yingcai xiao
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Yingcai Xiao
SCATTERED DATA VISUALIZATION
Scattered Data: sample points distributed unevenly and non-uniformly Scattered Data: sample points distributed unevenly and non-uniformly throughout the volume of interest. throughout the volume of interest.
Example Data: chemical leakage at a tank-farm.Example Data: chemical leakage at a tank-farm.
Method of Approach : Interpolation-based Two-step Approach (Foley & Lane, 1990)
Sparse DataInterpolation
ModelingIntermediate Grid
Rendering
Grid-BasedRendered 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. 1996Xiao etc. 1996
Interpolation Methods Example: 1D Global and Exact
Interpolation Methods Example: 1D Global and Exact
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,
f x y z b d d c c x c y c zi i ii
n
( , ) = ( ) + + + + =
, log2
1 2 31
4di 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
, + + + = ),,(1=
4321
3 n
iii zcycxccdbzyxf
f x y z b d d c c x c y c zi i ii
n
( , ) = ( ) + + + + =
, log2
1 2 31
4
Volume Spline f x y z b d c c x c y c zi ii
n
( , ) = + + + + =
, 31 2 3
14
Shepard
Multiquadric
Thin-plate Spline f x y z b d d c c x c y c zi i ii
n
( , ) = ( ) + + + + =
, log2
1 2 31
4
Volume Spline f x y z b d c c x c y c zi ii
n
( , ) = + + + + =
, 31 2 3
14
Shepard method f x y z
n
i
d v
n
i
di i i( , ) =
=
=
,
1 1
1 1
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
Deficiencies of the Interpolation-based Two-step Approach (Xiao et. Al., 1996)
Zero-value dilemma
Negative-value dilemma
Correctness dilemma
Three Dilemmas and Three Constraints (Xiao & Woodbury, 1999)
Point Constraint
Value Constraint
Local Constraint
Point Constraint
d
v
sample points
extrapolated values
d
v
sample pointsconstraining points
Value Constraint v
v fxyz v
fxyz
v fxyz v
min,
max,
,
,
,
if (, ) <
(, ),
if (, ) >
min,
max.
Local Constraint
p6
p1
p2
p8
p7
p4
p3
p5
ConclusionsConclusions
• 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.
ConclusionsConclusions
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
XIAO & ZIEBARTH, 2000
The Finite Element Based Approach The Finite Element Based Approach
(1) Tessellation
(2) Computation
(3) Rendering
The Finite Element Based ApproachThe Finite Element Based Approach
Sparse Data VolumeTriangulation
TessellationElement Network
Computation
FEM Element-Based
Node Values Rendered VolumeRendering
TessellationTessellation
Three-Dimensional Triangulation: Tetrahedronization
Delaunay Triangulation: Sphere Criterion
discontinuity surface
discontinuity points
refinement points
input sample points discontinuity surfacediscontinuity points
refinement points triangulated network
input sample points
Data PointsTriangulation
Element Network
The Double Layer TechniqueThe Double Layer Technique
double layersdiscontinuity points
refinement points triangulated network
input sample points
discontinuity surfacediscontinuity points
refinement points triangulated network
input sample points
Physical Discontinuity Logical Discontinuity
The Finite Element MethodThe Finite Element Method
(1) Problem Definition:
Boundary Value Problem
Governing equation:
Boundary Condition:
(2) Element Definition:
Shape: Tetrahedron
Order: Basis Function
L f
p S on
eje
je
j
x y z N x y z( , , ) ( , , )
1
4
The Finite Element MethodThe Finite Element Method
(3) System Formulation
Ritz Method Galerkin's method
(4) Sparse Sample Data
(5) System Solution
Gaussian Elimination Householder's Method
F L f f( ) , , , 12
12
12
r L f
{} = {i, i=1,2,...,n}T
k p k ( )
Rendering : Modifying Conventional MethodsRendering : Modifying Conventional Methods
(1) Hexahedron => Tetrahedron
(2) (ijk) Indexing => Neighbor-to-Neighbor Traversal
Advantages of the Finite Element Based ApproachAdvantages of the Finite Element Based Approach
(1) Meaningful Results
Y
X
Z
1000
2000
0
1000
1000
Ground Surface
A Pollution Problem Exact Grid-based FEM-based
Advantages of the Finite Element Based ApproachAdvantages of the Finite Element Based Approach
(2) Complicate Geometry: Non-Gridable Volumes
Advantages of the Finite Element Based ApproachAdvantages of the Finite Element Based Approach
(3) Discontinuity: Internal Discontinuity Surface
Advantages of the Finite Element Based ApproachAdvantages of the Finite Element Based Approach
(3) Discontinuity: Discontinuous Regions
Advantages of the Finite Element Based ApproachAdvantages of the Finite Element Based Approach
(4) Error Estimation and Iterative Refinement
E he 12
2| |'' h E 2 lim ''/| |
h 1.0 0.5 0.25Error 1.0 0.25 0.0625
Z
0
1
2
3
4
0 500 1000 1500 2000
Advantages of the Finite Element Based ApproachAdvantages 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 ApproachAdvantages 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 ApproachAdvantages of the Finite Element Based Approach
(8) Honoring Original Sample Data
Advantages of the Finite Element Based ApproachAdvantages of the Finite Element Based Approach
(9) Flexible, Fast and Interactive
Modification of an Existing Sample Point
Advantages of the Finite Element Based ApproachAdvantages of the Finite Element Based Approach
(9) Flexible, Fast and Interactive
Addition of a New Sample Point
Advantages of the Finite Element Based ApproachAdvantages of the Finite Element Based Approach
(10) Consistent Basis Function
e
jje
iex y z N x y z( , , ) ( , , )
1
4
N x y zi j
i jje
j j j ij( , , )
1
0
Future WorkFuture 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, ...)
SummarySummary
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.