Guided Filter-Based Fuzzy Clustering for General Data Analysis

Abstract

In recent years, benefiting from the abilities in implementation of the smooth operation on noises and preservation of the original gradient information on the texture edges, the guided filter (GF)-based fuzzy clustering has achieved inspiring performance in the task of image segmentation. However, different to image pixels, the general data samples do not have the spatial adjacency relations. Current GF-based methods suffer some limitations in general data clustering. To solve this issue, a series of improved GF-based fuzzy clustering algorithms are proposed in this paper for general data analysis. In the most basic algorithm, a new data filtering window is first defined according to the neighbor relationships between data samples, and a pruning mechanism is designed to ensure the symmetry of neighbor samples. Then, the GF-based Fuzzy C-Means algorithm is put forward for the general data clustering. In addition, a weighted version is presented to process high-dimensional data, in which each dimension is assigned a weight, and the entropy regularization method is applied to optimize the weight assignment and highlight the important dimensions on both filtering and clustering. Furthermore, the kernelization of the above methods are realized for the non-linear data. Experimental results on synthetic and real-world datasets demonstrate better performance of the proposed methods in comparison with some existing FCM-type clustering approaches.

Appendix

In this appendix, we prove that the filtered fuzzy memberships still retain the constraint $\sum _{i=1}^C u_{ij}' = 1$ which is equivalent to

$$\begin{aligned} \sum _{i=1}^{C}\frac{\sum _{m=1}^{M}\left( \bar{a}_{ijm}x_{jm} + \bar{b}_{ijm}\right) }{M} = 1. \end{aligned}$$

(38)

By using the general multi-channel guided filtering formula to carry out the backward calculation, we have

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{C}u_{ij}'&= \frac{1}{M}\sum _{i=1}^{C}\sum _{m=1}^{M}\left( \bar{a}_{ijm}x_{jm}+\bar{b}_{ijm}\right) \\&= \frac{1}{M}\sum _{i=1}^{C}\sum _{m=1}^{M}\left( \frac{1}{\left| \omega _j\right| }\sum _{k\in \omega _j} a_{ikm}x_{jm} + \sum _{k\in \omega _j} b_{ikm}\right) \\&= \frac{1}{M\left| \omega _j\right| } \sum _{k\in \omega _j}\sum _{i=1}^{C}\sum _{m=1}^{M}\left( a_{ikm}x_{jm} + b_{ikm}\right) \\&= \frac{1}{M\left| \omega _j\right| } \sum _{k\in \omega _j}\sum _{i=1}^{C}\sum _{m=1}^{M} \left( a_{ikm}x_{jm} + \bar{u}_{ik} - a_{ikm}\mu _{km}\right) . \end{aligned} \end{aligned}$$

(39)

As the sum of a data sample’s fuzzy membership degrees is constant 1, i.e., $\sum _{i=1}^{C}u_{ij}=1$, we obtain

$$\begin{aligned} \sum _{i=1}^{C} \bar{u}_{ik} = \sum _{i=1}^{C}\frac{1}{\left| \omega _k\right| }\sum _{j\in \omega _k}u_{ij} = \frac{1}{\left| \omega _k\right| }\sum _{j\in \omega _k}\sum _{i=1}^{C}u_{ij} = 1. \end{aligned}$$

(40)

We substitute (40) into (39) and then arrive at

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{C}u_{ij}' =&\frac{1}{M\left| \omega _j\right| } \sum _{k\in \omega _j}\sum _{i=1}^{C}\sum _{m=1}^{M} \left( a_{ikm}x_{jm} + \bar{u}_{ik} - a_{ikm}\mu _{km}\right) \\ =&\frac{1}{M\left| \omega _j\right| } \sum _{k\in \omega _j}\sum _{m=1}^{M}1 \\&+ \frac{1}{M\left| \omega _j\right| } \sum _{k\in \omega _j}\sum _{m=1}^{M}\left( x_{jm}-\mu _{km}\right) \sum _{i=1}^{C}a_{ikm}. \end{aligned} \end{aligned}$$

(41)

Considering that the guidance data $x_{jm}$ and the mean value $\mu _{km}$ are irrelevant to the cluster, we have

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{C}a_{ikm}&= \sum _{i=1}^{C}\frac{\frac{1}{\left| \omega _k\right| }\sum _{j\in \omega _k}x_{jm}u_{ij}-\mu _{km}\bar{u}_{ik}}{\sigma _{ikm}^2 + \epsilon } \\&= \frac{\frac{1}{\left| \omega _k\right| }\sum _{j\in \omega _k}\sum _{i=1}^{C} x_{jm}u_{ij} - \sum _{i=1}^{C}\mu _{km}\bar{u}_{ik}}{\sigma _{ikm}^2 + \epsilon } \\&= \frac{\frac{1}{\left| \omega _k\right| }\sum _{j\in \omega _k}x_{jm}\sum _{i=1}^{C}u_{ij} - \mu _{km}\sum _{i=1}^{C}\bar{u}_{ik}}{\sigma _{ikm}^2 + \epsilon } \\&= \frac{\frac{1}{\left| \omega _k\right| }\sum _{j\in \omega _k}x_{jm} - \mu _{km}}{\sigma _{ikm}^2 + \epsilon } \\&= 0. \end{aligned} \end{aligned}$$

(42)

Finally, bu substituting (42) into (41), we obtain

$$\begin{aligned} \sum _{i=1}^{C}u_{ij}' = \frac{1}{M\left| \omega _j\right| } \sum _{k\in \omega _j}\sum _{m=1}^{M}\left( 1 + \left( x_{jm}-\mu _{km}\right) *0\right) = 1. \end{aligned}$$

(43)

The above formula proves that the membership degrees still retains its inherent property after general guided filtering. In other words, the sum of filtered membership degrees of each data point belonging to all clusters is still 1.