Thesis (PDF) - Signal & Image Processing Lab

5.2. IMAGE PROCESSING ON TREE SEMILATTICES 69
exists, and, even though it is not well defined for all complete inf-semilattice elements,
it is always well-defined for elements that are mapped by the erosion, and γ = δε.
In the case of the above tree-representation flat erosion, the adjoint dilation is
given by:
δ(T ) = (t, �t {Mb|b ∈ B}) . (5.9)
5.2 Image Processing on Tree Semilattices
5.2.1 Proposed Approach
Our goal is to process a given grayscale image f, which we assume to be an integer-
valued function on E, i.e., f ∈ Fun(E, Z). We propose the following approach for
processing f, using the complete inf-semilattice of tree representations.
1. Transform f into a pair (T, ℓ), where T = (t, M) ∈ T L
r is a tree representation,
and ℓ : L ↦→ Z is a function that maps labels into graylevels,
2. Perform one or more morphological operations on T to obtain a processed tree
representation ˆ T = (t, ˆ M).
3. Transform ( ˆ T , ℓ) back into a new image ˆ f ∈ Fun(E, Z), using:
� �
ˆf(x) = ℓ ˆM(x) . (5.10)
Notice that the format for the inverse transform (item 3) is fixed, and does not
depend on the exact definition of the tree transform τ. The user however has a great
deal of freedom for choosing τ itself. A number of tree transforms are discussed
in sections 2.2 and in section 6.1. Because there are many possible alternatives for
defining τ, designing the forward transform is the main issue for applied research in
that field, where the goal is to find the best tree representation of an image in some
sense. In this chapter, however, this issue is not addressed; instead, a tree transform
is assumed given, and we focus on the corresponding morphological operators and
the methodology to obtain them.