1. 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. Frequency domain standard analysis tool
Intuition Analysis Application
Motivation
Frequency domain standard analysis tool
Assumption of band-limitedness
we know how to sample in the spatial domain
Given by Nyquist frequency f

4. Sampling in Frequency domain
Intuition Analysis Application
Sampling in Frequency domain  F
f(x) f  F
Wie komme ich mit den wenigsten samplen aus??  F x f

5. Convolution Theorem Spatial Domain: Frequency Domain:  F Convolution:
Intuition Analysis Application
Convolution Theorem
Spatial Domain:
Frequency Domain:  F
Convolution:
Multiplication:

6. Combining 2 different signals
Intuition Analysis Application
Combining 2 different signals
Convolution / Multiplication:
E.g. filtering in the spatial domain => multiplication in the frequency domain
Compositing: What about

7. Opacity f Transfer Function g
Map data value f to optical properties, such as opacity and colour
Then shading+compositing
g(f(x))
Opacity f g

8. Estimates for band-limit of h(x)
Intuition Analysis Application
Estimates for band-limit of h(x)
Considering
M. Kraus et al.
Can be a gross over-estimation
Our solution

9. Example of g(f(x)) Original function f(x) g(f(x)) sampled by
Intuition Analysis Application
Example of g(f(x))
Original function f(x)
g(f(x)) sampled by
Transfer function g(y)
g(f(x)) sampled by
The actual sampling rate is actual four times as much - that’s good for doing linear interpolation …

10. Analysis of Composition in Frequency Domain

11. Composition in Frequency Domain
Intuition Analysis Application
Composition in Frequency Domain y y

12. Composition as Integral Kernel
Intuition Analysis Application
Composition as Integral Kernel

13. Intuition Analysis Application
Visualizing P(k,l)

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

15. Contribution insignificant for rapidly changing
Intuition Analysis Application
Analysis of P(k,l)
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:

16. Second order Taylor expansion Exponential drop-off
Intuition Analysis Application
Exponential decay
Second order Taylor expansion
Exponential drop-off

17. Adaptive Sampling for Raycasting
Application
Adaptive Sampling for Raycasting

18. Compute the gradient-magnitude volume
Intuition Analysis Application
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

19. Adaptive sampling - 25% less samples
Intuition Analysis Application
Adaptive Raycasting
Adaptive sampling - 25% less samples
Uniform sampling
Fig. 5. Examples of hipiph data set sampled at a fixed rate (0.5) (a) and sampled with adaptive stepping (b). The adaptive method in (b) uses about 25% fewer samples than (a) only measuring in areas of non-zero opacity to not account for effects of empty-space skipping. The similarity of both images indicates that visual quality is preserved in the adaptive, reduced sampling.

20. Adaptive Raycasting Same number of samples
Intuition Analysis Application
Adaptive Raycasting
Same number of samples

21. Adaptive Raycasting SNR
Intuition Analysis Application
Adaptive Raycasting SNR
Ground-truth: computed at a fixed sampling distance of
unit grid point spacing of 1. The result of the evaluation is shown in Figure 6. The error plot is based on the signal to noise ratio (SNR), computed as Fig. 6. Quality vs. performance, where quality is measured using signal to noise ratio (SNR) and performance is indicated by the number of samples taken along all rays cast into the volume. Adaptive sampling clearly outperforms the uniform (fixed) sampling. Only samples in areas of non-zero opacity are taken into account, i.e., both sampling schemes equally make use of empty space skipping.

22. Pre-integration approach
Intuition Analysis Application
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

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. Proper sampling of combined signal g(f(x)):
Intuition Analysis Applications
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

25. Acknowledgements
NSERC Canada
BC Advanced Systems Institute
Canadian Foundation of Innovation

26. Thanks…
… for your attention!
Any Questions?

27. Transfer Functions (TFs)
Intuition Analysis Application
Transfer Functions (TFs)
a(g)
RGB(g) a
Simple (usual) case: Map data value g to color and opacity g
Shading,
Compositing…
Human Tooth CT
For the most part, TFs map from the data values to color and opacity:RGBa.
So in this TF, I have three ranges of values which receive opacity, according to these tents,
And within those ranges I’m assigning constant colors: red, cyan, yellow.
Here’s a sample dataset.
When we map the values through the transfer function, we get
Colors and opacities, since, the data values are the DOMAIN of the transfer function, and the
Colors and opacities are the range.
But the important thing is that color and opacity are something tangible, as far as graphics operations,
So then we can apply shading and compositing algorithms to produce a final rendering.
This is a volume rendering of this dataset, using this transfer function.

28. Motivation - Volume Rendering
Intuition Analysis Application
Motivation - Volume Rendering
Convolution used all the time: interpolation
ray-casting
multi-resolution pyramids
gradient estimation
Compositing used all the time: transfer functions
Given
Needed

29. Assume a linear function f(x) = ax
Intuition Analysis Application
Analysis of P(k,l)
Assume a linear function f(x) = ax
If phase is
zero - integral infinite
Non-zero - integral is zero

30. Intuition Analysis Application
Analysis of P(k,l)

31. Proper sampling of g(f(x))
Intuition Analysis Application
Proper sampling of g(f(x))
Our solution: