CIS 467/602-01: Data Visualization

Transcription

1 CIS 467/602-01: Data Visualization Vector Field Visualization Dr. David Koop

2 Fields Tables Networks & Trees Fields Geometry Clusters, Sets, Lists Items Items (nodes) Grids Items Items Attributes Links Positions Positions Attributes Attributes 2

3 Fields and Grids Fields: values come from a continuous domain, infinitely many values - Sampled at certain positions to approximate the entire domain - Positions are often aligned in grids uniform rectilinear structured unstructured [ Weiskopf/Machiraju/Möller] Geometry: the spatial positions of the data (points) Topology: how the points are connected (cells) 3

4 Fields in Visualization Scalar Fields Vector Fields Tensor Fields Each point in space has an associated... 4

5 Fields in Visualization Scalar Fields Vector Fields Tensor Fields (Order-0 Tensor Fields) (Order-1 Tensor Fields) (Order-2+) Each point in space has an associated s 0 4 v v 1 v Scalar Vector Tensor 3 5 4

6 Isosurfacing [J. Kniss, 2002] 5

7 Marching Cubes Break isovalue crossings into cases [Lorensen and Cline, 1987] # Positive Vertices 4A 4B 4C 4D 4E Zero Five 0 5A 5B 5C One Six 1 6A 6B 6C Two Seven 2A 2B 2C 7 Three Eight 3A 3B 3C 8 Four 4A 4B 4C 4D 4E 4F [R. Wenger, 2013] 6

8 Volume Rendering [J. Kniss, 2002] 7

9 Volume Rendering vs. Isosurfacing (a) Direct volume rendered (b) Isosurface rendered [Kindlmann, 1998] 8

10 Object Volume Ray Casting order approach Image Plane Data Set Eye [Levine] 9

11 Transfer Functions Need to map scalar value to a color (c) and opacity (α) Remember, we composite using the over operator Can use other information (e.g. gradient density) in multidimensional transfer functions α RGB Simp value 10

12 Assignment 5 Isosurfacing, Volume Rendering, Streamlines, and Glyphs (602-01) Hurricane Katrina Dataset Not D3, use ParaView - One visualization, 2 files - Turn in state file (pvsm) - Turn in screenshot Questions? 11

13 Final Exam Thursday, May 7, 11:30am-2:30pm Cumulative with emphasis on topics covered after the midterm Topics since midterm: Maps, Color & Perception, Interaction, Multiple Views, Aggregation, Focus+Context, Sets, Text, Scalar Fields, Vector Fields Textbook: Chapters 8, We have covered more details than in the book on: Sets, Text, Maps, Scalar & Vector Fields Format: Multiple Choice, Matching + Free Response 12

14 Fields in Visualization Scalar Fields Vector Fields Tensor Fields (Order-0 Tensor Fields) (Order-1 Tensor Fields) (Order-2+) Each point in space has an associated s 0 4 v v 1 v Scalar Vector Tensor

15 Examples of Vector Fields Wind [earth.nullschool.net, 2014] 14

16 Examples of Vector Fields Wind [earth.nullschool.net, 2014] 14

17 Examples of Vector Fields Computational Fluid Dynamics [newmerical] 15

18 Examples of Vector Fields Earthquake Ground Surface Movement [H. Yu et. al., SC2004] 16

19 Examples of Vector Fields Gradient Vector Fields 17

20 Examples of Vector Fields Wildfire Modeling [E. Anderson] 18

21 Visualizing Vector Fields Direct: Glyphs, Render statistics as scalars Geometry: Streamlines and variants Textures: Line Integral Convolution (LIC) Topology: Extract relevant features and draw them 19

22 Glyphs Represent each vector with a symbol Hedgehogs are primitive glyphs (glyph is a line) ParaView Example 20

23 Glyphs Represent each vector with a symbol Hedgehogs are primitive glyphs (glyph is a line) Glyphs that show direction and/or magnitude can convey more information If we have a separate scalar value, how might we encode that? Clutter issues 21

24 Glyphs For vector fields, can encode - Direction - Magnitude - Scalar value Good: - Show precise local measures - Can encode scalar information as color Bad: - Possible sampling issues - Clutter (Occlusion): Can remove some points to help - Clutter is worse in higher dimensions 22

25 Rendering Vector Field Statistics as Scalars Many statistics we can compute for vector fields: - Magnitude - Vorticity - Curvature These are scalars, can color with our scalar field visualization techniques (e.g. volume rendering) [Color indicates vector magnitude] 23

26 Streamlines & Variants Trace a line along the direction of the vectors Streamlines are always tangent to the vector field Basic Particle Tracing: 1. Set a starting point (seed) 2. Take a step in the direction of the vector at that point 3. Adjust direction based on the vector where you are now 4. Go to Step 2 and Repeat 24

27 Example Elliptical path Suppose we have the actual equation Given point (x,y), the vector is at that point is [vx, vy] where - vx = -y - vy = (1/2)x Want a streamline starting at (0,-1) y x [LIC (not streamlines!) via Levine] 25

28 Some Glyphs x/2 : [x,y] [-y, (1/2)x], Step: m [via Levine] 26

29 Streamlines (Step 1) [x,y] [-y, (1/2)x], Step: [via Levine] 27

30 Streamlines (Step 2) [x,y] [-y, (1/2)x], Step: [via Levine] 28

31 Streamlines (Step 3) [x,y] [-y, (1/2)x], Step: [via Levine] 29

32 Streamlines (Step 4) [x,y] [-y, (1/2)x], Step: [via Levine] 30

33 Streamlines (Step 10) [x,y] [-y, (1/2)x], Step: [via Levine] 31

34 Streamlines (Step 19) [x,y] [-y, (1/2)x], Step: [via Levine] 32

35 Euler Method Seeking to approximate integration of the velocity over time Euler method is the starting point for approximating this Problems? 33

36 Euler Method Seeking to approximate integration of the velocity over time Euler method is the starting point for approximating this Problems? - Choice of step size is important 33

37 Euler Method Seeking to approximate integration of the velocity over time Euler method is the starting point for approximating this Problems? - Choice of step size is important - Choice of seed points are important 33

38 Euler Method Seeking to approximate integration of the velocity over time Euler method is the starting point for approximating this Problems? - Choice of step size is important - Choice of seed points are important Also remember that we have a field we don't have measurements at every point (interpolation) 33

39 Euler Quality by Step Size [via Levine] 34

40 Numerical Integration How do we generate accurate streamlines? Solving an ordinary differential equation dl dt = v(l(t)) L(0) = L 0 where is the streamline, is the vector field, and is time Solution: L v t L(t + t) =L(t)+ Z t+ t t v(l(t))dt 35

41 Higher-order methods Z t+ t t v(l(t))dt Euler method (use single sample) Higher-order methods (Runge- Kutta) (use more samples) v v [A. Mebarki] 36

42 Higher-Order Comparison [via Levine] 37

43 ParaView Example 38