Connected operators filtering

The resulting binary image presents a number of artifacts that are best expressed and handled in terms of regions and their properties. This can be formalized using connected morphological operators, as described by Heijmans [72].
The binary image is considered as a partition P(X) of the set X of pixels into black and white regions. As illustrated in Figure  , the zonal graph of the image is the graph that takes the regions of P(X) as vertices and whose arcs represent the adjacency of the regions. The graph also specifies the color of the regions it represents. Given two partitions P and P' of the image, P is coarser than P' if $P' \subseteq P$. A morphological operator $\psi$ is called connected if the resulting partition $P(\psi(X))$ is coarser than P(X), for any set X. In other words, connected zones are either left untouched or changed altogether. In the common case where connectivity is based on adjacency, connected operators can be described and implemented by re-coloring and merging vertices in the zonal graph.

  

Figure 4.5: The area operator flips zones with an area of less than 10 in the original image ( left). It can be seen as a re-coloring ( center) and merging ( right) of vertices in the zonal graph.

A well know connected morphological operator is the morphological opening by reconstruction, where objects that are too small to contain the structural element of the original erosion are deleted, while the other objects are left unchanged. More complex criteria can be of course defined, either considering each zone separately (it is then called a grain operator) or considering the relationships between zones and their neighbors. We use both hereafter.
Different connectivities yield different zonal graphs. In our case, we use 8-adjacency for foreground pixels and 4-adjacency for background pixels. This defines a topology similar to the continuous case, and in particular the zonal graph is then a tree, i.e. a graph without cycles. The following connected operators are applied:

• Noise in the original image creates small mis-labeled areas in the binary image. Those are removed by applying the area operator of figure for all areas smaller than the smallest axons ( $0.5 \mu m^2$). Unfortunately, this operator is not stable. Applied iteratively it can fail to converge and oscillate between two solutions. Thus, we restrict its action to the leaves of the zonal tree, i.e. the regions with only one neighbor.
• Fibers have a bright center surrounded by a black ring i.e. a black region with two neighboring white ones. Thus, all black leaves in the zonal tree do not represent a useful feature and are removed.
• Fixation and coloration problems can separate the myelin sheath in two parts (see figure ). It appears as a white ring surrounded by two black rings. These white rings should be re-colored in black. White rings are detected because the gravity center of the zone is located outside of it. Two cases are possible: either the ring is open and it is a leaf of the zonal graph, it is then merged with its only neighbor; either the ring is closed and has 2 neighbors in the zonal graph, the three vertices of the graph are then merged together.

After this filtering, axon candidates are identified as white leaves in the zonal tree satisfying both a size criterion ( $1 \mu m < d < 12 \mu m$) and a shape criterion ensuring the compactness and approximate circularity of the axon, depending on the perimeter²/area ratio.