3D vector fields. Contents. Introduction 3D vector field topology Representation of particle lines. 3D LIC Combining different techniques

Transcription

1 3D vector fields Scientific Visualization (Part 9) PD Dr.-Ing. Peter Hastreiter Contents Introduction 3D vector field topology Representation of particle lines Path lines Ribbons Balls Tubes Stream tetrahedra 3D LIC Combining different techniques 2 Applied Visualization, SS10

2 Introduction Extension Many methods of 2D vector fields can be extended to 3D, in particular as far as ODE and cell searching is concerned Main problem in 3D Effective mapping to graphical primitives Main aspects Occlusion Amount of (visual) data Depth perception 3 Applied Visualization, SS10 Introduction Approaches to occlusion issue Sparse representations Animation Color differences to distinguish separate objects Continuity Reduction of visual data Sparse representations Clipping Importance of semi-transparency 4 Applied Visualization, SS10

3 Introduction We are concerned with enhancements and special features for 3D vector fields Glyphs Like 2D, more special icons Particle tracing The same as in 2D Time lines time surfaces P-space vs. C-space LIC Curvilinear grids like 2D Calculation is no problem, but rendering is difficult (display on 2D-screen, see next section) Qualitative analysis at critical points 5 Applied Visualization, SS10 3D vector field topology 6 Applied Visualization, SS10

4 3D vector field topology Assumption Critical point x 0 r r r v( x0) = 0 J r v ( x0 ) : Jacobian matrix Classification of critical points In 3D, using the 3 eigenvalues of the Jacobian There are 2 main cases 3 real eigenvalues 2 complex conjugate and 1 real eigenvalue 7 Applied Visualization, SS10 3D vector field topology Eigenvalues: each is real λ 1 < λ 2 < λ 3 <0 λ 1 < λ 2 <0<λ 3 λ 1 <0<λ 2 < λ 3 0 < λ 1 < λ 2 < λ 3 attracting node saddle node saddle node repelling node Eigenvalues: 1 real & 2 imaginary (like spiral in bathing tube) λ 1 < 0: attracting relative to 0 Re(λ 2 ) = Re(λ 3 ) < 0 attracting focus Re(λ 2 ) = Re(λ 3 ) > 0 repelling focus Orientation of rotation depending on imaginary part λ 1 > 0: repelling relative to 0 Re(λ 2 ) = Re(λ 3 ) < 0 attracting focus Re(λ 2 ) = Re(λ 3 ) > 0 repelling focus Orientation of rotation depending on imaginary part 8 Applied Visualization, SS10

5 3D vector field topology Nodes Node-Saddles Focus Focus-Saddles Repelling variants Attracting variables 3 real eigenvalues 2 complex eigenvalues Images: Saddle Connectors An approach to visualizing the topological skeleton of complex 3D vector fields, Theisel, Weinkauf, Hege, and Seidel 9 Applied Visualization, SS10 3D vector field topology Critical points Attracting and repelling point Flow perpendicular to surface Tangential component of vector field 0 Particle line start / end there Field vortex (ger.: Wirbel) Motion of flow swirling rapidly around a center Important since these are locations of loss of energy Regions of concentrated vorticity (i.e. flow rotation) Vorticity (ger.: Wirbelstärke or Vortizität) Central measure in fluid mechanics an meteorology Measures tendency for elements of the fluid to "spin" More formally Related to amount of "rotation" (i.e. local angular rate of rotation) in a flow (Pseudo-) vector field ω r r r r defined as rotation of the velocity field v r : ω = v 10 Applied Visualization, SS10

6 3D vector field topology Examples for vortices Spiral motion with closed streamlines is vortex flow Air vortex: wake vortex (color encoded) Displacement of vortices behind moving rod Vortex in 11 impeller Applied Visualization, SS10 3D vector field topology Strategy for 3D vector field algorithms Find critical points Classify eigenvalues of Jacobian J v (x) Attracting / repelling node Attracting / repelling focus Saddle, center Combine critical points with particle lines 12 Applied Visualization, SS10

7 3D vector field topology Separatrices Occlusion effects of stream surfaces cluttered and hardly interpretable visualization Iconic representation Visualization with separation surfaces Images: Saddle Connectors An approach to visualizing the topological skeleton of 13 complex 3D vector fields, Theisel, Weinkauf, Hege, and Seidel Applied Visualization, SS10 3D vector field topology Separatrices (cont.) Instead use saddle connectors Separation surfaces of the saddles Intersection of the separation surfaces is the saddle connector Images: Saddle Connectors An approach to visualizing the topological skeleton of 14 complex 3D vector fields, Theisel, Weinkauf, Hege, and Seidel Applied Visualization, SS10

8 3D vector field topology Example using saddle connectors 13 critical points and 9 saddle connectors LIC shows correspondence of skeleton and flow 15 Applied Visualization, SS10 3D vector field topology Stream line oriented topology of a 2D time-dependent vector field LIC images at 3 different time slices Tracking the locations of critical points as stream lines (red / blue / yellow) Images: Saddle Connectors An approach to visualizing the topological skeleton of complex 3D vector fields, Theisel, Weinkauf, Hege, and Seidel 16 Applied Visualization, SS10

9 Path lines (Representation of particle lines) 17 Applied Visualization, SS10 Path lines Approach Randomly generate particles in the flow and trace the path of these particles over a specified time interval Choose randomly Starting positions Starting time Life time Encode scalar values by Color Line width of path Stream balls Etc. 18 Applied Visualization, SS10

10 Path lines Including shading of lines 19 Applied Visualization, SS10 Path lines Including shading of lines 20 Applied Visualization, SS10

