Vector Visualization Chap. 6 March 7, 2013 March 26, Jie Zhang Copyright

Transcription

1 ector isualization Chap. 6 March 7, 2013 March 26, 2013 Jie Zhang Copyright CDS 301 Spring, 2013

2 Outline 6.1. Divergence and orticity 6.2. ector Glyphs 6.3. ector Color Coding 6.4. Displacement Plots (skip) 6.5. Stream Objects 6.6. Texture-Based ector isualization 6.7. Simplified Representation of ector Fields (skip)

3 CH 6.1 Divergence and orticity

4 ector Function f: R 3 R 3 (usually in 3- D) f: R 2 R 2 (simpler case : 2 - D )

5 ector versus Scalar ),, ( ),, ( ),, ( ),, ( ˆ ˆ ˆ : ector z y x z y x f z y x f z y x f or or k j i z y x z y x z y x ),, ( :s Scalar z y x f s s

6 Example in 2-D x y ), ( ), ( : ector y x y x f y x f y x y x s e s y x f s y x : exp ), ( :s Scalar ) ( 2 2

7 Gradient of a Scalar ) ( ) ( ) ( D : Exp ),, ( vector a a scalar is Gradient of y x y y x x y x ye y s xe x s e s z s y s x s s

8 Gradient of a Scalar Exp : 2D s x y s x s y x y 1 1

9 Divergence of a ector 1 lim ( n ) ds 0 Γ is closed hypersurface (closed curve in 2D and closed surface in 3D) Γ is the area enclosed by Γ (area in 2D and volume in 3D)

10 Divergence of a ector Divergence computes the flux that the vector field transports through the imaginary boundary Γ, as Γ0 Divergence of a vector is a scalar A positive divergence point is called source, because it indicates that mass would spread from the point (in fluid flow) A negative divergence point is called sink, because it indicates that mass would get sucked into the point (in fluid flow) A zero divergence denotes that mass is transported without compression or expansion.

11 Divergence of a ector x x y x x z Exp : (x, y) 11 2 Exp : (y,x) 0 0 Divergence 0 Free Positive divergence Exp : (-x,-y) 11 2 Negative divergence :source :sink

12 Divergence of a ector

13 (March 07, 2013 Stops Here)

14 March 19, 2013

15 Review: Divergence of a ector x x y x x z Exp : (x, y) 11 2 Exp : (y,x) 0 Divergence 0 0 Free Positive divergence Exp : (-x,-y) 11 2 Negative divergence :source :sink

16 orticity of a ector lim 0 1 ( ds ) is closed hypersurface (curve in 2D and surface in 3D) Γ is the area enclosed by (area in 2D and volume in 3D)

17 orticity of a ector orticity computes the rotation flux around a point orticity of a vector is a vector The magnitude of vorticity expresses the speed of angular rotation The direction of vorticity indicates direction perpendicular to the plane of rotation orticity signals the presence of vortices in vector field

18 orticity of a ector y x x z z y x y z x y z ) ( 0 ) ( 0 ) ( y,0) (x, For 0 0 : for 2D planar velocity Exp y x z z z

19 orticity of a ector y x x z z y x y z x y z ) ( 0 ) ( 0 ) ( (-y,x,0) For 0 0 2D planar velocity :for Exp y x z z z

20 orticity of a ector Stream line: vector direction Color: vorticity

21 CH 6.2 ector Glyph

22 ector Glyph l ( x, x k ( x)) x ector glyph mapping technique associates a vector glyph (or icon) with the sampling points of the vector dataset The magnitude and direction of the vector attribute is indicated by the various properties of the glyph: direction, orientation, size and color

23 ector Glyph Line glyph, or hedgehog glyph: Orientation, Length and Color Sub-sampled by a factor of 8 (32 X 32) Original (256 X 256) elocity Field of a 2D Magnetohydrodynamic Simulation

24 ector Glyph Line glyph, or hedgehog glyph Sub-sampled by a factor of 4 (64 X 64) Original (256 X 256) elocity Field of a 2D Magnetohydrodynamic Simulation

