Constrained Shepard Method for Data Modeling and Visualization

1. Constrained Shepard Method for Modeling and Visualization of Scattered Databy G. Mustafa, A. A. Shah and M. R. Asim WSCG 2008

2. OUTLINE • INTRODUCTION • RELATED WORK • THE CONSTRAINED SHEPARD METHOD • IMPLEMETATION RESULTS • CONCLUSIONS & FUTURE DIRECTION • QUESTIONS & ANSWERS

3. INTRODUCTION • What is Visualization? • Why Visualization? • Visualization Process Image Geometric Model Rendering Interpolation Empirical Model Visualization Mapping DATA

4. DATA Interpolation Empirical Model Introduction (continued)Empirical Modeling/Reconstruction SCATTERED DATA METODS • MESH BASED Triangulation/Tetra Based Natural Neighborhood based • MESHLESS Radial Basis Function Shepard Family Large, multidimensional data sets

5. Introduction (continued)MODIFIED QUADRATIC SHEPARD METHOD (MQS)

6. Introduction (continued)Weight Functions

7. (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

8. RELATED WORK • Previous Work [1, 2, 3, 4] • Problem with the previous methods Efficiency Accuracy Continuity Scalar invariance

9. (RELATED WORK continued)Minima Free Algorithms • Negative Value to Zero (Xiao & Woodbury[7]) • Basis Function Truncation • Dynamic Scaling Algorithm

10. 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 function Blending Algorithm (Brodlie, Asim & Unsworth[3]) • Fixed Point Scaling • Dynamic Scaling

11. (Related work continued)Scaling Solutions(Fixed Point Scaling)

12. Current Work (continued)Scaling Factor varies between 0 and 1

13. RELATED WORK (continued )Execution Time N=30 25x25 grids

14. RELATED Work (continued)Minima of Quadratic Basis Function

15. Previous Work (continue)The ProblemMinima Searching • Computationally Intensive • Difficult to implement • Convergence Problem

16. Current WorkThe Constrained Shepard Method

17. Current Work (continued) The K Value

18. Current Work (continued) The K Value

19. Current Work (continued)Approximation for constraints Functions • Maximum and minimum in the whole domain • Use nearest from the Maxima and minima in the whole domain

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

21. IMPLEMENTATION & RESULTSExample :Lancaster Function Plot

22. IMPLEMENTATION & RESULTS (continued)Performance Measures (Accuracy)Root Mean Square (RMS) and Absolute Maximum (AM) Deviations)

23. IMPLEMENTATION & RESULTS (continued)Performance Measures (Accuracy)RMS and AM Jackknifing Errors

24. IMPLEMENTATION & RESULTS (continued)Performance Measures (Accuracy)

25. IMPLEMENTATION & RESULTS (continued)Performance Measures (Accuracy)

26. Sample Size (VS) Preprocessing Time

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

28. Grids (VS) Execution Time

29. CONCLUSION & FUTURE WORK • Achievement • Efficient Solution • Accurate • Easy to implement for n-D data • C1 Continuity • Scalar invariant • Drawbacks • No more quadratic precision

30. 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.

31. Q & A session Thanks for Patience

32. (RELATED WORK continued)Basis Function Truncation

34. θ = -4Q+1 Grad F(Xi) = Grad Qi(Xi) Ri(X) = (1.0 − θ)Q(X) + θ RELATED WORK (continued ) Blending Algorithm (Most recent work) (Brodlie, Asim & Unsworth[3])