Myelin sheath thickness evaluation

Unless fibers are very sparse, some of them are connected in the binary image. In the zonal tree, this corresponds to several white leaves that share the same neighboring black zone. This section deals with the division of this black zone into sub-regions that are either myelin sheaths surrounding axon candidates or artifacts to be merged with the background.
For instance, let us consider the example of figure . The black zone includes 9 white leaves, numbered 1-7,x,y. Leaves x and y were discarded at the previous stage, because they lack circularity to be proper axon candidates. Among the 7 axon candidates, areas 1 to 6 are true axons while area 7 is an artifact.
Let us first consider a single white area. We evaluate the thickness of the myelin sheath around it as follows: we define X_d as the set of pixels at a distance d of a set X of pixels

$$X_{d} = (X \oplus B_{d+1}) \backslash (X \oplus B_{d})$$ (4.1)

with B_d a ball of size d, $\oplus$ the morphological dilation and $\backslash$ the set difference. We define the thickness of the myelin sheath around a white area X as the smallest distance d for which there are more white than black pixels in X_d.
This is very efficiently implemented using the algorithm of section for the morphological dilation. In particular, the set X_d is composed of the pixels present within the buckets structure when bucket(d) is being processed. Therefore, the termination criterion can be computed for all values of d during the dilation, and the propagation process can be stopped as soon as needed.

  

Figure 4.6: Axon separation by distance transform. From left to right: original image; result of the connected operators filtering; distance map corresponding to the dilation process; detected fibers.

Let us now consider all the axon candidates that are leaves of the same black area. We apply the previous procedure to each candidate, by order of decreasing size. In figure , this separates fibers numbered from 1 to 5. For area number 6, the propagation process reaches pixels that were previously considered as belonging to the myelin sheath around area number 2. These pixels are re-labeled as belonging to the sheath around the axon they are closer to. The resulting edge between the two fibers corresponds either to the thickness of the smallest fiber, or to the iso-distance line between the two axons. Area number 7 is entirely included inside the myelin sheath surrounding area 1. Therefore, it must be an artifact and it is discarded.