A lot of the atrium segmentation computations depend on how the MRI image is subdivided into subdivisions (or basic components) and also partly on the location and spatial distribution of the local maximum and saddle points. Subdivisions are the collection of voxels each of which lead to a local maximum by following a path of increasing EDT values. Now, although intuitively these subdivisions could be thought of as one group of neighboring voxels, however, as I just found out this is not the case. The subdivisions can actually be a disjoint group of voxels as shown below in the figure. Notice that this is one single subdivision that is centered by the local maximum voxel marked in green (by the red arrow). The trail of points indicate the path that leads to the local maximum for a point in the smaller disjoint subdivision.
Here is another image of the same subdivision with the opacity changed to show the opaqued surface.
This is the main reason behind why the boundaries between certain subdivisions dont have a saddle point. For example if we look at the smaller disjoint group in the subdivision shown above in the figure, this smaller group may not have a saddle point with its neighboring subdivisions, since the other larger disjoint subdivision have satisfied the saddle point requirement by having a saddle point with a neighboring subdivision. Ideally we wouldn't have wanted subdivisions to be disjointed in this manner. As explained, this causes some subdivison boundaries to not have saddle points. This has implications in the work we are currently doing where we are trying to locate ostium centers automatically.
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