1. Scientific Visualization Data Modelling for Scientific Visualization CS 5630 / 6630 August 28, 2007

2. Recap: The Vis Pipeline

4. Types of Data in SciVis: Functions http://lambda.gsfc.nasa.gov/product/cobe/firas_image.cfm

5. Types of Data in SciVis: Functions on Circles E. Anderson et al.: Towards Development of a Circuit Based Treatment for Impaired Memory

6. Types of Data in SciVis: 2D Scalar Fields

7. Types of Data in SciVis: Scalar Fields on Spheres http://lambda.gsfc.nasa.gov/product/cobe/firas_image.cfm

8. Types of Data in SciVis: 3D, time-varying Scalar Fields http://background.uchicago.edu/~whu/beginners/introduction.html

9. Types of Data in SciVis: 2D Vector Fields

10. Types of Data in SciVis: 3D Vector Fields

11. Tensors Tensors are “multilinear functions”  rank 0 tensors are scalars  rank 1 tensors are vectors  rank 2 tensors are matrices, which transform vectors  rank 3..n tensors have no nice name, but they transform matrices, rank-3 tensors, etc. We are not going to see these

12. DTI Tensors DTI Tensors are symmetric, positive definite  SPD: scale along orthogonal directions  More specifically, they approximate the rate of directional water diffusion in tissue

13. Types of Data in SciVis: 2D, 3D Tensor Fields Kindlmann et al. Super-Quadric Tensor Glyphs and Glyph-packing for DTI vis.

14. Computers like discrete data, but world is continuous

15. Sampling Continuous to discrete Store properties at a finite set of points

16. Sampling Continuous to discrete Store properties at a finite set of points

17. Sampling Continuous to discrete Store properties at a finite set of points

18. Interpolation Discrete to continuous  Reconstruct the illusion of continuous data, using finite computation

19. Nearest Neighbor Interpolation Pick the closest value to you

20. Linear Interpolation Assume function is linear between two samples

21. Linear Interpolation Assume function is linear between two samples v1 v2 01 u f(x) = ax + b v1 = a.0 + b = b v2 = a.1 + b = a + b b = v1 a = v2 – b = v2 - v1 f(x) = v1+ (v2 – v1).x sometimes written as f(x) = v2.x + v1.(1-x)‏

22. Cubic Interpolation Linear reconstruction is better than NN, but it is not smooth across sample points Let's use a cubic  Two more parameters: we need constraints  Constrain derivatives

23. Cubic Interpolation Same as with linear 012 v0 v1 v3 v2 f(x) = a+b.x+c.x^2+d.x^3 f'(x) = b + 2cx + 3dx^2 f(0) = v1 f(1) = v2 f'(0) = (v2 – v0)/2 f'(1) = (v3 – v1)/2... a = v1 b = (v2-v0) / 2 c = v0 – 5.v1/2 + 2v2 – v3/2 d = -v0/2 + 3.v1/2 – 3.v2/2 + v3/2

24. (VisTrails Demo) Linear vs Higher-order interpolation in plotting

25. Might make a big difference! Kindlmann et al. Geodesic-loxodromes... MICCAI 2007

26. 1D vs n-D Most common technique: separability Interpolate dimensions one at a time

27. (VisTrails Demo) 2D Interpolation in VTK images

28. Implicit vs Explicit Representations Explicit is parametric  Domain and range are “explicit” Implicit stores domain... implicitly  Zero set of a explicit domain  Pro: it's easy to change topology of domain: just change the function  Con: it's harder to analyze and compute with

29. Implicit vs. explicit representations Explicit: y(t) = sin(t)‏ x(t) = cos(t)‏ s = (x(t), y(t)), 0 < t <= 2  Implicit: f(x,y) = x^2 + y^2 - 1 s = (x,y): f(x,y) = 0

30. Regular vs Irregular Data Regular data: sampling on every point of an integer lattice Irregular data: more general sampling

31. Curvilinear grid Like a regular grid, but on curvilinear coordinates  Here, radius and angle

32. Triangular and Tetrahedral Meshes Completely arbitrary samples  Need to store topology: How do samples connect with one another?

33. Quadrilateral and Hexahedral Meshes Basic element is a quad or a hex  Element shape is better for computation  Much, much harder to generate

34. Tabular Data Most common in information visualization Relational DBs

35. ... etc. Node vs cell data: do we store values on nodes (vertices) or on cells (tets and tris)? Pure-quad vs quad-dominant: mixing types of elements Linear vs high-order: different interpolation modes on elements