Constrained Shepard Method for Modeling and Visualization

of Scattered Databy

G. Mustafa, A. A. Shah and M. R. Asim

WSCG 2008

OUTLINE1. INTRODUCTION2. RELATED WORK3. THE CONSTRAINED

SHEPARD METHOD4. IMPLEMETATION RESULTS5. CONCLUSIONS & FUTURE

DIRECTION6. QUESTIONS & ANSWERS

INTRODUCTION

• What is Visualization?

• Why Visualization?

• Visualization Process

DATAEmpirical

Model

GeometricModelInterpolation

VisualizationMapping

Rendering Image

Introduction (continued)

Empirical Modeling/Reconstruction

DATAEmpirical

ModelInterpolation

SCATTERED DATA METODS• MESH BASED

Triangulation/Tetra Based

Natural Neighborhood based

• MESHLESSRadial Basis Function

Shepard FamilyLarge, multidimensional data sets

Introduction (continued)

MODIFIED QUADRATIC SHEPARD METHOD (MQS)

N

ii

N

ii

Xw

XQXwXF

1

1i

)(

)()()(

Introduction (continued)

Weight Functions

2)( ii XXXd

2

)(

)()(

XRd

XdRXw

i

i

otherwise

XdRifXdRXdR ii

i 0

)()()]([

(introduction (continued)

Loss of Positivity using MQS Method

Time (sec) 0 20 40 100 280 300 320 Oxygen (%) 20.8 8.8 4.2 .5 3.9 6.2 9.6

Table . Oxygen Levels in Flue Gases From a Boiler

0 50 100 150 200 250 300 350-5

0

5

10

15

20

25

Time (sec)

Oxy

gen

Lev

el (

%)

RELATED WORK

• Previous Work [1, 2, 3, 4]

• Problem with the previous methods

Efficiency

Accuracy

Continuity

Scalar invariance

(RELATED WORK continued)

Minima Free Algorithms

• Negative Value to Zero (Xiao &

Woodbury[7])

• Basis Function Truncation • Dynamic Scaling

Algorithm

RELATED WORK (continued )

Minima/Zero Searching Algorithms

• Modified Positive Basis Function (Asim[1])• Scaling & Shifting Algorithm (Asim[1])• Constraining Radius of Participation• Hybrid Algorithms

Piecewise continuous basis functionBlending Algorithm (Brodlie, Asim & Unsworth[3])

• Fixed Point Scaling• Dynamic Scaling

(Related work continued)

Scaling Solutions(Fixed Point Scaling)

)()()( imiT

imimTiimi XXAXXXXgfXQ

iii fXQ )(

Current Work (continued)

Scaling Factor

varies between 0 and 1

)( mii

ii XQf

fK

iK

iK

RELATED WORK (continued )

Execution Time

)( mii

ii XQf

fK

N=30

25x25 grids

MQS Fixed Point

Scaling

(Positive)BlendingMethod

Execution Time (sec) .0170 .0640 .0670

RELATED Work (continued)

Minima of Quadratic Basis Function

1 2 3 4 5 6 7-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

y

0 50 100 150 200 250 300 350-10

0

10

20

30

40

50

Time (sec)

Oxygen (

%)

1 2 3 4 5 6 7-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

y

Previous Work (continue)

The Problem

Minima Searching

• Computationally Intensive• Difficult to implement • Convergence Problem

Current Work

The Constrained Shepard Method

N

ii

N

ii

Xw

XRXwXF

1

1i

)(

)(ˆ)()(ˆ

otherwiseXDXXC

fXQifXDXXCXR

LiLimL

iiUiUimUi

)(ˆ)()(

)(ˆ)(ˆ)()()(ˆ

Current Work (continued)

The K Value

0 100 200 300-0.5

0

0.5

1

1.5

2

2.5

X

Y K=1/10R1

R2

R3

0 100 200 300-0.5

0

0.5

1

1.5

2

2.5

X

Y

K=K0 R1

R2

R3

0 100 200 300-0.5

0

0.5

1

1.5

2

2.5

X

Y

K=1/3 R1

R2

R3

0 100 200 300-0.5

0

0.5

1

1.5

2

2.5

X

Y

K=1/100R1

R2

R3

Current Work (continued)

