Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

4 Automated pit pattern classification
4.3.2.4 The mapping function
To construct the mapping function U from section 4.3.2.2, which maps a vertex v i to a
unique number u i , we use the formula
u i =
M∑
5 j p j where p j ∈ P vi (4.36)
j=1
where M is the number of nodes along the path to the node v i . j is nothing more than the
depth level in the tree of the j-th node in the path to node v i . Hence equation (4.36) is just
a p-adic number with p = 5 which is a unique representation of a number with base 5.
Base 5 is required since at each node there are four possible ways to traverse down the tree.
If we used base 4 we would only have the digits 0 to 3, which is not useful since this would
result in a tree with multiple nodes having a unique id of 0.
To ensure that every u i is unique for all v i ∈ V the requirement that a path sequence
P vi for a node v i must be unique among the quadtree must be met. And since a quadtree
contains no cycles and every node in a tree has only one parent node, this requirement is
met.
4.3.2.5 The metric
We are now able to express the similarity between to quadtrees by measuring the distance
between them. Distance, in this context, is a function d(T 1 , T 2 ) which returns the real valued
distance between two quadtrees T 1 and T 2 , which are elements of the so-called metric set T.
More formally this can be written as
d : T × T −→ R (4.37)
Now the following properties for metrics hold for all T 1 , T 2 ∈ T:
d(T 1 , T 2 ) > 0 ⇔ T 1 ≠ T 2 (4.38)
d(T 1 , T 2 ) = 0 ⇔ T 1 = T 2 (4.39)
d(T 1 , T 2 ) = d(T 2 , T 1 ) (4.40)
Therefore this distance between two quadtrees can be used to express the similarity between
two quadtrees.
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