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Functional Approximation

Functional Approximation. Yun Jang Swiss National Supercomputing Centre Data Management, Analysis and Visualization. Overview. Introduction Functional approximation system Generalized basis functions Time series encoding Conclusion. Motivation. Goal:

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Functional Approximation

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  1. Functional Approximation Yun Jang Swiss National Supercomputing Centre Data Management, Analysis and Visualization

  2. Overview • Introduction • Functional approximation system • Generalized basis functions • Time series encoding • Conclusion

  3. Motivation • Goal: • Interactive visualization, exploration, and analysis of datasets on desktop PCs • Challenge: volume rendering and exploration • Large scattered or unstructured volume datasets

  4. Approach • Functional approximation • Unified representation for arbitrary volumetric data • Eliminate dependence on computational grids • Reduce data storage by approximation • Basis functions • Spherical shape basis functions • Radial basis functions (RBFs) • Non-spherical shape basis functions • Ellipsoidal basis functions (EBFs)

  5. Functional Approximation

  6. Problem Statement • Find a function that provides a good approximation • Input data, • : Spatial locations • : Data values • Weighted sum of M basis functions (Gaussians) • Accuracy vs. number of basis functions

  7. Encoding System Input (x, y) Find Centers Calculate Widths Compute Weights Compute Errors Output (μ, σ, λ) Add Basis Functions Residual Data emax>et Encoding System

  8. Spherical vs. Ellipsoidal Functions • Spherical basis functions (RBFs) • Quick approximation and evaluation • Appropriate for blobby shape volume • Ellipsoidal basis functions (EBFs) • More computation • More texture lookups • Smaller number of basis functions • Appropriate for any volume Spherical basis Functions 59 RBFs Ellipsoidal basis Functions 13 EBFs

  9. General Gaussians • Basic expression using Mahalanobis distance

  10. ry ry r rx rx y x Comparison of Basis Functions • Approximation of grey data • White lines: basis functions • Blue lines: Influence ranges • Red lines: Axis of basis function Spherical Gaussian Axis aligned ellipsoidal Gaussian Arbitrary directional ellipsoidal Gaussian

  11. Cost Functions & Errors • Using L2-norm based error • Data values only • Using H1-norm based error • Data values & gradients • Error criteria • Maximum error: 5% of data value

  12. Functional Approximation Results

  13. 4 4 4 3 3 2 2 Spatial Data Structure • Speed up the rendering • Use influence of basis function • Example, Max number of basis functions per cell = 4

  14. Results • Rendering performance • Measured on • Intel Bi-Xeon 5150, 2.66GHz • NVDIA 8800 GTS graphics board • Setting • 130 slices for volume rendering • One slice for texture advection visualization • 400x400 viewport

  15. Basis Function Comparison Convection 70th 237 RBFs 10 fps 101 EBFs 16 fps 90 EBFs 9 fps 150th 266 RBFs 16 fps 199 EBFs 21 fps 162 EBFs 13 fps Axis aligned ellipsoidal Gaussian L2-norm Arbitrary directional ellipsoidal Gaussian L2-norm Spherical Gaussian L2-norm

  16. Basis Function Comparison X38 Density 554 EBFs 16 fps 3,343 EBFs 8 fps 3,084 RBFs 7 fps Axis aligned ellipsoidal Gaussian Arbitrary directional ellipsoidal Gaussian Spherical Gaussian

  17. Basis Function & Error Comparison Marschner-Lobb L2-norm 2,092 RBFs 4 fps 208 EBFs 21 fps 112 EBFs 13 fps H1-norm 1,009 RBFs 7 fps 148 EBFs 24 fps 78 EBFs 13 fps Axis aligned ellipsoidal Gaussian Arbitrary directional ellipsoidal Gaussian Spherical Gaussian

  18. Basis Function & Error Comparison Bluntfin L2-norm 891 RBFs 21 fps 264 EBFs 32 fps 282 EBFs 8 fps H1-norm 256 RBFs 31 fps 121 EBFs 32 fps 148 EBFs 13 fps Arbitrary directional ellipsoidal Gaussian Axis aligned ellipsoidal Gaussian Spherical Gaussian

  19. Time Series Data • Using temporal coherence • Coefficient of variation • Error from previous encoding result

  20. Time Series Rendering System

  21. Time Series Results 57th 58th Number of basis function Comparison Encoding time Comparison

  22. Time Series Results Number of basis function Comparison Encoding time Comparison

  23. Conclusion • Effective procedural encoding of scalar and multi-field data • Novel approach for interactive reconstruction, visualization, and exploration of arbitrary 3D fields • Encoding based on • Rendering using graphics boards • Both statistical and visual accuracy

  24. Future Work • Investigate various basis functions and cost functions • Reduce computation of nonlinear optimization • Data specific basis function • Feature comparisons between input data and encoded data • Time series encoding with moving grid datasets

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