Spectral Analysis of Function Composition and Its Implications for Sampling in Direct Volume Visualization

Steven Bergner GrUVi-Lab/SFU

Torsten Möller

Daniel Weiskopf

David J Muraki Dept. of Mathematics/SFU

2

Overview

Frequency domain intuition Function Composition in Frequency Domain Application to Adaptive Sampling Future Directions

3

Motivation

Frequency domain standard analysis tool

Assumption of band-limitedness• we know how to sample in the spatial domain

Given by Nyquist frequency f

Intuition Analysis Application

R

ix dxexfFxf

)(

2

1)()(

4

Sampling in Frequency domain

x

f(x)

F

F

F

f

f

Intuition Analysis Application

5

Spatial Domain:

f t

g x t dt

Frequency Domain:

F G Multiplication:Convolution:

Convolution Theorem

Intuition Analysis Application

F

6

Combining 2 different signals

Convolution / Multiplication:• E.g. filtering in the spatial domain

=> multiplication in the frequency domain

Compositing: What about

Intuition Analysis Application

GFgf

?))(( xfgfg

7

Transfer Function g

Map data value f to optical properties, such as opacity and colour

Then shading+compositing

f

Opacity

g(f(x))

g

8

Considering

M. Kraus et al.

• Can be a gross over-estimation

Our solution

Intuition Analysis Application

Estimates for band-limit of h(x)

)())(()( kHxfgxh

gfh 2

gh f |'|max

9

Example of g(f(x))

Original function f(x)

Transfer function g(y)

g(f(x)) sampled by

g(f(x)) sampled by

Intuition Analysis Application

gf2

gf |'|max

Analysis of Composition in Frequency Domain

11

Composition in Frequency Domain

R

xfil dlelGxfgxh )()(2

1))(()(

Intuition Analysis Application

yy

dxedlelGkH xik

R R

xfil )()(2

1)(

dldxeelGkH

R R

xikxfil )()(2

1)(

R

xkxfli dxelkP ))((),(

12

Composition as Integral Kernel

Intuition Analysis Application

R

xkxfli dxelkP ))((),(

dllkPlGkHR ),()(

2

1)(

),(),(

2

1)( kPGkH

13

Visualizing P(k,l)

Intuition Analysis Application

R

xkxfli dxelkP ))((),( ),(),(2

1)( kPGkH

14

Visualizing P(k,l)

Intuition Analysis Application

Slopes of lines in P(k,l) are related to 1/f‘(x) Extremal slopes bounding the wedge are 1/max(f’)

15

For general• Contribution insignificant for rapidly

changing• Contributions large when

These points are called points of stationary phase:

The largest such k is of interest:

Analysis of P(k,l)

Intuition Analysis Application

R

xkxfli dxelkP ))((),(

kfl |'|max

16

Exponential decay

Intuition Analysis Application

R

xkxfli dxelkP ))((),(kfl |'|max

Second order Taylor expansion

Exponential drop-off

Application

Adaptive Sampling for Raycasting

18

Adaptive Raycasting

Compute the gradient-magnitude volume For each point along a ray - determine max|f’| in a

local neighborhood Use this to determine stepping distance

Intuition Analysis Application

19

Adaptive Raycasting

Uniform sampling Adaptive sampling -25% less samples

Intuition Analysis Application

20

Adaptive Raycasting

Same number of samples

Intuition Analysis Application

21

Adaptive Raycasting SNR

Ground-truth:computed at a fixedsampling distanceof 0.06125

Intuition Analysis Application

22

Pre-integration approach

Standard fix for high-quality rendering• Assumes linearity of f between sample points

Fails for• High-dynamic range data• Multi-dimensional transfer function• Shading approximation between samples

A return to direct computation of integrals is possible

Intuition Analysis Application

23

Future directions

Exploit statistical measures of the data contained in P(k,l)

Combined space-frequency analysis Other interpretations of g(f(x)) • change in parametrization of g • activation function in artificial neural networks

Fourier Volume Rendering

24

Summary

Proper sampling of combined signal g(f(x)):

Solved a fundamental problem of rendering Applicable to other areas Use the ideas for better algorithms

Intuition Analysis Applications

25

Acknowledgements

NSERC Canada BC Advanced Systems Institute Canadian Foundation of Innovation

26

Thanks…

… for your attention!

Any Questions?