Pattern Recognition Techniques - Biomedical Signal Analysis: Contemporary Methods and Applications - page 178

Figure 6.9
Delaunay triangulation.

To obtain perfectly topology-preserving maps, we employ a powerful
structure from computational geometry: the Delaunay triangulation ,
which is the dual of the Voronoi diagram [212]. In a plane, the Delaunay
triangulation is obtained if we connect all pairs w j by an edge if and only
if their Voronoi polyhedra are adjacent. Figure 6.9 shows an example of
a Delaunay triangulation. The Delaunay triangulation arises as a graph
matching the given pattern manifold.

The “neural-gas” algorithm is simple and is described below.

1. Initialization : Randomly initialize the weight vectors { w j | j =1 , 2 ,...,N }

and the training parameters ( λ i ,λ f ,ε i ,ε f ), where λ i ,ε i are initial values
of λ ( t )and ε ( t )and λ f ,ε f are the corresponding final values.
2. Sampling :Drawasample x from the input data; the vector x represents
the new pattern that is presented to the “neural-gas” network.
3. Distortion : Determine the distortion set D x between the input vector x
and the weights w j at time n , based on the minimum-distance Euclidean
criterion:

D x = || x ( n ) − w j ( n ) || , j =1 , 2 ,...,N (6.18)

Then order the distortion set in ascending order.
4. Adaptation : Adjust the weight vectors according to