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

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

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,
, ) = n  i = 1 b d i i
2 log d i c c x + c y + c z
1 2 3 4 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

Global Exact Interpolation Functions
(Foley & Lane, 1990; Nielson, 1993)
Thin-plate spline
Volume Spline
, ) = n  i = 1 b d i i
2 log d i c c x + c y + c z
1 2 3 4 f ( f x y z
3
) = x , y , z ) = i n 
= 1
, b i d i
+ c
1
+ c n  i 2 x
= 1 c
3 b d y

3 c
4 i z , c
1 c x + c y + c z
Multiquadric
Shepard

Thin-plate Spline , ) = n  i = 1 i
2 log ( d i c
1 c x + c y + c z

Volume Spline
, ) = i n 
= 1 i
3 c
1 c x + +

Shepard method f ( , ) =



 i n

= 1 d i

1 v i







 i n

= 1 d i

1




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

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

Point Constraint v extrapolated values sample points d v sample points constraining points d

Value Constraint v
 v min,
if ( , ) < v min,
( , ), v max,
if ( , ) > v max.

Local Constraint p
7 p
2 p
3 p
4 p
1 p
5 p
6 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
 preserve input data values;
 produce meaningful output values;
 provide error estimations;
 accept additional constraints;
 reduce the requirement on the sampling intensity.

A FINITE ELEMENT BASED APPROACH

(1) Tessellation
(2) Computation
(3) Rendering

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

 Three-Dimensional Triangulation: Tetrahedronization
 Delaunay Triangulation: Sphere Criterion input sample points refinement points
Data Points discontinuity points discontinuity surface input sample points discontinuity points refinement points
Triangulation discontinuity surface triangulated network
Element Network

input sample points discontinuity points refinement points discontinuity surface triangulated network
Physical Discontinuity input sample points discontinuity points refinement points double layers triangulated network
Logical Discontinuity

(1) Problem Definition:
 Governing equation:
Boundary Value Problem
L
  f
  p on S
 Boundary Condition:
(2) Element Definition:
 Shape: Tetrahedron
 Order: Basis Function
 e 
4 j


1 e
N x y z
 e j

(3) System Formulation
 Ritz Method
 Galerkin's method
 
(4) Sparse Sample Data
(5) System Solution
 Gaussian Elimination
 Householder's Method
F (
 
)


1 L
    
1
 
, r

L

2
 f
2 f

1
2 f ,
 
{

} = {
 i , i=1,2,...,n }T
 k
 p k

(1) Hexahedron => Tetrahedron
(2) (ijk) Indexing => Neighbor-to-Neighbor Traversal

Advantages of the Finite Element Based Approach
(1) Meaningful Results
Z
Ground Surface

2000


1000
0
1000
1000

Y

X
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
E e

1
2
|

''
| h
2 h

2 E lim /|

'' |
4

2
3
1
0
0 500 1000
Z
1500 2000 h
Error
1.0
1.0
0.5
0.25
0.25
0.0625

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
 e  j
4 

1
N e j
( , , )
 i e e
N x y z
  ij
 
1
0 i
 j i
 j

(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, ...)

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.