c2: Fuzzy C-Means

The Fuzzy C-Means (FCM) clustering algorithm [11] is an extension of the k-Means algorithm. The main difference is that Fuzzy C-Means allows each pattern of the image to be associated with every cluster using a fuzzy membership function (in contrast with k-Means, where each pattern belongs to one and only one cluster). The introduction of such fuzziness has two effects [78]:

• The FCM algorithm has a poor convergence performance compared to k-Means.
• The FCM, in general, has an improved segmentation performance.

In our implementation, the function criterion minimized by the algorithm is defined by:

$$e^2(I,\Xi) = \sum_{k=1}^K\sum_{i=1}^{N}u_{ik}^m\vert\vert p_i-c_k\vert\vert^2$$ (2.10)

where $u_{it}$ represents the membership of pattern $p_i$ to belong to cluster $k$ , which is centred at

$$c_k=\frac{\sum_{i=1}^{N}u_{ik}^mp_i}{\sum_{i=1}^{N}u_{ik}^m},$$ (2.11)

$N$ is the number of patterns in the whole image (i.e. the number of pixels), $K$ the number of clusters, which has to be known a priori, and $m$ the degree of fuzzyness (a number greater than $1$ ).

For mammographic mass segmentation purposes, the Fuzzy C-Means algorithm has been applied by Velthuizen [185] and Chen and Lee [29] who again only used grey-level features. As for the c1 approach, the c2 implementation is based on additional features. To perform a realistic comparison, we used the same features for both algorithms.