Thesis (PDF) - Signal & Image Processing Lab

6.2. MORPHOLOGICAL OPERATIONS ON THE EXTREMA-WATERSHED TREE 87
6.2.2 Opening by reconstruction
We can define an opening by reconstruction operator in the extrema-watershed tree.
The classical version of this morphology operator is defined in [3] as a single erosion
operation followed by a sequence of conditional dilation operations. The sequence is
stopped when stability is reached - meaning that a dilation followed by the minimum
operation does not change the resulting image.
Using an opening operator defined on a tree, an opening by reconstruction operator
is equivalent to tree pruning, where the pruning criteria is whether the vertex in
question exists in the opened image. The tree pruning for opening by reconstruction
is done using the following steps:
1. Apply an opening operator on the source image.
2. In a tree, for every label remaining in the tree, mark all vertices of the path
starting from this label until the tree root, as still existing in the tree.
3. Replace every label in the input image by the closest parent label that was
marked as still existing in the tree.
Let’s demonstrate the method using an example illustrated in Fig. 6.13. In this
example a source image consists of three rectangles located on a background. In the
corresponding tree, those shapes have indexes V1, V2 and V3. In the example, a
regular opening operator and an opening by reconstruction are applied on the image.
The structuring element is a square that is bigger than V2, but smaller than the other
shapes. The regular opening, shown in the figure, will leave the root label V5 shown
in the image instead of V2. However, the opening by reconstruction will remove V2,
as shown, and then reconstruct label V4 instead.
Additional examples can be seen on Figs. 6.14 and 6.15.