25 ector Glyph Sub-sampled by a factor of 2 (128 X 128) Original (256 X 256) Problem with a dense Representation using glyph: (1) clutter (2) miss-representation

26 Random Sub-sampling Is better ector Glyph

27 ector Glyph: 3D Simulation box: 128 X 85 X 42; or 456,960 data point 100,000 glyphs Problem: visual occlusion

28 ector Glyph: 3D Simulation box: 128 X 85 X 42; or 456,960 data point 10,000 glyphs: less occlusion

29 ector Glyph: 3D Simulation box: 128 X 85 X 42; or 456,960 data point 100,000 glyphs, 0.15 transparency: less occlusion

30 ector Glyph: 3D Simulation box: 128 X 85 X 42; or 456,960 data point 3D velocity isosurface

31 ector Glyph Glyph method is simple to implement, and intuitive to interpretation High-resolution vector datasets must be sub-sampled in order to avoid overlapping of neighboring glyphs. Glyph method is a sparse visualization: does not represent all points Occlusion Subsampling artifacts: difficult to interpolate Alternative: color mapping method is a dense visualization

32 Matlab: ector We will use Matlab to implement the vector visualization. For one well-illustrated example, refer to wind.m

33 Matlab: ector Question Use Matlab to visualize the 2-D vector function F, which is defined as Fx(x,y) = x Fy(x,y) = y

34 quiver plot %example [x,y]=meshgrid(-2:0.2:2) u = x % fx, or x-component of the vector v = y % fy, or y-component of the vector quiver(x,y,u,v)

35 divergence plot %example [x,y]=meshgrid(-2:0.2:2) u = x % fx, or x-component of the vector v = y % fy, or y-component of the vector quiver(x,y,u,v) %calculate and display the divergence div=divergence(x,y,u,v) hold on imagesc([-2.0,2.0],[-2.0,2.0],div) alpha(0.5)

36 gradient and curl [FX, FY, FZ] = gradient(f) [curlx, curly, curlz, cav] = curl (X, Y, Z, U,, W)

37 (March 19, 2013 Stops Here)

38 March 21, 2013

39 Review (1) Gradient, Divergence and orticity (2) ector Glyph

40 Review Gaussian_gradient_divergence.m %Author: Jie Zhang %Date: March 20, 2013 %Purpose: CDS301 class CH6: isualize the gradient, divergence and %vorticity %Scalar function: a double peak 2-D Gaussian function %ector function: the gradient of the Gaussian function %Divergence function: divergence of the gradient %orticity: the curl of the vector function %clear the variables and figure space clear clf reset %define the domain %x: [-2.0, 2.0], and y: [-2.0,2.0] [x,y]=meshgrid([-2.0:0.2:2.0],[-2.0:0.2:2.0]); %define the Gaussian function z=x.*exp(-x.^2-y.^2); %surf(x,y,z) %get the idea of the data

41 Review Gaussian_gradient_divergence.m (continued) %display the image hf_1=figure(1) %hi_1=imagesc([-2.0,2.0],[-2.0,2.0],z) hi_1=surface(x,y,z) %get(hi_1) %find the handle values set(hi_1,'edgealpha',0) %remove the wireframe lines, or edge lines set(hi_1,'facecolor','interp') %smooth color from the surface colorbar %colorbar indicates the values %calculate the gradient [gx,gy]=gradient(z,0.2,0.2) %note that spaceing between points is 0.1 hf_2=figure(2) %surf(x,y,gy) quiver(x,y,gx,gy)

42 Review Gaussian_gradient_divergence.m (continued) %calcuate the divergence div=divergence(x,y,gx,gy) hf_3=figure(3) surf(x,y,div) %calculate the vorticity [curlz,cav]=curl(x,y,gx,gy) hf_4=figure(4) surf(x,y,curlz)

