1. Yingcai Xiao SCATTERED DATA VISUALIZATION

2. Scattered Data: sample points distributed unevenly and non-uniformly throughout the volume of interest. Example Data: chemical leakage at a tank-farm.

3. Method of Approach : Interpolation-based Two-step Approach (Foley & Lane, 1990) Sparse Data Interpolation Modeling Intermediate Grid Rendering Grid-Based Rendered Volume

4. Interpolation Methods (Nielson, 1993) Xiao etc. 1996 Global: all sample points are used to interpolated a grid value. Local: only nearby sample points are used to interpolated a grid value. Exact: the interpolation function can exactly reproduce the data values on the sample points. Problems: Xiao etc. 1996

5. Defining a Global Exact Interpolant (Foley & Lane, 1990; Nielson, 1993) n sample points: (x i,y i,z i,v i ) for i = 1,2,..n One interpolation function, e.g., Thin-plate spline, d i is the distance between sample point i and the point to be interpolated p(x,y,z). d i = ((x-x i ) 2 +(y-y i ) 2 +(z-z i ) 2 ) 1/2 bi,c1,c2,c3,c4 are n+4 constants to be solved by enforcing the following conditions: f (x i,y i,z i ) = v i for i = 1,2,..n

6. Global Exact Interpolation Functions (Foley & Lane, 1990; Nielson, 1993) Thin-plate spline Volume Spline Shepard Multiquadric

7. Thin-plate Spline

8. Volume Spline

9. Shepard method

10. l Misinterpretation (Negative Concentration) l Ambiguity in Selecting Interpolation Methods l Inconsistent Interpolations in Modeling and Rendering l Visualizing Secondary Data Instead of the Original Data l No Error Estimation l Unable to Add Known Information l Not Efficient Deficiencies of the Interpolation-based Two-step Approach (Xiao et. Al., 1996)

11. l Zero-value dilemma l Negative-value dilemma l Correctness dilemma Three Dilemmas and Three Constraints (Xiao & Woodbury, 1999) lPoint Constraint l Value Constraint l Local Constraint

12. Point Constraint

13. Value Constraint

14. Local Constraint p6p6 p1p1 p2p2 p8p8 p7p7 p4p4 p3p3 p5p5

15. Conclusions Two-step approach faces three dilemmas. Constrained interpolations can alleviate the dilemmas. The problems are far from being solved. Data modeling is import to data visualization, just as geometry modeling is important to geometry visualization.

16. Conclusions To visualize scattered data, we are challenged to find modeling techniques that l preserve input data values; l produce meaningful output values; l provide error estimations; l accept additional constraints; l reduce the requirement on the sampling intensity.

17. A FINITE ELEMENT BASED APPROACH XIAO & ZIEBARTH, 2000

18. The Finite Element Based Approach (1) Tessellation (2) Computation (3) Rendering

19. The Finite Element Based Approach Sparse Data Volume Triangulation Tessellation Element Network Computation FEM Element-Based Node Values Rendered Volume Rendering

20. Tessellation Three-Dimensional Triangulation: Tetrahedronization Delaunay Triangulation: Sphere Criterion Data Points Triangulation Element Network

21. The Double Layer Technique Physical DiscontinuityLogical Discontinuity

22. The Finite Element Method (1) Problem Definition: Boundary Value Problem Governing equation: Boundary Condition: (2) Element Definition: Shape: Tetrahedron Order: Basis Function

23. The Finite Element Method (3) System Formulation Ritz Method Galerkin's method (4) Sparse Sample Data (5) System Solution Gaussian Elimination Householder's Method

24. Rendering : Modifying Conventional Methods (1) Hexahedron => Tetrahedron (2) (ijk) Indexing => Neighbor-to-Neighbor Traversal

25. Advantages of the Finite Element Based Approach (1) Meaningful Results A Pollution ProblemExactGrid-basedFEM-based

26. Advantages of the Finite Element Based Approach (2) Complicate Geometry: Non-Gridable Volumes

27. Advantages of the Finite Element Based Approach (3) Discontinuity: Internal Discontinuity Surface

28. Advantages of the Finite Element Based Approach (3) Discontinuity: Discontinuous Regions

29. Advantages of the Finite Element Based Approach (4) Error Estimation and Iterative Refinement

30. Advantages of the Finite Element Based Approach (5) Efficient Add One Point => Add O(1) Tetrahedrons O(n 2 ) Times More Efficient Than Grid-Based Approaches.

31. Advantages of the Finite Element Based Approach (6) No Whittaker-Shannon Sampling Rate Interpolation Problem ==> Boundary Value Problem (7) No Ambiguity in Selecting Modeling Methods

32. Advantages of the Finite Element Based Approach (8) Honoring Original Sample Data

33. Advantages of the Finite Element Based Approach (9) Flexible, Fast and Interactive Modification of an Existing Sample Point

34. Advantages of the Finite Element Based Approach (9) Flexible, Fast and Interactive Addition of a New Sample Point

35. Advantages of the Finite Element Based Approach (10) Consistent Basis Function

36. Future Work (1) Other Types of Problems: Initial Value Problems (2) Other Types of Elements: Polyhedrons (3) Higher-Order Elements: P-Version (4) Automated Tessellation: Densification (5) Thinning (6) Curved Discontinuity Surfaces (7) Delaunay Triangulation near Discontinuity Surfaces (8) Higher-Order Rendering Method (9) Fast Searching Algorithms (10) Technique Issues (e.g., Solving Sparse Matrices,...)

37. Summary The finite element based approach is a new framework for scattered data visualization. Many challenging problems can be solved easily within this framework. This approach revealed a promising direction and brought many interesting research topics into the field of sparse data volume visualization.