Abstract: In this work, we propose a methodology for automatically correcting the spherical topology of any segmentation under any digital connectivity.

In an initial work (MICCAI 2003), a multiple region growing process, concurrently acting on the foreground and the background, divides the segmentation into connected components and successive minimum cost decisions guarantee convergence to correct spherical topology. In contrast to existing procedures that suppose specific initial segmentation (full connectivity, no cavities... ) and are designed for a particular task (cortical representation), no assumption is made on the initial image. Our method applied to subcortical segmentations allows us to correct the topology of fourteen deep nuclei in less than a minute (see figure below).

We are currently investigating different extensions by integrating statistical and geometrical information into the localization and the correction of the topological defects. We have introduced the concept of "multisimple" point, which facilitates the merging of multiple digital regions under topological constraints. With the help of this new concept, we are currently developping different approaches. (topological watershed techniques, multi-lesetset topology correction (see [1]) ), phrased within a general Bayesian framework. This general approach is appropriate to the topology correction of more complex structures, such as the cortex.

Excluding pathological cases, most subcortical brain structures are fully connected and do not possess any topological artifact such as handles or cavities: they have the simple topology of a sphere. Many recent segmentation algorithms are able to identify and precisely locate these structures, although without constraining the topology. Being able to achieve accurate and topologically correct representation of different brain structures is certainly of valuable interest in Medical Imaging (Shape Analysis, Visualization...)

Our approach uses classical results of digital topology (see the work of G. Bertrand for more details [2]). We have introduced the concept of 'multisimple' point (see corresponding publications for more details), which allows us to concurrently work on different parts of the digital object, but still control the whole topology of the resulting segmentation.  This digital concept is particularly usefull for multiple region growing techniques (watershed, multi-levelsets).

Similarly to the approaches developed by Han et al. (see [3]) and Kriegeskorte and Goeble (see [4]), we segment the initial digital volume  S_o (the foreground object and its inverse object : the background) into a set of connected components (body components of known topology and residual components - see figure below) and build two interacting connected graphs. Schematically, the topology correction amounts to finding the components that should be deleted, i.e. added to the invese object. Using a Bayesian framework (we search for the map estimate of p(S|I) where S and I represent the estimated digital segmentation and the original M.R.Image respectively), iterative minimal graph decisions correct the topology of the binary volume and convergence is guaranteed.

The initial binary segmentation S, its joint probability map p(I,S) , and the resulting segmentation into connected components
residual components are in green and blue - body components are in pink and black).

We have implemented a preliminary version of the proposed method, based on previous work (MICCAI 2003).

In order to validate the proposed algorithm, we have apply our method to 25 brain segmentations, manually and automatically labeled. The Whole Brain Segmentation algorithm proposed by Fischl et al. in [5] was used to generate automatic brain segmentations and to generate the joint probabilities p(I(x),S(x)) for all voxels x. The pair of compatible digital connectivities that we use for all the experiences was (6,18).

We present some results of the proposed method on subcortical segmentations and white matter segmentations. Topology correction of an individual structures takes a few seconds on a current machine. Most subcortical segmentations have few topological defects.  Results show that addition and deletion of very few voxels is necessary to correct the topology of each structure. Manual segmentations are corrected by changing the labels of approximately 0.05% of the total number of voxels. Automatic segmentations requires of the order of 0.1% of labels to be changed. A typical example is given in the figure below, which shows the segmentation of the right pallidum before and after topology correction. The initial segmentation had 3 topological defects ( 3 handles: euler characteristic of the tessellation X = -4 ); the final surface has the correct spherical topology.

Correction of the topology of the cortical surface is a much more challenging task: its highly convoluted nature often causes numerous topological defects, which interact with each other, and are difficult to precisely locate and correct. We have applied our method to 25 brains in order to generate white matter segmentations with a correct spherical topology. Before applying
the algorithm, we merge the ventricles into the white matter segmentation in order to avoid topological defects to be introduced in this area. We note that the following medical structures: caudate, putamen and pallidum nucleus, are considered to be part of the white matter segmentation. We apply the algorithm on each hemisphere separately. A binary segmentation S _o of an hemisphere white matter contains on the order of 10 ⁵ voxels (100 ³ image domain). Similarly to subcortical segmentation results, approximately 0.1% of the total number of white matter voxels (approximately 100 voxels) need to have their label changed to achieve topology correction.

Initial Surface and Final Surface. The initial surface, generated under (n,n) = (6,18), has an
euler characteristic of X = -236. Some topological defects are circled in red.
 

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