OF S ECOND O RDER T ENSOR F IELDS AND M ATRIX D ATA, IEEE V ISUALIZATION 1992

41. D IFFUSION T ENSOR I MAGING Diffusion Tensor (DT)-MRI measures anisotropic (directional) diffusion properties of biological tissue (e.g., brain) Diffusion tensor is symmetric positive definite (positive eigenvalues) Objective: use tensor information to reconstruct the path of tissue fibers Problems: (very) noisy data + isotropy

42. B RAIN S TRUCTURE - F IBER T RACKS

43. DT MRI V ISUALIZATION

44. W HITE M ATTER T RACTS P ARK, W ESTIN, AND K IKINIS, BWH, H ARVARD M EDICAL S CHOOL, 2003

45. D IFFUSION IN B IOLOGICAL T ISSUE Motion of water through tissue Faster in some directions than others Anisotropy: diffusion rate depends on direction isotropic anisotropic NewspaperKleenex

46. D IFFUSION MRI OF THE B RAIN Anisotropy high along white matter fiber tracts

47. 11/13/15 2.1 -0.1 -0.2 -0.1 2.0 -0.0 -0.2 -0.0 2.1 3.7 0.3 -0.8 0.3 0.6 -0.1 -0.8 -0.1 0.8 1.7 0.1 -0.1 0.1 2.3 -0.3 -0.1 -0.3 0.3 D IFFUSION MRI OF THE B RAIN Anisotropy high along white matter fiber tracts

48. F IBER T RACING Moving Least Squares: Apply Gauss filter mask whose support is determined by current path orientation and local anisotropy Trace fiber path along filtered eigenvector L. Z HUKOV, A. B ARR, O RIENTED T ENSOR R ECONSTRUCTION : T RACING N EURAL P ATHWAYS FROM D IFFUSION T ENSOR MRI, IEEE V ISUALIZATION 2002

49. F IBER T RACING White matter L. Z HUKOV, A. B ARR, O RIENTED T ENSOR R ECONSTRUCTION : T RACING N EURAL P ATHWAYS FROM D IFFUSION T ENSOR MRI, IEEE V ISUALIZATION 2002

50. F IBER T RACING White matter L. Z HUKOV, A. B ARR, O RIENTED T ENSOR R ECONSTRUCTION : T RACING N EURAL P ATHWAYS FROM D IFFUSION T ENSOR MRI, IEEE V ISUALIZATION 2002

51. F IBER T RACING Heart L. Z HUKOV, A. B ARR, H EART F IBER R ECONSTRUCTION FROM D IFFUSION T ENSOR MRI, IEEE V ISUALIZATION 2003