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

OUTLINE 1.INTRODUCTION 2.RELATED WORK 3.THE CONSTRAINED SHEPARD METHOD 4.IMPLEMETATION RESULTS 5.CONCLUSIONS & FUTURE DIRECTION 6.QUESTIONS & ANSWERS

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

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

Introduction (continued) MODIFIED QUADRATIC SHEPARD METHOD (MQS)

Introduction (continued) Weight Functions

(introduction (continued) Loss of Positivity using MQS Method Time (sec) Oxygen (%) Table. Oxygen Levels in Flue Gases From a Boiler

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])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

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

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

RELATED WORK (continued ) Execution Time N=30 25x25 grids MQS Fixed Point Scaling (Positive) Blending Method Execution Time (sec)

RELATED Work (continued) Minima of Quadratic Basis Function

Previous Work (continue) The Problem Minima Searching Computationally Intensive Difficult to implement Convergence Problem

Current Work The Constrained Shepard Method

Current Work (continued) The K Value

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)

IMPLEMENTATION & RESULTS Example : Lancaster Function Plot

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

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

IMPLEMENTATION & RESULTS (continued) Performance Measures (Accuracy)

Sample Size (VS) Preprocessing Time

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

Grids (VS) Execution Time

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, [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 [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 [5]Renika R. J., “Multivariate Interpolation of Large Set of Scattered Data. ACM Transactions on Mathematical Software, 14 (2), 1988, pp [6]Shepard, D., “A two-dimensional interpolation function for irregularly spaced data,” Proceedings of 23rd National Conference, New Yark, ACM, 1968, pp [7]Xiao, Y and C. Woodbury, “Constraining Global Interpolation Methods for Sparse Data Volume Visualization,” International Journal of Computers and Applications, 21(2), 1999, [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 [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 [10] Fuhrmann A. and E. Gröller, “Real-Time Techniques for 3D Flow Visualization,” Proceedings of the IEEE Visualization 98 (VIZ’98), 1998, pp [11] Wagner, T. C., M.O. Manuel, C. T. Silva and J. Wang, “Modeling and Rendering of Real Environments,” RITA, 9(2), 2002, pp [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, [14] W. J. Gordon & J. A. Wixom, “Shepard's method of ‘Metric Interpolation’ to bivariate and multivariate Interpolation,” Mathematics of Computation, 32(141), 1978,

Q & A session Thanks for Patience

(RELATED WORK continued) Basis Function Truncation

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