11 Path lines Glyphs along path line 21 Applied Visualization, SS10 Path lines Glyphs along path line 22 Applied Visualization, SS10

12 Path lines Glyphs along path-line 23 Applied Visualization, SS10 Ribbons (Representation of particle lines) 24 Applied Visualization, SS10

13 Ribbons Approach Show rotation and divergence Trace two close particles Fill with polygon in between Disadvantages 2 traces Difficult in case of separation (better keep the width constant) Better: direct calculation of rotation Map rotation of vector field directly on particle line Vorticity Measure for rotation of vector field Streamwise vorticity Projection of vorticity on vector of velocity 25 Applied Visualization, SS10 Ribbons Decomposition of the Jacobian r = 1 r r T + ( ) + 1 r r J x( v) J x( v) J x ( v) J x ( v) J x( v) 2 2 = Λ + Ω T ( ) ( ( ) ) symmetric (deformation or stretching tensor) anti-symmetric (local rotation) Furthermore r = 1 r J x ( v) spur J x( v) Id 3 ( ) + Σ + Ω 1 spur( J x( v r )) Id 3 : can be interpreted as the expansion, i.e. local convergence or divergence of the flow field Σ : symmetric matrix, representing the local shear 26 Applied Visualization, SS10

14 Ribbons Local rotation Ω This is a skew-symmetric matrix, called vorticity or spin matrix Form of the rotation matrix where the vector field ω r = Ω = 1 2 streamwise vorticity 0 ω3 ω2 ( ω ω ) T 1 2 ω3 ω 0 ω 1 3 ω2 ω1 0 is the rotation of the velocity, i.e. r r ω = v r = rot v vorticity 27 Applied Visualization, SS10 Ribbons rotation Angle of stream ribbon Assuming Euler s method (for simplicity only!) for integrating the orbit x r τ v( x 1) n = xn 1 + n Let the band vector b (determines direction of the ribbon), be a small vector perpendicular to the tangent vector of the orbit Then b is transformed according to r b n r r r = + + r bn 1 τ J x( v( xn 1)) bn 1 O bn Neglecting higher order terms, expansion, and shear, it follows r b r n = bn 1 + τ Ω( xn 1) bn 1 Local rotation perpendicular to the velocity direction Project the new band vector b after each integration step onto plane perpendicular to tangent vector of the orbit Neglect rotation in path direction (already revealed by orbit itself) particle line 28 normal plane Applied Visualization, SS10 r 1 2

15 Ribbons Divergence of vector field Width of ribbon Advantage Curls of the flow along the flow s direction can easily be recognized 29 Applied Visualization, SS10 Ribbons 30 Applied Visualization, SS10

16 Ribbons 31 Applied Visualization, SS10 Ribbons 32 Applied Visualization, SS10

17 Ribbons Aortic Blood Turbulence (by: R. Boon et al.) Velocities and turbulence due to a pump in the aorta Helps to understand how the pump affects blood flow near a patient's heart in order to better design the pump 33 Applied Visualization, SS10 Ribbons Alternative approach Trace two close-by particles Keep distance constant 34 Applied Visualization, SS10

18 Balls (Representation of particle lines) 35 Applied Visualization, SS10 Balls Distance of balls Velocity Adaptive step size Radius of balls Scalar value Disadvantage Many triangles 36 Applied Visualization, SS10

19 Balls 37 Applied Visualization, SS10 Balls 38 Applied Visualization, SS10

20 Stream tubes (Representation of particle lines) 39 Applied Visualization, SS10 Stream tubes Polygonal object Spatial impression Radius Scalar value Calculation Circle trace all elements Trace line then place circles and combine them Advantage Curls of the flow along the flow s direction can easily be recognized 40 Applied Visualization, SS10

21 Stream tubes 41 Applied Visualization, SS10 Stream tubes 42 Applied Visualization, SS10

22 Stream tubes 43 Applied Visualization, SS10 Stream tubes 44 Applied Visualization, SS10

23 Stream tetrahedra 45 Applied Visualization, SS10 Stream tetrahedra Combines advantages of balls and ribbons Local rotation + velocity + divergence (rotation) (density) (size) Very simple geometry 46 Applied Visualization, SS10

24 Stream tetrahedra 47 Applied Visualization, SS10 3D LIC 48 Applied Visualization, SS10

25 3D LIC Calculation No problem the same as in 2D Representation Difficult interpretation Requires volume visualization Additional tools Clipping (plane, box, ) Animation Transparency 49 Applied Visualization, SS10 3D LIC Missing continuity 50 Applied Visualization, SS10

26 3D LIC Color differences to identify connected structures 51 Applied Visualization, SS10 3D LIC Animation 52 Applied Visualization, SS10

27 3D LIC Reduction of visual data Restrict to volume of interest 53 Applied Visualization, SS10 3D LIC Reduction of visual data Application of transparency 54 Applied Visualization, SS10

28 3D LIC Reduction of visual data Deform clip object based on vector field 55 Applied Visualization, SS10 3D LIC Reduction of visual data Clipping Masking 56 Applied Visualization, SS10

29 3D LIC Reduction of visual data Clipping 57 Applied Visualization, SS10 Combining different techniques 58 Applied Visualization, SS10

30 Combining different techniques Examples Frequent combination of different techniques Requirement of interactivity Efficient implementation 59 Applied Visualization, SS10 Combining different techniques 60 Applied Visualization, SS10

31 Combining different techniques 61 Applied Visualization, SS10 Combining different techniques Visualization of blood flow 62 Applied Visualization, SS10