Histograms &
Isosurface Statistics
Hamish Carr, Brian Duffy & Barry Denby
University College Dublin
Motivation
Overview
Mathematical Analysis
Analytical Functions
• where we know the correct answer
Experimental Results
• where we don’t know the correct answer
Isosurface Complexity
• a related problem
Conclusions
3
Mathematics of Histograms
Histograms represent distributions
• the proportion at each value
H h     h  f xi 


i


f xi  h
1
Fundamentally discrete
But volumetric functions are
continuous
4
Continuous Distributions
Continuous distributions use:
 f h     h  f x dx
D


1 dx
f 1 h 
 h 
 Size f
1
The area of the isosurface
5
Nearest Neighbour
Nearest Neighbour Interpolant
 f xi  x Vor xi 
F x   
otherwise
 0
Regular grids use uniform Voronoi
cells
• all of the same size ζ
Let’s look at the distribution of F
6
Histograms use
Nearest Neighbour
 f xi  x Vor xi 
F x   
otherwise
 0
 F h   Size F 1 h 



Size Vor xi 


f xi  h
f xi  h




f xi  h
1
   H h 
7
Isosurface Statistics
Histogram (Count)
Active Cell Count
Triangle Count
Isosurface Area
• Marching Cubes approximation
• (Montani & al., 1994)
8
Analytic Functions
Can be sampled at various resolutions
All statistics should converge at limit
Distribution
Sampling
Isovalue
9
Marschner-Lobb
10
Experimental Results
11
Experimental Results
12
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
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
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
Complexity Results
16
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
17
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
Acknowledgements
Science Foundation Ireland
University College Dublin
Anonymous reviewers
Sources of data (www.volvis.org &c.)
19