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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 MurakiDept. of Mathematics/SFU

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

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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

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

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

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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

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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

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

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

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Analysis of Composition in Frequency Domain

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11 Composition in Frequency Domain Intuition Analysis Application y y

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12 Composition as Integral Kernel Intuition Analysis Application

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13 Visualizing P(k,l) Intuition Analysis Application

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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)

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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

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16 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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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

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

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20 Adaptive Raycasting Same number of samples Intuition Analysis Application

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

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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

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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

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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

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

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26 Thanks… … for your attention! Any Questions?

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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

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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

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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

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

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

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