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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.

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Presentation on theme: "Spectral Analysis of Function Composition and Its Implications for Sampling in Direct Volume Visualization Steven Bergner GrUVi-Lab/SFU Torsten Möller."— Presentation transcript:

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 MurakiDept. of Mathematics/SFU

2 2 Overview Frequency domain intuition Function Composition in Frequency Domain Application to Adaptive Sampling Future Directions

3 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

4 4 Sampling in Frequency domain x f(x) F F F f f Intuition Analysis Application

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

6 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

7 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 8 Considering M. Kraus et al. Can be a gross over-estimation Our solution Intuition Analysis Application Estimates for band-limit of h(x)

9 9 Example of g(f(x)) Original function f(x) Transfer function g(y) g(f(x)) sampled by Intuition Analysis Application

10 Analysis of Composition in Frequency Domain

11 11 Composition in Frequency Domain Intuition Analysis Application y y

12 12 Composition as Integral Kernel Intuition Analysis Application

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

14 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 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

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

17 Application Adaptive Sampling for Raycasting

18 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 19 Adaptive Raycasting Uniform sampling Adaptive sampling - 25% less samples Intuition Analysis Application

20 20 Adaptive Raycasting Same number of samples Intuition Analysis Application

21 21 Adaptive Raycasting SNR Ground-truth: computed at a fixed sampling distance of Intuition Analysis Application

22 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 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 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 25 Acknowledgements NSERC Canada BC Advanced Systems Institute Canadian Foundation of Innovation

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

27 27 Human Tooth CT Transfer Functions (TFs) (g) RGB(g)RGB(g) g Shading, Compositing… Simple (usual) case: Map data value g to color and opacity Intuition Analysis Application

28 28 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 Intuition Analysis Application

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

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

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


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