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
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: 1D Global and Exact
Interpolation Methods Example: 1D Global and Exact
2
f (x) =å bi x
i=1
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,
n
f ( x, y, z ) =  bi d log( di ) + c1 + c2 x + c3 y + c4 z
i=1
2
i
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,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
Global Exact Interpolation Functions
(Foley & Lane, 1990; Nielson, 1993)
Thin-plate spline
n
f ( x, y, z ) =  bi d log( di ) + c1 + c2 x + c3 y + c4 z
2
i
i=1
n
Volume Spline
f ( x, y, z ) =  bi d 3 + c1 + c2 x + c3 y + c4 z
n
f ( x, y, z ) =
b d
i
i =1
Multiquadric
Shepard
3
i
+ c1 + c2 x + c3 y  c4iz,
i =1
n
Thin-plate Spline
f ( x, y, z ) =  bi d log( di ) + c1 + c2 x + c3 y + c4 z
i =1
2
i
Volume Spline
n
f ( x, y, z ) =
b d
i
i =1
3
i
+ c1 + c2 x + c3 y + c4 z
Shepard method
f ( x, y, z ) =
 n



1
  d i vi 


i =1


 n


1 
  di 


i = 1

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
lPoint Constraint
l Value Constraint
l Local Constraint
Point Constraint
v
sample points
constraining points
d
v
extrapolated values
sample points
d
Value Constraint

v

v min, if f ( x , y , z ) < vmin,
f ( x , y , z ),
v max, if f ( x , y , z ) > vmax.
p6
p1
Local Constraint
p2
p7
p3
p4
p5
p8
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
lThree-Dimensional Triangulation: Tetrahedronization
lDelaunay Triangulation: Sphere Criterion
in put samp le p oint s
input sample point s
discont inuit y point s
refinem ent point s
discont inuit y surface
Data Points
disco nt inuit y point s
disco nt inuit y surface
refin em ent po int s
t rian gulat ed n et work
Triangulation
Element Network
The Double Layer Technique
in put samp le p oint s
in put samp le p oint s
disco nt inuit y point s
disco nt inuit y surface
refin em ent po int s
t rian gulat ed n et work
Physical Discontinuity
disco nt inuit y point s
do uble layers
refin em ent po int s
t rian gulat ed n et work
Logical Discontinuity
The Finite Element Method
(1) Problem Definition:
l
Boundary Value Problem
L  f
Governing equation:
  p on S
l
Boundary Condition:
(2) Element Definition:
l
Shape: Tetrahedron
4
l
Order: Basis Function
 ( x, y, z)   N ej ( x, y, z) ej
e
j 1
The Finite Element Method
(3) System Formulation
l
l
Ritz Method
Galerkin's method
[K]{f } = {C}.
(4) Sparse Sample Data
(5) System Solution
l
l
Gaussian Elimination
Householder's Method
F ( )  21 L ,   21  , f  21 f , 
r  L  f
{} = {i, i=1,2,...,n}T
 k  p( k )
Rendering : Modifying Conventional Methods
(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
  
1000
0
Y
  
1000
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  21 | '' | h 2
h  2 E lim /| '' |
4
3

2
1
0
0
500
1000
1500
2000
Z
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(n2) 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
4
 ( x , y , z )   N ej ( x , y , z ) ie
e
j 1
1
N ej ( x j , y j , z j )   ij  
0
i j
i j
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, ...)
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.