Histograms & Isosurface Statistics Hamish Carr, Brian Duffy & Barry Denby University College Dublin.

Histograms & Isosurface Statistics

Hamish Carr, Brian Duffy & Barry DenbyUniversity College Dublin

Motivation

3

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

3

4

Mathematics of Histograms

Histograms represent distributions• the proportion at each value

Fundamentally discreteBut volumetric functions are continuous• by assumption, analysis or reconstruction

4

H h( ) = δ h− f xi( )( )i∑

= 1f xi( )=h∑

5

Continuous Distributions

Continuous distributions use:

The area of the isosurface

5

π f h( ) = δ h − f x( )( )dxD∫

= 1 dxf −1 h( )

∫

= Size f −1 h( )( )

6

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

6

F x( ) =f xi( ) x∈Vor xi( )0 otherwise

⎧⎨⎩

7

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( )

8

Isosurface Statistics

Histogram (Count)Active Cell CountTriangle CountIsosurface Area• Marching Cubes approximation

• (Montani & al., 1994)

8

9

Analytic Functions

Can be sampled at various resolutionsAll statistics should converge at limit

9

IsovalueSampling

Distribution

10

Marschner-Lobb

11

Experimental Results

12

Experimental Results

13

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

13

14

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

14

15

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

5

16

Complexity Results

17

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

17

18

Future Work

Accurate trilinear isosurface areaHigher-order interpolantsMore data setsEffects of data typeUse for quantitative measurements2D Histogram PlotsMultivariate & Derived Properties

18

19

Acknowledgements

Science Foundation IrelandUniversity College DublinAnonymous reviewersSources of data (www.volvis.org &c.)

19