Functional Approximation

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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
• 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)

Basis

Functions

Residual

Data

emax>et

Encoding System
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
• 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