Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

4.3 Structure-based classification
4.3.2.2 Renumbering the nodes
As already mentioned above renumbering is necessary since we want a unique number for
each node in the tree depending only on the path to the node. Normally the nodes in a tree
would be numbered arbitrarily. By renumbering the vertices of the quadtree we get a list of
vertices which is defined very similar to equation (4.27):
V = {v u1 , . . . , v un } with |V | = n (4.29)
where the numbers u 1 , . . . , u n are the unique numbers of the respective nodes.
These numbers can be mapped through a function U
U : V −→ N (4.30)
where U must have the properties
and
U(v i ) = U(v j ) ⇔ i = j (4.31)
U(v i ) ≠ U(v j ) ⇔ i ≠ j (4.32)
4.3.2.3 Unique number generation
A node in a quadtree can also be described by the path in the tree leading to the specific
node. As we saw in section 4.3.1, if a node in a quadtree has child nodes there are always
exactly four children. This means that for each node in the tree having child nodes we have
the choice between four possible child nodes to traverse down the tree.
By defining the simple function s
⎧
1, if v l is the lower left child node of the parent node
⎪⎨
2, if v
s(v l ) =
l is the lower right child node of the parent node
3, if v l is the upper left child node of the parent node
⎪⎩
4, if v l is the upper right child node of the parent node
, (4.33)
we can express the path to a node v i as sequence P vi which is defined as
P vi = (p 1 , . . . , p M ) (4.34)
where M is the length of the path and for 1 ≤ j ≤ M the values p j are given by
p j = s(v m ) (4.35)
where m is the index of the j-th node of the path into the vertex set V .
45