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Yingcai Xiao

SCATTERED DATA VISUALIZATION. Yingcai Xiao. Scattered Data: sample points distributed unevenly and non-uniformly throughout the volume of interest. Example Data: chemical leakage at a tank-farm. Method of Approach : Interpolation-based Two-step Approach (Foley & Lane, 1990). Rendering.

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Yingcai Xiao

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  1. SCATTERED DATA VISUALIZATION • Yingcai Xiao

  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) Rendering Modeling Intermediate Grid Sparse Data Rendered Volume Interpolation Grid-Based

  4. Interpolation Methods (Nielson, 1993) 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. Interpolation Methods Example: 1D Global and Exact

  6. Interpolation Methods Example: 1D Global and Exact

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

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

  9. Thin-plate Spline

  10. Volume Spline

  11. Shepard method

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

  13. Three Dilemmas and Three Constraints (Xiao & Woodbury, 1999) • Zero-value dilemma • Negative-value dilemma • Correctness dilemma • Point Constraint • Value Constraint • Local Constraint

  14. Point Constraint

  15. Value Constraint

  16. p6 p4 p3 p1 p2 p8 p7 p5 Local Constraint

  17. 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.

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

  19. A FINITE ELEMENT BASED APPROACH • XIAO & ZIEBARTH, 2000

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

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

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

  23. The Double Layer Technique Physical Discontinuity Logical Discontinuity

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

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

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

  27. Advantages of the Finite Element Based Approach • (1) Meaningful Results A Pollution Problem Exact Grid-based FEM-based

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

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

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

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

  32. Advantages of the Finite Element Based Approach • (5) Efficient Add One Point => Add O(1) Tetrahedrons O(n2) Times More Efficient Than Grid-Based Approaches.

  33. 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

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

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

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

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

  38. 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, ...)

  39. 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.

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