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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:

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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

Encoding System

Input

(x, y)

Find

Centers

Calculate

Widths

Compute

Weights

Compute

Errors

Output

(μ, σ, λ)

Add

Basis

Functions

Residual

Data

emax>et

### 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

### General Gaussians

• Basic expression using Mahalanobis distance

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

### 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

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

### Time Series Results

57th

58th

Number of basis function

Comparison

Encoding time

Comparison

### Time Series Results

Number of basis function

Comparison

Encoding time

Comparison

### 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