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

Motivation

3 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 3

4 Mathematics of Histograms Histograms represent distributions the proportion at each value Fundamentally discrete But volumetric functions are continuous by assumption, analysis or reconstruction 4

5 Continuous Distributions Continuous distributions use: The area of the isosurface 5

6 Nearest Neighbour Nearest Neighbour Interpolant Regular grids use uniform Voronoi cells all of the same size ζ Lets look at the distribution of F 6

7 Histograms use Nearest Neighbour

8 Isosurface Statistics Histogram (Count) Active Cell Count Triangle Count Isosurface Area Marching Cubes approximation (Montani & al., 1994) 8

9 Analytic Functions Can be sampled at various resolutions All 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 histogram Area 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(N 2/3 ) Itoh & Koyamada,

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 15

16 Complexity Results

17 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(N 0.82 ) worse than expected but further testing needed with more data 17

18 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 18

19 Acknowledgements Science Foundation Ireland University College Dublin Anonymous reviewers Sources of data ( &c.) 19