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)
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 We have tried to give attribution to the people who originally made the data available in the NRRD header, as a comment.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.
This is work in progress - let me know what you think! Draft of talk slides.
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.