The K Value

1 2 3 4 5 6 7-0.5

0

0.5

1

X

Y

R1

R2

R3K=K0

1 2 3 4 5 6 7-0.5

0

0.5

1

X

Y

R1

R2

R3K=1/3

1 2 3 4 5 6 7-0.5

0

0.5

1

R1

R2

R3

K=1/100

1 2 3 4 5 6 7-0.5

0

0.5

1

X

Y

R1

R2

R3

K=1/10

• Maximum and minimum in the whole domain

• Use nearest from the Maxima and minima in the whole domain

Current Work (continued)

Approximation for constraints Functions

Example 1: Graph of z=sin2(x)sin2(y)

01

23

0

0.5

1

1.50

0.5

1

xy

z

IMPLEMENTATION & RESULTS Example : Lancaster Function Plot

01

2

0

0.5

1

0

0.5

1

XY

Z

0

1

2

0

0.5

1

0

0.5

1

1.5

XY

Z

01

2

0

0.5

1

0

0.5

1

XY

Z

0100

200

0

50

100

0

0.5

1

XY

Z

IMPLEMENTATION & RESULTS (continued)

Performance Measures (Accuracy)Root Mean Square (RMS) and Absolute Maximum (AM) Deviations)

Deviations

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Data

Set 1

Data

Set 2

Data

Set 3

Data

Set 4

A. M

. D

evia

tions

MQS

Empirical

Brodlie

Deviations

0

0.05

0.1

0.15

0.2

0.25D

ata

Set

1

Data

Set

2

Data

Set

3

Data

Set

4

R.M

.S.

Devia

tions MQS

Empirical

Brodlie

IMPLEMENTATION & RESULTS (continued)

Performance Measures (Accuracy)RMS and AM Jackknifing Errors

Jackknifing Errors

0

0.05

0.1

0.15

0.2

0.25

0.3

Data

Set

1

Data

Set

2

Data

Set

3

Data

Set

4

R.M

.S. err

ors

MQSEmpiricalBrodlie

Jackknifing Errors

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Data

Set

1

Data

Set

2

Data

Set

3

Data

Set

4

A.M

. err

ors

MQSEmpiricalBrodlie

IMPLEMENTATION & RESULTS (continued)

Performance Measures (Accuracy)

Deviations (vs) Sample Size

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 100 200 300 400 500Sample Size(N)

A. M

. Dev

iatio

ns

MQS

Empirical

Brodlie

Deviations (vs) Sample Size

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 100 200 300 400 500Sample Size(N)

R.M

.S. D

evia

tions

MQS

Brodlie

Empirical

IMPLEMENTATION & RESULTS (continued)

Performance Measures (Accuracy)

Jackknifing Errors (vs) Sample Size

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 100 200 300 400 500Sample Size (N)

R.M

.S. E

rrors

MQS

Brodlie

Empirical

Jackknifing Errors (vs) Sample Size

0

0.02

0.04

0.06

0.08

0.1

0.12

0 100 200 300 400 500Sample Size(N)

A. M

. Er

rors

MQS

Brodlie

Empirical

Sample Size (VS) Preprocessing Time

Preprocessing time (vs) sample size

0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 400 500Sample size (N)

Prep

roce

ssin

g tim

e (s

ec)

MQS

Brodlie

Empirical

Components of Execution Time(N=30 and 25x25grids)

Time Division

0

0.05

0.1

0.15

0.2

0.25

MQ

S

Empi

rical

Brod

lie

Tota

l Tim

e (s

ec)

Ex ecutionSetupPositiv ity

Grids (VS) Execution Time

Execution time (vs) Grids

0

2

4

6

8

10

12

14

0 50 100 150 200Grid Dimensions

Exe

cu

tio

n T

ime (

sec

)

MQS

Brodlie

Empirical

CONCLUSION & FUTURE WORK

• Achievement• Efficient Solution

• Accurate

• Easy to implement for n-D data

• C1 Continuity

• Scalar invariant

• Drawbacks– No more quadratic precision

References

• [1] Asim M. R., “Visualization of Data Subject to Positivity Constraint,” Doctoral thesis, School of Computer Studies, University of Leeds, Leeds, England, 2000.

