# Revisiting Histograms and Isosurface Statistics

### Carlos Scheidegger, John Schreiner, Brian Duffy, Hamish Carr,
Claudio Silva

## Abstract

Recent results have shown a link between geometric
properties of isosurfaces and statistical properties of the
underlying sampled data. However, this has two defects: not all of
the properties described converge to the same solution, and the
statistics computed are not always invariant under
isosurface-preserving transformations.
We apply Federer's Coarea Formula from geometric measure theory to
explain these discrepancies. We describe an improved substitute for
histograms based on weighting with the inverse gradient magnitude,
develop a statistical model that is invariant under
isosurface-preserving transformations, and argue that this provides
a consistent method for algorithm evaluation across multiple
datasets based on histogram equalization.
We use our corrected formulation to reevaluate recent results on
average isosurface complexity, and show evidence that noise is one
cause of the discrepancy between the expected figure and the
observed one. PDF Version (manuscript, 7MB)

## Dataset

We are making available the set of volumes we collected to
compute the histograms and isosurface statistics. Note, however,
that it is a fairly large dataset (~1GB).

Download volumes. Files are in NRRD
format. We have tried to give attribution to the people who
originally made the data available in the NRRD header, as a
comment. We couldn't find the authors for a few of those files, so
if you see your file here and it is incorrectly attributed, do not
hesitate to let us know.

## Talk slides

**This is work in progress - let me know what you
think!** Draft of talk slides.

## Source code, scripts

We implemented the gradient weighted isosurface statistics
collection as part
of Afront. Note that you
need to use the CVS version, since the binary releases do not yet
include the necessary features.

The scripts to execute the software and generate the results will
be made available soon.