Tensor Field Visualization 9-1 Ronald Peikert SciVis 2007 - Tensor - PowerPoint PPT Presentation

Tensor Field Visualization 9-1 Ronald Peikert SciVis 2007 - Tensor Fields

Tensors "Tensors are the language of mechanics" T Tensor of order (rank) f d ( k) 0: scalar 1: vector 1: vector 2: matrix (example: stress tensor) … j ij Tensors can have "lower" and "upper" indices, e.g. , a a a , , ij i indicating different transformation rules for change of coordinates indicating different transformation rules for change of coordinates. 9-2 Ronald Peikert SciVis 2007 - Tensor Fields

Tensors Visualization methods for tensor fields: • tensor glyphs • t tensor field lines, hyperstreamlines fi ld li h t li • tensor field topology • fiber bundle tracking fiber bundle tracking Tensor field visualization only deals with 2 nd order tensors (matrices). → eigenvectors and eigenvalues contain full information. Separate visualization methods for symmetric and nonsymmetric tensors. tensors. 9-3 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs In 3D, tensors are 3x3 matrices. The velocity gradient tensor is nonsymmetric → 9 degrees of freedom for the local change of the velocity vector. f d f th l l h f th l it t A glyph developed by de Leeuw and van Wijk can visualize all these 9 DOFs: these 9 DOFs: • tangential acceleration (1): green "membrane" • orthogonal acceleration (2): curvature of arrow orthogonal acceleration (2): curvature of arrow • twist (1): candy stripes • shear (2): orange ellipse (gray ellipse for ref.) ( ) g p (g y p ) • convergence/divergence (3): white "parabolic reflector" 9-4 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs Example: NASA "bluntfin" dataset, glyphs shown on points on a streamline. 9-5 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs S Symmetric 3D tensors have real eigenvalues and orthogonal t i 3D t h l i l d th l eigenvectors → they can be represented by ellipsoids. Three types of anisotropy: Anisotropy measure: yp py py ( ) ( ) = λ − λ λ + λ + λ c • linear anisotropy l 1 2 1 2 3 ( ( ) ( ) ( ) ) = λ − λ λ + λ + λ c 2 • planar anisotropy p p 2 2 3 3 1 1 2 2 3 3 ( ) = λ λ + λ + λ • isotropy (spherical) c 3 s 3 1 2 3 ( ( ) ) λ λ ≥ ≥ λ λ ≥ ≥ λ λ 1 2 3 Images: G. Kindlmann 9-6 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs Problem of ellipsoid glyphs: • shape is poorly recognized in projected view Example: 8 ellipsoids, 2 views 9-7 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs Problems of cylinder glyphs: Problem of cuboid glyphs: • discontinuity at c l = c p • small differences in eigenvalues are over- • artificial orientation at c s = 1 emphasized 9-8 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs Combining advantages: superquadrics Superquadrics with z as primary axis: ⎛ ⎞ α θ β φ cos sin ⎜ ⎟ q ( ( ) ) α β θ φ θ φ = ⎜ = ⎜ θ θ φ φ q , sin sin sin sin ⎟ ⎟ z ⎜ ⎟ β φ cos ⎝ ⎠ ≤ ≤ θ θ ≤ ≤ π π ≤ ≤ φ φ ≤ ≤ π π 0 0 2 2 , 0 0 cos α θ with used as shorthand for Superquadrics for some pairs ( α β) pairs ( α , β) α θ θ cos sgn(cos ) Shaded: subrange used for glyphs 9-9 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs Superquadric glyphs (Kindlmann): Given c l , c p , c s compute a base superquadric using a sharpness value γ : • ( ) ⎧ ⎧ γ ( ) ( ) γ ≥ θ φ α = − β = − if c c : q , with 1 c and 1 c ⎪ ( ) l p z p l θ φ = ⎨ q , ( ( ) ) γ ( ( ) ) ( ( ) ) γ ⎪ ⎪ < < θ φ θ φ α α = = − β β = = − if if c c c c : : q q , with with 1 1 c c and and 1 1 c c ⎩ ⎩ l p x l p • scale with c l , c p , c s along x,y,z and rotate into eigenvector frame c s = 1 c = 1 c = 1 c s = 1 c = 1 c s = 1 c l = 1 c p = 1 c l = 1 c p = 1 c l = 1 c p = 1 γ = 1.5 γ = 3.0 γ = 6.0 9-10 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs Comparison of shape perception (previous example) • with ellipsoid glyphs • with superquadrics glyphs 9-11 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor glyphs Comparison: Ellipsoids vs. superquadrics (Kindlmann) ⎛ ⎞ 1 e ⎛ ⎞ ⎛ ⎞ R 1 ⎜ ⎟ x ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ( ( ) ) ⎜ ⎜ ⎟ ⎟ + = + − 1 1 color map: (with e 1 = major eigenvector) 1 l G G c c e e 1 1 c c 