histograms & isosurface statistics hamish carr, brian duffy & barry denby university college...
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Histograms & Isosurface Statistics
Hamish Carr, Brian Duffy & Barry DenbyUniversity College Dublin
Motivation
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Overview
Mathematical AnalysisAnalytical Functions• where we know the correct answer
Experimental Results• where we don’t know the correct answer
Isosurface Complexity• a related problem
Conclusions
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Mathematics of Histograms
Histograms represent distributions• the proportion at each value
Fundamentally discreteBut volumetric functions are continuous• by assumption, analysis or reconstruction
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H h( ) = δ h− f xi( )( )i∑
= 1f xi( )=h∑
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Continuous Distributions
Continuous distributions use:
The area of the isosurface
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π f h( ) = δ h − f x( )( )dxD∫
= 1 dxf −1 h( )
∫
= Size f −1 h( )( )
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Nearest Neighbour
Nearest Neighbour Interpolant
Regular grids use uniform Voronoi cells• all of the same size ζ
Let’s look at the distribution of F
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F x( ) =f xi( ) x∈Vor xi( )0 otherwise
⎧⎨⎩
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Histograms use Nearest Neighbour
F x( ) =f xi( ) x∈Vor xi( )0 otherwise
⎧⎨⎩
π F h( ) =Size F−1 h( )( )
= Size Vor xi( )( )f xi( )=h∑
= ζf xi( )=h∑
=ζ 1f xi( )=h∑
=ζ ⋅H h( )
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Isosurface Statistics
Histogram (Count)Active Cell CountTriangle CountIsosurface Area• Marching Cubes approximation
• (Montani & al., 1994)
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Analytic Functions
Can be sampled at various resolutionsAll statistics should converge at limit
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IsovalueSampling
Distribution
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Marschner-Lobb
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Experimental Results
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Experimental Results
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Experimental Results
94 Volumetric Data sets tested• various sources / types
Histograms systematically:• underestimate transitional regions• miss secondary peaks• display spurious peaks
Noisy data smoothes histogramArea is the best distribution• but cell count & triangle count nearly as good
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Isosurface Complexity
Isosurface acceleration relies on• N - number of point samples• k - number of active cells / triangles
What is the relationship?• Worst case:
• k = Θ(N)
• Typical case (estimate):• k = O(N2/3)
– Itoh & Koyamada, 1994
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Experimental Relationship
For each data set• normalize to 8-bit• compute triangle count for each isovalue• average counts over all isovalues• generates a single value (avg. triangle count)
For all data sets• plot N (# of samples) vs. k (# of triangles)• plot as log-log scatterplot• find least squares line• slope should be 2/3 1
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Complexity Results
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Conclusions
Histograms are BAD distributionsIsosurface area is much better• it takes interpolation into account
Even active cell count is acceptableIsosurface complexity is k ≈ O(N0.82)• worse than expected• but further testing needed with more data
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Future Work
Accurate trilinear isosurface areaHigher-order interpolantsMore data setsEffects of data typeUse for quantitative measurements2D Histogram PlotsMultivariate & Derived Properties
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Acknowledgements
Science Foundation IrelandUniversity College DublinAnonymous reviewersSources of data (www.volvis.org &c.)
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