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

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Functional approximation

Functional Approximation

Yun Jang

Swiss National Supercomputing Centre

Data Management, Analysis and Visualization


Overview

Overview

  • Introduction

  • Functional approximation system

  • Generalized basis functions

  • Time series encoding

  • Conclusion


Motivation

Motivation

  • Goal:

    • Interactive visualization, exploration, and analysis of datasets on desktop PCs

  • Challenge: volume rendering and exploration

    • Large scattered or unstructured volume datasets


Approach

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)


Functional approximation1

Functional Approximation


Problem statement

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

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


Spherical vs ellipsoidal functions

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

General Gaussians

  • Basic expression using Mahalanobis distance


Comparison of basis functions

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

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


Functional approximation results

Functional Approximation Results


Spatial data structure

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

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

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 comparison1

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

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 comparison1

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

Time Series Data

  • Using temporal coherence

    • Coefficient of variation

    • Error from previous encoding result


Time series rendering system

Time Series Rendering System


Time series results

Time Series Results

57th

58th

Number of basis function

Comparison

Encoding time

Comparison


Time series results1

Time Series Results

Number of basis function

Comparison

Encoding time

Comparison


Conclusion

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

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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