1. Scalar Visualization

2. Outline • Visualizing scalar data • A number of the most popular scalar visualization techniques • Color mapping • Contouring • Height plots • Color mapping • Design of effective colormaps • Contouring in 2D and 3D • Height plots

3. Visualization Pipeline 1. Data Importing 2. Data Filtering 3. Data Mapping 4. Date Rendering

4. Scalar Function

5. Color Mapping • color look-up table • Associate a specific color with every scalar value • The geometry of Dv is the same as D

6. Luminance Colormap • Use grayscale to represent scalar value • Most scientific data (through measurement, observation, or simulation) are intrinsically grayscale, not color Luminance Colormap Legend

7. Rainbow Colormap • Red: high value; Blue: low value • A commonly used colormap Luminance Map Rainbow Colormap

8. Rainbow Colormap • Construction • f<dx: R=0, G=0, B=1 • f=2: R=0, G=1, B=1 • f=3: R=0, G=1, B=0 • f=4: R=1, G=1, B=0 • f>6-dx: R=1, G=0, B=0

9. Rainbow Colormap Implementation void c(float f, float & R, float & G, float &B) { const float dx=0.8 f=(f<0) ? 0: (f>1)? 1 : f //clamp f in [0,1] g=(6-2*dx)*f+dx //scale f to [dx, 6-dx] R=max(0, (3-fabs(g-4)-fabs(g-5))/2); G=max(0,(4-fabs(g-2)-fabs(g-4))/2); B=max(0,(3-fabs(g-1)-fabs(g-2))/2); }

10. Colormap: Designing Issues • Choose right color map for correct perception • Grayscale: good in most cases • Rainbow: e.g., temperature map • Rainbow + white: e.g., landscape • Blue: sea, lowest • Green: fields • Brown: mountains • White: mountain peaks, highest

11. Rainbow Colormap http://atmoz.org/img/weatherchannel_national_temps.png

12. Exp: Earth map http://www.oera.net/How2/PlanetTexs/EarthMap_2500x1250.jpg

13. Exp: Sun in green-white colormap

14. Exp: Coronal loop http://media.skyandtelescope.com/images/SPD+on+CME+image+5+--+TRACE.gif

15. Color Banding Effect Caused by a small number of colors in a look-up table

16. Contouring • A contour line C is defined as all points p in a dataset D that have the same scalar value, or isovalue s(p)=x • A contour line is also called an isoline • In 3-D dataset, a contour is a 2-D surface, called isosurface

17. Contouring Cartograph

18. Contouring One contour at s=0.11 S > 0.11 S < 0.11 Contouring and Color Banding

19. Contouring Contouring and Colormapping: Show (1) the smooth variation and (2) the specific values 7 contour lines

20. Properties of Contours • Indicating specific values of interest • In the height-plot, a contour line corresponds with the interaction of the graph with a horizontal plane of s value

21. Properties of Contours • The tangent to a contour line is the direction of the function’s minimal (zero) variation • The perpendicular to a contour line is the direction of the function’s maximum variation: the gradient Contour lines Gradient vector

22. Constructing Contours V=0.48 Finding line segments within cells

23. Constructing Contours • For each cell, and then for each edge, test whether the isoline value v is between the attribute values of the two edge end points (vi, vj) • If yes, the isoline intersects the edge at a point q, which uses linear interpolation • For each cell, at least two points, and at most as many points as cell edges • Use line segments to connect these edge-intersection points within a cell • A contour line is a polyline.

24. Constructing Contours V=0.37: 4 intersection points in a cell -> Contour ambiguity

25. Implementation: Marching Squares • Determining the topological state of the current cell with respect to the isovalue v • Inside state (1): vertex attribute value is less than isovalue • Outside state (0): vertex attribute value is larger than isovalue • A quad cell: (S3S2S1S0), 24=16 possible states • (0001): first vertex inside, other vertices outside • Use optimized code for the topological state to construct independent line segments for each cell • Merge the coincident end points of line segments originating from neighboring grid cells that share an edge

26. Implementation: Marching Squares Topological State of a Quad Cell

27. Implementation: Marching Cube Topological State of a hex Cell Marching cube generates a set of polygons for each contoured cell: triangle, quad, pentagon, and hexagon

28. Contours in 3-D • In 3-D scalar dataset, a contour at a value is an isosurface Isosurface for a value corresponding to the skin tissue of an MRI scan 1283 voxels

29. Contours in 3-D Two nested isosurface: the outer isosurface is transparent

30. Height Plots • The height plot operation is to “warp” the data domain surface along the surface normal, with a factor proportional to the scalar value

31. Height Plots Height plot over a planar 2-D surface

32. Height Plots Height plot over a nonplanar 2-D surface

33. Vector Visulization • Divergence and Vorticity • Vector Glyphs • Vector Color Coding • Displacement Plots • Stream Objects • Texture-Based Vector Visualization • Simplified Representation of Vector Fields

34. Vector Function

35. Vector versus Scalar

36. Example in 2-D

37. Gradient of a Scalar

38. Gradient of a Scalar

39. Divergence of a Vector

40. Divergence of a Vector • Divergence computes the flux that the vector field transports through the imaginary boundary Γ, as Γ0 • 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.

41. Divergence of a Vector

42. Divergence of a Vector

43. Vorticity of a Vector

44. Vorticity of a Vector • Vorticity computes the rotation flux around a point • Vorticity 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 • Vorticity signals the presence of vortices in vector field

45. Vorticity of a Vector Color: Glyph:

46. Vector Glyph • Vector 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: location, direction, orientation, size and color

47. Vector Glyph Line glyph, or hedgehog glyph Sub-sampled by a factor of 8 (32 X 32) Original (256 X 256) Velocity Field of a 2D Magnetohydrodynamic Simulation

48. Vector Glyph Line glyph, or hedgehog glyph Sub-sampled by a factor of 4 (64 X 64) Original (256 X 256) Velocity Field of a 2D Magnetohydrodynamic Simulation

49. Vector 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

50. Vector Glyph Random Sub-sampling Is better