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### 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:
- Interactive visualization, exploration, and analysis of datasets on desktop PCs
- Challenge: volume rendering and exploration
- Large scattered or unstructured volume datasets

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)

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

Input

(x, y)

Find

Centers

Calculate

Widths

Compute

Weights

Compute

Errors

Output

(μ, σ, λ)

Add

Basis

Functions

Residual

Data

emax>et

Encoding SystemSpherical 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

General Gaussians

- Basic expression using Mahalanobis distance

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

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

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

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

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

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

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

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

Time Series Data

- Using temporal coherence
- Coefficient of variation
- Error from previous encoding result

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

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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