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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