43 CH 6.3 ector Color Coding

44 ector Color Coding Similar to scalar color mapping, vector color coding is to associate a color with every point in the data domain Typically, use HS system (color wheel) Hue is used to encode the direction of the vector, e.g., angle arrangement in the color wheel alue of the color vector is used to encode the magnitude of the vector Saturation is set to one (e.g., no white color)

45 ector Color Coding 2-D elocity Field of the MHD simulation: Orientation: color Magnitude: brightness

46 ector Color Coding 2-D elocity Field of the MHD simulation: Orientation only; no magnitude

47 ector Color Coding Dense visualization Lacks of intuitive interpretation take time to be trained to interpret the image

48 CH 6.4 Stream Objects

49 Stream Objects ector glyph plots show the straight trajectories over a short time (fixed velocity) of trace particles released in the vector fields Shows the direction of the flow at a given point Stream objects show the curved trajectories for longer time intervals (varying velocity with time) of trace particles released in the vector field. Follow the flow and trace out the flow

50 Streamlines Streamline is a curved path over a given time interval of a trace particle passing through a given start location or seed point S { p( ), [0, T]} p(τ) ( p) dt where p(0) t 0 p 0, the seed point

51 Streamlines All lines are traced up to the same time T Seed points (gray ball) are uniformly sampled Color is used to reinforce the vector magnitude

52 Streamlines: Issues Require numerical integration, which accumulates errors as the integration time increases Euler integration / t p(τ) ( p) dt ( pi) t where p p i t i1 0 Euler integration: fast but less accurate Runge-Kutta integration: slower but more accurate Need to find optimal value of time step Δt Choose number and location of seed points Trace to maximum time or maximum length Trace upstream or downstream Saved as a polyline on an unstructured grid i1 t i0

53 Stream tubes Add a circular cross section along the streamline curves, making the lines thicker Tracing downstream: the seed points are on a regular grid

54 Stream tubes Tracing upstream: the arrow heads are on a regular grid

55 Stream Objects in 3-D Input: 128 X 85 X 42 Undersampling: 10 X 10 X 10 Opacity 1 Maximum Length

56 Stream Objects in 3-D Input: 128 X 85 X 42 Undersampling: 3 X 3 X 3 Opacity 1 Maximum Length

57 Stream Objects in 3-D Input: 128 X 85 X 42 Undersampling: 3 X 3 X 3 Opacity 0.3 Maximum Time

58 Stream Objects in 3-D Stream tubes Seed area at the flow inlet

59 Stream Ribbons Two thick Ribbons orticity is color coded ector Glyth

60 Stream Ribbons A stream ribbon is created by launching two stream lines from two seed points close to each other. The surface created by the lines of minimal length with endpoints on the two streamlines is called a stream ribbon

61 Stream Surface Given a seed curve Γ, a stream surface S Γ is a surface that contains Γ and its streamlines Everywhere tangent to the vector field Flow can not cross the surface Stream tube is a particular case of a stream surface: the seed curve is a small closed curve Stream ribbon is also a particular case of a stream surface: the seed curve is a short line

62 (March 21, 2013 Stops Here)

63 March 26, 2013

64 ector Color Coding Review

65 Stream Tubes Review

66 CH 6.6 Texture-Based ector isualization

67 Texture-Based ector is. Discrete or sparse visualizations can not convey information about every point of a given dataset domain Similar to color plots, texture-based vector visualization is a dense representation The vector field (direction and magnitude) is encoded by texture parameters, such as luminance, color, graininess, and pattern structure

68 Texture-Based ector is. ector magnitude: Color direction: Graininess

69 Texture-Based ector is. LIC principle: Line Integrated Convolution Principle T ( p) k( s) e L L s 2 N( S( p, s)) k( s) ds L L k( s) ds N : noise texture S(p,s) :streamline of seed point P k(s) : weighting or blurring function L : width of blurring function

70 Texture-Based ector is. LIC is a process of blurring or filtering the texture (noise) image along the streamlines Due to blurring, the pixels along a streamline are getting smoothed; the graininess of texture is gone However, between neighboring streamlines, the graininess of texture is preserved, showing contrast.

71 End of Chap. 6 Note: skip 6.7