1 1 ( ith j i t ) ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ l y l ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ B ⎠ ⎝ ⎠ 1 1 e ⎝ ⎠ z 9-12 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field lines Let T ( x ) be a (2 nd order) symmetric tensor field. → real eigenvalues, orthogonal eigenvectors Tensor field line: by integrating along one of the eigenvectors Important: Eigenvector fields are not vector fields! • eigenvectors have no magnitude and no orientation (are bidirectional) • the choice of the eigenvector can be made consistently as long as eigenvalues are all different • tensor field lines can intersect only at points where two or more eigenvalues are equal, so-called degenerate points. 9-13 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field lines Tensor field lines can be rendered as hyperstreamlines: tubes with elliptic cross section, radii proportional to 2 nd and 3 rd eigenvalue. eigenvalue. Image credit: W. Shen 9-14 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field topology Based on tensor field lines, a tensor field topology can be defined, in analogy to vector field topology. Degenerate points play the role of critical points: At degenerate points, infinitely many directions (of eigenvectors) exist. For simplicity, we only study the 2D case. For locating degenerate points: solve equations ( ) ( ) ( ) ( ) ( ) ( ) − = = T x T x 0, T x 0 11 22 12 9-15 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field topology It can be shown: The type of the degenerated point depends on δ = − ad bc where ( ) ( ) ∂ − ∂ − T T T T 1 1 = = 11 22 11 22 a b ∂ ∂ 2 x 2 y ∂ ∂ T T = = 12 12 c d ∂ ∂ x y for δ <0 the type is a trisector • for δ >0 the type is a wedge • for δ =0 the type is structurally unstable • 9-16 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field topoloy Types of degenerate points illustrated with linear tensor fields Types of degenerate points, illustrated with linear tensor fields. trisector double wedge single wedge − + + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 2 x y 1 2 x 3 y 1 x y = ⎜ = ⎜ = ⎜ T ⎟ T ⎟ T ⎟ − ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ y y 1 1 y y 1 1 y y 1 1 x x ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ y + + + − 2 2 2 2 x x 9 y x y x = ⎜ = ⎜ = ⎜ ⎟ ⎟ ⎟ e e e ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + − 2 2 ⎝ ⎠ ⎝ ⎠ y 3 y x y x ⎝ ⎠ δ = − δ = δ = 1 1 3 1 9-17 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field topology Separatrices are tensor field lines converging to the degenerate point with a radial tangent. They are straight lines in the special case of a linear tensor field. Double wedges have one "hidden separatrix" and two other separatrices which actually separate regions of different field line p y p g behavior. Single wedges have just one separatrix. Si l d h j t t i 9-18 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field topology The angles of the separatrices are obtained by solving: + + + − − = 3 2 dm ( ( c 2 ) b m ) ( (2 a d m ) ) c 0 m ∈ � If , the two angles θ = ± arctan m are angles of a separatrix. The two choices of signs correspond to the two choices of tensor field lines (minor and major eigenvalue). g ) If d = 0, an additional solution is θ = ± ° 90 There are in general 1 or 3 real solutions: • 3 separatrices for trisector and double wedge • 1 separatrix for single wedge 1 separatrix for single wedge 9-19 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field topology Saddles, nodes, and foci can exists as nonelementary (higher- order) degenerate points with δ =0. They are created by merging trisectors or wedges. They are not structurally stable and break g y y up in their elements if perturbed. 9-20 Ronald Peikert SciVis 2007 - Tensor Fields

Tensor field topology The topological skeleton is defined as the set of separatrices of trisector points. Example: Topological transition of the stress tensor field of a flow Example: Topological transition of the stress tensor field of a flow past a cylinder Image credit: T. Delmarcelle 9-21 Ronald Peikert SciVis 2007 - Tensor Fields