Thesis (PDF) - Signal & Image Processing Lab

5.2. IMAGE PROCESSING ON TREE SEMILATTICES 71
mappings belong to that subtree. But x lies outside a shape left by an erosion
operation on �
v ′ �v R(v′ ) using the structure element B. It means that when B was
applied to x, it included M−B(x) vertices, and one of that vertices was outside the
subtree of v. So, ˆ M(x) could not belong to �
v ′ �v ˆ R(v ′ ).
In the second case, x belongs to �
v ′ �v R(v′ ) but doesn’t belong to �
v ′ �v ˆ R(v ′ ). In
a similar way as described above, all source M−B(x) vertices belong to �
v ′ �v R(v′ ),
so that ˆ M(x) must belong to the same subtree - �
v ′ �v ˆ R(v ′ ).
The above proposition suggests that one way of implementing the tree-representation
erosion (assume the list of all flat zones of f and a spanning tree t are given) is by:
1. Associating to each item v of the list the set of points ¯ R(v) △ = �
v ′ �v R(v′ ), i.e.,
the union of all flat zones that are equal or greater than v in the tree t.
2. Eroding ¯ R(v) for each v in the list.
3. Assigning the gray level of v for all pixels in ˆ R(v) △ = ¯ R(v) \ �
v ′ �v ¯ R(v ′ ).
5.2.2 Examples and Particular Cases
RAG’s and Spanning Trees
In order for the tree transform to be invertible, τ should be such that it assigns a
common label to each flat zone of f. This is because τ −1 maps each label to a single
gray value. This suggests that special attention should be paid to the flat zones of f.
One way of addressing the flat zones of a given image is by considering its Regional
Adjacency Graph (RAG). The RAG is a graph, where V is the set of all flat zones of
the image, and E contains all pairs of flat zones that are adjacent to each other.
A spanning tree is a subgraph of a RAG that should, obviously, be a tree, and
have the same vertex set V as the RAG. A spanning tree creates a hierarchy in the
RAG, defining father/son relationships between adjacent flat zones.
BTV and Shape Trees
The BTV tree described in chapter 3, is built from the RAG using minimal topo-
graphic distance criteria for building the tree t. The vertex set of the BTV tree is the