11th Annual OHBM Meeting Abstract Number: 640 Submitted By: Lee Singleton Last Modified: 11 Jan 05

Comparison of Variations in the Marching Cubes Algorithm for Cortical Surface Reconstructions
Lee Singleton, Monica K. Hurdal
Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510

Objective: Cortical surface reconstructions are increasingly being used to study the anatomy and functional processing of the brain and to make comparisons across individuals. Although the geometry of the surface of the brain varys across individuals, its topology is always equivalent to a two-sphere with an Euler characteristic (c) of 2. Many different algorithms exist for reconstructing cortical surfaces. We use the marching cubes algorithm[1], one of the most common algorithms used for constructing a surface from volume data, namely because of its speed. We use variations in this algorithm to compare geometric and topological properties in the resulting cortical surface. To our knowledge, this is the first comparison of geometric and topological cortical surface properties to be performed with the marching cubes algorithm.
Methods: The marching cubes algorithm and its variations rely on a lookup table to reconstruct a triangulated surface corresponding to a given value known as the isovalue. The properties of the triangulated brain surface can be quite different, depending on the triangulation premise[2][3] and isovalue chosen. In order to facilitate more efficiency in downstream applications such as handle removal and flattening, we wish to optimize the surface reconstruction algorithm as much as possible. We have compared the effects of using different lookup tables and isovalues, as well as perturbing data values that are equal to the isovalue, to find the best combination of properties on MRI data from 11 different subjects. Some important surface characteristics we are investigating include the number of generated triangles, surface area, c, and the maximum degree of the vertices (largest number of triangles at one vertex). A smaller maximum degree is desired since fewer triangles at a single vertex result in fewer problems in later applications. A smaller number of triangles in the surface will mean a smaller number of processing steps further downstream.
Results & Discussion: A subset of our results is shown in Table 1.
Conclusions: The manner in which the marching cubes algorithm is run does have an effect on the resulting surface. The low separation table and asymptotic table result in the best combination of triangles, c, and vertex degree. However, using an asymptotic decider with marching cubes has a slightly longer computational time and has varying c results, depending on the isovalue used. Using integer isovalues and perturbing data values equal to the isovalue also gives better results than using non-integers for the isovalue. In the future, we will investigate new algorithms of data perturbation in order to give topologically correct surfaces.
References & Acknowledgements:
[1] W. Lorensen and H. Cline. Marching Cubes: a high resolution 3D surface construction algorithm. Computer Graphics, 21(4):163-170, 1987.
[2] G. Nielson and B. Hamann. The Asymptotic decider: resolving the ambiguity in marching cubes. Proceedings of Visualization, 29–38, 1991.
[3] T. Lewiner, et. al. Efficient implementation of marching cubes' cases with topological guarantees. J.Graphics Tools, 8(2):1-15, 2003.

This work is supported by grants NSF:DMS-0101329, NIH:P20-EB02013. We thank Dr. David Rottenberg, Radiology and Neurology Departments, U.Minnesota for the MRI data.

 Marching Cubes Results for a Single Subject M.C. Lookup Table Isoval. Triangles Surf. Area Euler Char. Max Deg. M.C. Lookup Table Isoval. Triangles Surf. Area Euler Char. Max Deg. Low 43 325900 88845 -638 12 Asymp 43 331678 89424 -630 12 Low 44 336796 91687 -674 12 Asymp 44 342602 92239 -736 12 Low 45 350204 95221 -792 12 Asymp 45 358180 96238 -852 12 High 43 333896 90642 -644 16 Low 43.1 342216 90728 -756 12 High 44 347236 93995 -762 16 Low 44.1 355064 93882 -854 12 High 45 364560 98504 -890 16 Low 45.1 370300 97637 -960 12 Tetra 43 332752 90252 -768 14 Asymp 43.7 346656 92173 -770 12 Tetra 44 344844 93350 -850 14 Asymp 44.1 357446 93829 -814 12 Tetra 45 361372 97637 -962 14 Asymp 44.4 359934 94928 -838 12 Table 1: Low and High lookup tables always separate data values lower and higher than the isovalue, respectively. Tetra table is based on a tetrahedral decomposition of the cubes. Asymp table uses the asymptotic decider to determine configurations.
NeuroImage, Volume 26, Supplement 1, Page S38, CD-Rom Abstract 640, 2005