Thesis (PDF) - Signal & Image Processing Lab

66 CHAPTER 5. TREE SEMILATTICES
Figure 5.5: An example of a projection of the mapping function. Mapping functions
M1(x) and M2(x) of point x are projected to a subtree t1 ∧ t2.
Proposition 2. The tree representation infimum is given by:
T1 ∧ T2 = (t1 ∧ t2, Pt1∧t2(M1) �t1∧t2 Pt1∧t2(M2)), (5.3)
where t1∧t2 = Cr(t1∩t2), Cr(·) stands for the connected graph containing r, and Pt ′(·)
means the projection onto t ′ , i.e., the closest vertex in the path linking the operand to
r, which belongs to V (t ′ ).
In words, the tree representation infimum is built in the following way: The
intersection of the trees is calculated, and the sub-tree containing r is extracted. This
is the infimum of the trees. For each point in E, the mapping function is obtained by
calculating the infimum vertex of the projections of the original mapping functions
onto the infimum tree.
Proof. The infimum must have the following properties:
T = T1 ∧ T2 is infimum iff :
1. T � T1 and T � T2.
2. If there exists T3 such that T � T3, T3 � T1 and T3 � T2, then T3 = T .
The graph part of the tree transform is created by a regular set intersection and
Cr(·) operator, thus t ⊆ t1 and t ⊆ t2 . The mapping function of T is calculated from
infimum of vertices in the tree t1 ∧ t2 , so M � M1 and M � M2.