SCATTERED DATA VISUALIZATION Yingcai Xiao Scattered Data sample

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 Intermediate Grid Sparse Data Rendered Interpolation Grid-Based Volume Modeling

Thin-plate Spline

Volume Spline

Shepard method

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

Interpolation Methods Example: 1 D Global and Exact

Interpolation Methods Example: 1 D Global and Exact

Defining a Global Exact Interpolant (Foley & Lane, 1990; Nielson, 1993) n sample points: (xi, yi, zi, vi) for i = 1, 2, . . n One 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)1/2 bi, c 1, c 2, c 3, c 4 are n+4 constants to be solved by enforcing the following conditions: f (xi, yi, zi) = vi for i = 1, 2, . . n

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

Thin-plate Spline

Volume Spline

Shepard method

Deficiencies of the Interpolation-based Two-step Approach (Xiao et. Al. , 1996) 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

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

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

Point Constraint

Value Constraint

p 6 p 1 Local Constraint p 2 p 7 p 3 p 4 p 5 p 8

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.

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.

A FINITE ELEMENT BASED APPROACH XIAO & ZIEBARTH, 2000

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

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

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

The Double Layer Technique Physical Discontinuity Logical Discontinuity

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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.