Overview: Reconstructing an accurate and topologically correct representation of the cortical surface of the brain is an important objective in various neuroscience applications. Most cortical surface reconstruction methods either ignore topology or correct it using manual editing or methods that lead to inaccurate reconstructions. In this work, we developed a new volumetric topology correction algorithm, which we refer to as the graph-based topology correction algorithm (GTCA). The method is fully automatic, and provides several advantages over existing approaches, including the use of arbitrary digital connectivities, a flexible morphology-based multiscale approach, and the option of foreground-only or background-only correction. The method has been applied on hundreds of data sets, and its success has been verified every time.

Introduction: Reconstruction of the brain cortical surface(s) from magnetic resonance images is an important goal in medicine and neuroscience, e.g, to study brain geometry and quantify geometric variations across populations. Producing the correct topology is an important part of the cortical surface reconstruction process. A reconstructed cortical surface without a correct topology may lead to incorrect interpretations of local structural relationships and will prevent cortical unfolding. 
Geometrically, the human cerebral cortex is a thin folded sheet of gray matter that lies inside the cerebrospinal fluid and outside the white matter of the brain. If the opening at the brain stem is artificially closed, the surface of the cortex is topologically equivalent to a sphere. The major topological defect is the presence of one or more handles on the reconstructed surface. 

Although a considerable amount of work has been dedicated to the automatic extraction of cortical surfaces, few of them produce the correct topology. As a result, a topology correction method is needed to detect and remove handles from an initial topologically incorrect surface reconstruction, while at the same time minimize the modification to the original segmentation.

Method: Our method performs the topology correction on the binary volume defined by the initial surface or volume segmentation.

Two types of corrections can be made to remove a handle: either break the handle directly on the foreground, or fill-up the tunnel associated with the handle on the background.
To ensure minimal modification to the original volume, our method sequentially applies both types of corrections (filters) at successively increasing scales until all handles are removed. 
The scale analysis is performed by an morphological opening operator, the size of whose structuring element determines the scale of the current filter. The opening breaks the original binary object into body and residue parts. A graph is then constructed by analyzing the connectivity of the body and residue pieces. If a handle exists whose size is less than the current filter scale, it will be broken into body and residue pieces, and the connectivity among these pieces will form a cycle in the graph. We can thus detect handles by detecting cycles through graph analysis, and remove handles by removing the smallest residue piece within the cycle. 

Results: 

Conclusion:We have developed and evaluated GTCA, a fully automatic method to remove handles in 3D digital images. GTCA is fundamentally based on the theory of digital topology and uses results from mathematical morphology, implicit surface tiling, and graph theory. GTCA is intrinsically 3D processing, works with any consistent pair of digital connectivities. We expect GTCA to be a useful automated tool for brain geometric and functional analysis studies across populations.

Acknowledgment: We thank Dr. S. Batman, Dr. J. Goutsias, Dr. D. Pham, and Dr. S. Resnick for their contributions. This work is supported in part by NSF/ERC Grant CISST9731748 and in part by NIH/NINDS Grant R01NS37747.

Publications: 

 X. Han, C. Xu, U. Braga-Neto, and J. L. Prince, "Graph-based topology correction for brain cortex segmentation," in Proc. XVIIth Int. Conf. Information Processing in Medical Imaging, June 2001.
 X. Han, C. Xu, U. Braga-Neto, and J. L. Prince, "Topology correction in brain cortex segmentation using a multiscale, graph-based approach," IEEE Trans. Med. Imag., 21(2): 109--121, 2002.