Thesis (PDF) - Signal & Image Processing Lab

72 CHAPTER 5. TREE SEMILATTICES
same vertex set of the RAG - a set of all flat zones in the image. In that sense, the
BTV tree is a particular case of a spanning tree.
The shape tree is defined in [20] and the resulting semilattice is defined in [17].
The shape tree differs from the BTV tree in one main aspect: its vertex set is not
the flat zones of the image. Its vertex set is built from the flat zones during the tree
generation. Each father vertex area includes all sons in addition to an area of itself.
For each point in the image there may be more than one vertex in the shape tree that
includes that point. In order to preserve the single value of the mapping function we
define a mapping to the biggest vertex of all possible vertices in the tree.
Max and Min Trees
Another group of trees is Max and Min trees. Those trees, as described in sec-
tion 2.2.1, are created from the RAG, sorted by the gray level of the flat zones.
Similar to the above shape tree, each vertex region in the picture contains all children
vertices. In Max Tree the father gray level is always bigger than the son. And, of
course, the opposite in a Min tree. When a father is always brighter/darker then a
son, the infimum operation always changes the gray level to the darker/brighter side.
This has the same effect as a regular erosion on a gray-level picture. For this reason
those trees are of no special interest for the proposed approach.
5.3 Semilattice of Images
5.3.1 Structure Induction
What we would really like is the complete inf-semilattice of tree representation (using
a tree τ) to induce a complete inf-semilattice in the image domain. That is, we
would like, for instance, that the composite operation of τ −1 ετ be an erosion in the
image domain. However, that is not guaranteed; for some tree transforms τ an image
semilattice is obtained and for others not. In fact, the partial ordering in the tree-
representation domain does induce a partial ordering for images, for any τ; however,
the infimum is not guaranteed to exist.
Let f be an image, τ(f) is a tree representation. Tε = ε(τ(f)) is the tree repre-
sentation of an erosion operator result. Let fε = τ −1 (ε(τ(f))) be the image obtained