A novel type-II intuitionistic fuzzy clustering algorithm for mammograms segmentation

Abstract

Fuzzy clustering has been gaining prominence in medical image segmentation but challenges still exist. This paper proposes a novel Type-II Intuitionistic Fuzzy C Means clustering algorithm by introducing a new membership degree called Intuitionistic Type-II membership. Intuitionistic Type-II membership combines Type-II membership with hesitation degree. Using Intuitionistic Type-II membership, the proposed algorithm shows following advantages: (1) defining clusters clearly, (2) robustness to noise and outliers, and (3) improving desired position of centroids. These advantages make Type-II Intuitionistic Fuzzy C Means clustering algorithm a preferred choice for mammogram segmentation. On some mammograms from Mammographic Image Analysis Society database, performance of Type-II Intuitionistic Fuzzy C Means clustering algorithm is compared with the performance of other fuzzy clustering algorithms such as Fuzzy C-Means, Intuitionistic Fuzzy C-Means, Type-II Fuzzy C-Means, Interval Type-II Fuzzy C-Means, and Particle Swarm Optimization Based Interval Type-II Fuzzy C-Means using Intuitionistic Fuzzy Sets. By qualitative analysis, results of Type-II Intuitionistic Fuzzy C Means are found to be better than the results of discussed algorithms as the proposed algorithm identifies shape and size of lumps in mammograms more accurately. On experimenting with synthetic data sets, it is observed that Type-II Intuitionistic Fuzzy C Means produces robust and stable results as outliers increase and average error is reduced by 84% on D15 dataset.

Appendix: Type-II intuitionistic fuzzy c means clustering

1.1 Proof of Type-II Intuitionistic fuzzy c means clustering

Proof: Proof of T2IFCM is given here in this Appendix. Minimization of Eq. (17), subject to constraint in Eq. (2) is done using Lagrangian method. Following is the Lagrangian equation with respect to Eq. (17):

$${J}_{T2IFCM}=\sum_{j=1}^{cluster\_n}\sum_{i=1}^{z}{{(u}_{ij}^{*})}^{m}dis{t}_{ij}^{2}+ \sum_{j=1}^{cluster\_n}{\pi }_{j}^{*}{e}^{1-{\pi }_{j}^{*}}+ \sum_{j=1}^{cluster\_n}{\lambda }_{j}\left(\sum_{i=1}^{z}{u}_{ij}^{*}-1\right)$$

(33)

where ${\lambda }_{j}$ for $j=1, 2, ..., cluster\_n$ are Lagrangian multipliers.

1.2 Partial derivation of ${J}_{T2IFCM}$ wrt ${u}_{ij}^{*}$

$$\frac{\partial {J}_{T2IFCM}}{\partial {u}_{ij}^{*}}=m{{u}_{ji}^{*}}^{(m-1)}dis{t}_{ij}^{2}+0+ {\lambda }_{i}-0$$

(34)

Equate Eq. (34) to zero:

$$\frac{\partial {J}_{T2IFCM}}{\partial {u}_{ij}^{*}}=0$$

(35)

$$m{{u}_{ji}^{*}}^{(m-1)}dis{t}_{ij}^{2}+ {\lambda }_{i}=0$$

(36)

$${u}_{ij}^{*}={\left(\frac{-{\lambda }_{i}}{m.dis{t}_{ij}^{2}}\right)}^{\frac{1}{(m-1)}}$$

(37)

To fulfil the constraint in Eq. (2)

$${u}_{ij}^{*}=\frac{{u}_{ij}^{*}}{\sum_{k=1}^{cluster\_n}{u}_{ik}^{*}}$$

(38)

In view of Eq. (38), Eq. (37) can be written as:

$${u}_{ij}^{*}=\frac{{\left(\frac{-{\lambda }_{i}}{m.dis{t}_{ij}^{2}}\right)}^{\frac{1}{(m-1)}}}{\sum_{k=1}^{cluster\_n}{\left(\frac{-{\lambda }_{i}}{m.dis{t}_{ij}^{2}}\right)}^{\frac{1}{(m-1)}}}$$

(39)

$${u}_{ij}^{*}=\frac{{\left(\frac{1}{dis{t}_{ij}^{2}}\right)}^{\frac{1}{(m-1)}}}{\sum_{k=1}^{cluster\_n}{\left(\frac{1}{m.dis{t}_{ij}^{2}}\right)}^{\frac{1}{(m-1)}}}$$

(40)

$${u}_{ij}^{*}={\left(\sum_{k=1}^{cluster\_n}{\left(\frac{dis{t}_{ij}^{2}}{dis{t}_{ik}^{2}}\right)}^{\left(\frac{1}{m-1}\right)}\right)}^{-1}$$

(41)

1.3 Partial derivation of ${J}_{T2IFCM}$ wrt ${c}_{j}$:

Rewrite Eq. (33) as:

$${J}_{T2IFCM}=\sum_{j=1}^{cluster\_n}\sum_{i=1}^{z}{{(u}_{ij}^{*})}^{m}{\Vert {d}_{i}-{c}_{j}\Vert }^{2}+ \sum_{j=1}^{cluster\_n}{\pi }_{j}^{*}{e}^{1-{\pi }_{j}^{*}}+ \sum_{j=1}^{cluster\_n}{\lambda }_{j}\left(\sum_{i=1}^{z}{u}_{ij}^{*}-1\right)$$

(42)

$$\frac{\partial {J}_{T2IFCM}}{\partial {c}_{j}}=\sum_{i=1}^{z}{{-2(u}_{ij}^{*})}^{m}{\left|{d}_{i}-{c}_{j}\right|}^{2}+0$$

(43)

Equate Eq. (43) to zero:

$$\sum_{i=1}^{z}{{-2(u}_{ij}^{*})}^{m}\left|{d}_{i}-{c}_{j}\right|=0$$

(44)

$$\sum_{i=1}^{z}{{-2(u}_{ij}^{*})}^{m}{d}_{i}+ \sum_{i=1}^{z}{{2(u}_{ij}^{*})}^{m}{c}_{j}=0$$

(45)

$${c}_{j}=\frac{\sum_{i=1}^{z}\left({{u}_{ij}^{*}}^{m}{d}_{i}\right)}{\sum_{i=1}^{z}\left({{u}_{ij}^{*}}^{m}\right)}$$

(46)

IT2m (mathematically notated as ${b}_{ij}^{*}$) is computed as per Eq. (26) and on the basis of T2IFCM concept, Eq. (46) is updated as:

$${c}_{j}=\frac{\sum_{i=1}^{z}\left({{b}_{ij}^{*}}^{m}{d}_{i}\right)}{\sum_{i=1}^{z}\left({{b}_{ij}^{*}}^{m}\right)}$$

(47)