Yingcai Xiao

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

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


input sample points

discontinuity surfacediscontinuity points


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


(2) Complicate Geometry: Non-Gridable Volumes


(3) Discontinuity: Internal Discontinuity Surface


(3) Discontinuity: Discontinuous Regions


(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


(5) Efficient

Add One Point => Add O(1) Tetrahedrons

O(n2) Times More Efficient Than Grid-Based Approaches.


(6) No Whittaker-Shannon Sampling Rate

Interpolation Problem ==> Boundary Value Problem

(7) No Ambiguity in Selecting Modeling Methods


(8) Honoring Original Sample Data


(9) Flexible, Fast and Interactive

Modification of an Existing Sample Point


(9) Flexible, Fast and Interactive

Addition of a New Sample Point


(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.

Yingcai Xiao

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