Histograms%20

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Histograms Isosurface Statistics

• Hamish Carr, Brian Duffy Barry Denby
• University College Dublin

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Motivation
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Overview

• Mathematical Analysis
• Analytical Functions
• where we know the correct answer
• Experimental Results
• where we dont 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 discrete
• But volumetric functions are continuous
• by assumption, analysis or reconstruction

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Continuous Distributions

• Continuous distributions use
• The area of the isosurface

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Nearest Neighbour

• Nearest Neighbour Interpolant
• Regular grids use uniform Voronoi cells
• all of the same size ?
• Lets look at the distribution of F

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Histograms use Nearest Neighbour
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Isosurface Statistics

• Histogram (Count)
• Active Cell Count
• Triangle Count
• Isosurface Area
• Marching Cubes approximation
• (Montani al., 1994)

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Analytic Functions

• Can be sampled at various resolutions
• All statistics should converge at limit

Distribution
Isovalue
Sampling
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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 histogram
• Area 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 T(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

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Complexity Results
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Conclusions

• Histograms are BAD distributions
• Isosurface area is much better
• it takes interpolation into account
• Even active cell count is acceptable
• Isosurface 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 area
• Higher-order interpolants
• More data sets
• Effects of data type
• Use for quantitative measurements
• 2D Histogram Plots
• Multivariate Derived Properties

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Acknowledgements

• Science Foundation Ireland
• University College Dublin
• Anonymous reviewers
• Sources of data (www.volvis.org c.)

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