Automated Topological Correction of Cortical Surfaces

Monica K. Hurdal Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, U.S.A.

Locations and patterns of functional brain activity are difficult to compare across subjects because individual differences in cortical folding and functional foci are often buried within cortical sulci. Cortical flat mapping can address these problems by taking advantage of the two-dimensional sheet topology of the cortical surface and may facilitate the recognition of structural and functional relationships that were not previously apparent. Many flat mapping algorithms have been applied to cortical data [1-4]. All require a topologically correct two-manifold (i.e. a topological sphere or disc) triangulated mesh representing the cortical surface. Few algorithms produce topologically correct surfaces and widely used algorithms, such as marching cubes/tetrahedra algorithms, produce surfaces with topological errors. Thus, there is a need for methods that detect and repair topological problems in surfaces. Algorithms which meet this need are presented here. Methods A piecewise flat, topologically correct triangulated surface consists of flat triangular faces connected along edges. Each edge is an interior edge (contained in exactly two triangles) or a boundary edge (contained in exactly one triangle). If there are no boundary edges, the surface is a topological sphere; if the boundary edges form a single closed boundary component, the surface is a topological (closed) disc. Typical topological problems include non-manifold edges (i.e edges which occur more than two times), holes (i.e. more than one boundary component), handles and multiply-connected components. A surface's Euler characteristic is defined by χ(S) = v - e + t; the genus of a surface, g(S), yields the number of handles and satisfies χ(S) = 2 - 2g(S) - m(S), where v, e, t and m(S) are the numbers of vertices, edges, triangles and boundary components of the surface respectively. If a surface is topologically correct, then it is a topological sphere if and only if χ(S) = 2; it is a topological disc if and only if χ(S) = 1. This paper presents algorithms that detect and repair topological problems by using these topological invariants. The algorithm for correcting non-manifold edges examines the surface complex; surface components are detected using a region growing algorithm and surface handles are detected by examining how the Euler characteristic changes during the region growing algorithm. Surface holes are detected and repaired by examining the extra boundary components. Results and Conclusions This paper presents algorithms which have been developed for automatically detecting and correcting topological errors in triangulated surfaces. These algorithms have been used successfully on cortical surfaces generated from a variety of algorithms. Software is available that can read in and output surfaces in a variety of file formats (including byu, obj, vtk, CARET and FreeSurfer). Applying these algorithms to cortical data will speed up the processing pipeline for creating surfaces which are topologically correct. References [1] Drury, H.A. et al. 1996. J. Cog. Neuro. 8:1--28. [2] Fischl, M.I. et al. 1999. Neuroimage 9:195--207. [3] Goebel, R. Neuroimage 11:S680. [4] Hurdal, M.K. et al. 1999. Lecture Notes in Computer Science 1679:279-286. Acknowledgments This work is supported in part by NSF grant DMS-0101329 and NIH grant P20 EB02013.

NeuroImage, Volume 26, Supplement 1, Page S45, CD-Rom Abstract WE-197, 2004