• [2] Asim M. R, G. Mustafa and K.W. Brodlie, “Constrained Visualization of 2D Positive Data using Modified Quadratic Shepard Method” Proceedings of The 12th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision, Czeck Republic, 2004, pp 9-13.

• [3] Brodlie, K. W., M.R. Asim, K. Unsworth, “Constrained Visualization Using the Shepard Interpolation Family,” Computer Graphics Forum, 24(4), Blackwell Synergy, 2005, pp. 809–820.

• [4] Franke, R. and G. Neilson, “Smooth Interpolation of Large set of Scattered Data,” International Journal of Numerical Methods in Engineering, 15, 1980, pp 1691-1704.

•[5] Renika R. J., “Multivariate Interpolation of Large Set of Scattered Data. ACM Transactions on Mathematical Software, 14 (2), 1988, pp 139-148.

• [6] Shepard, D., “A two-dimensional interpolation function for irregularly spaced data,” Proceedings of 23rd National Conference, New Yark, ACM, 1968, pp 517-523.

• [7] Xiao, Y and C. Woodbury, “Constraining Global Interpolation Methods for Sparse Data Volume Visualization,” International Journal of Computers and Applications, 21(2), 1999, 56-64.

• [8] Xiao, Y., J.P Ziebarth, B. Rundell, and J. Zijp, “The Challenges of Visualizing and Modeling Environmental Data,” Proceedings of the Seventh IEEE Visualization (VIS'96), San Francisco, California, 1996, pp 413-416.

• [9] William F. G., F. Henry, C. W. Mary and S. Andrei, “Real-Time Incremental Visualization of Dynamic Ultrasound Volumes Using Parallel BSP Trees,” Proceedings of the 7th IEEE Visualization Conference (VIS’96), San Francisco, California, 1996, page 1070-2385.

• [10] Fuhrmann A. and E. Gröller, “Real-Time Techniques for 3D Flow Visualization,” Proceedings of the IEEE Visualization 98 (VIZ’98), 1998, pp 0-8186-9176.

• [11] Wagner, T. C., M.O. Manuel, C. T. Silva and J. Wang, “Modeling and Rendering of Real Environments,” RITA, 9(2), 2002, pp 127-156.

• [12] Park S.W., L. Linsen, O. Kreylos, J. D. Owens, B. Hamann, “A Framework for Real-time Volume Visualization of Streaming Scattered Data,” 10th International Fall Workshop on Vision, Modeling and Visualization (VMV 2005), 2005, Erlangen, Germany.

• [13] Nagarajan H., “Software for Real Time Systems,” Real Time Systems Group, Centre for Development of Advanced Computing, Bangalore, 2002.

• [14] W. J. Gordon & J. A. Wixom, “Shepard's method of ‘Metric Interpolation’ to bivariate and multivariate Interpolation,” Mathematics of Computation, 32(141), 1978, 253-264.

Q & A session

Thanks for Patience

(RELATED WORK continued)

Basis Function Truncation

0 100 200 300 400-5

0

5

10

15

20

25

Time (sec)

Oxy

gen

Lev

el (

%)

RELATED WORK (continued )

Blending Algorithm (Most recent work) (Brodlie, Asim & Unsworth[3])

• θ = -4Q+1• Grad F(Xi) = Grad Qi(Xi)

0 50 100 150 200 250 300 350-20

0

20

40

60

80

100

Time (sec)

Oxy

gen

Leve

l (%

)Unscaled MQSScaling & ShiftingBlending method

• Ri(X) = (1.0 − θ)Q(X) + θ )(ˆ XQ