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

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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

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Overview

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

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Motivation
Intuition Analysis
Application

• Frequency domain standard analysis tool
• Assumption of band-limitedness
• we know how to sample in the spatial domain
• Given by Nyquist frequency ?f

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Sampling in Frequency domain
Intuition Analysis
Application
f(x)
?f
x
?f
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Convolution Theorem
Intuition Analysis
Application
Frequency Domain
Spatial Domain
Multiplication
Convolution
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Combining 2 different signals
Intuition Analysis
Application

• Convolution / Multiplication
• E.g. filtering in the spatial domaingt
  multiplication in the frequency domain
• Compositing What about

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Transfer Function g

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

g(f(x))
Opacity
f
g
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Estimates for band-limit of h(x)
Intuition Analysis
Application

• Considering
• M. Kraus et al.
• Can be a gross over-estimation
• Our solution

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Example of g(f(x))
Intuition Analysis
Application
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Analysis of Composition in Frequency Domain
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Composition in Frequency Domain
Intuition Analysis
Application
y
y
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Composition as Integral Kernel
Intuition Analysis
Application
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Visualizing P(k,l)
Intuition Analysis
Application
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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)

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Analysis of P(k,l)
Intuition Analysis
Application

• For general
• Contribution insignificant for rapidlychanging
• Contributions large when
• These points are called points of stationary
  phase
• The largest such k is of interest

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Exponential decay
Intuition Analysis
Application

• Second order Taylor expansion
• Exponential drop-off

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Application

• Adaptive Sampling for Raycasting

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Adaptive Raycasting
Intuition Analysis
Application

• Compute the gradient-magnitude volume
• For each point along a ray - determine maxf in
  a local neighborhood
• Use this to determine stepping distance

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Adaptive Raycasting
Intuition Analysis
Application
Adaptive sampling -25 less samples
Uniform sampling
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Adaptive Raycasting
Intuition Analysis
Application
Same number of samples
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Adaptive Raycasting SNR
Intuition Analysis
Application
Ground-truthcomputed at a fixedsampling
distanceof 0.06125
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Pre-integration approach
Intuition Analysis
Application

• 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

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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

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Summary
Intuition Analysis
Applications

• Proper sampling of combined signal g(f(x))
• Solved a fundamental problem of rendering
• Applicable to other areas
• Use the ideas for better algorithms

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Acknowledgements

• NSERC Canada
• BC Advanced Systems Institute
• Canadian Foundation of Innovation

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Thanks

• for your attention!
• Any Questions?

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Transfer Functions (TFs)
Intuition Analysis
Application
a
Simple (usual) case Map datavalue g to color
and opacity
g
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Motivation - Volume Rendering
Intuition Analysis
Application

• Convolution used all the time interpolation
• ray-casting
• multi-resolution pyramids
• gradient estimation
• Compositing used all the timetransfer functions

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Analysis of P(k,l)
Intuition Analysis
Application

• Assume a linear function f(x) ax
• If phase is
• zero - integral infinite
• Non-zero - integral is zero

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Analysis of P(k,l)
Intuition Analysis
Application
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Proper sampling of g(f(x))
Intuition Analysis
Application

• Our solution