# Kernel-Based Robust Bias-Correction Fuzzy Weighted C-Ordered-Means Clustering Algorithm

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Introduction of Fuzzy Clustering

#### 1.2. Introduction to FCM Clustering with Local Spatial Information

#### 1.3. Introduction of Clustering Algorithm Based on Ordered Means

#### 1.4. Introduction of Fuzzy Clustering Based on Kernel Method

- (1)
- The introduction of kernel function avoids the ”dimensional curse” and significantly reduces the computation. It is a perfect trick to shift the ”disaster” effect from the low dimension of the input space to the processing of high-dimensional vector product, thus reducing the computation.
- (2)
- There is no need to know the form and parameters of the non-linear mapping function.
- (3)
- The kernel function method can select different kernel functions and algorithms according to different applications and combine them together to form a variety of different kernel function-based techniques, while the design of kernel functions and algorithms can be performed separately.
- (4)
- The kernel function implicitly turns the originally complicated computation into a mapping computation and changes the initial form and parameters of computation into the mapping from input space to feature space and thus having an effect on the properties of feature space, finally achieving the purpose of computation.

- (1)
- By using robust kernel, a class of robust non-euclidean distance measure is introduced. A new class of robust non-euclidean distance measure is introduced into the input space and then the robust second-order metric (kernel) is used to replace the non-robust Euclidean distance metric. Thus, data clustering or image segmentation can be carried out more effectively.
- (2)
- Maintain the computational simplicity of FCM.
- (3)
- It is more likely to reveal the inherent non-euclidean structure of the data.
- (4)
- The clustering results can be intuitively explained.
- (5)
- Data sets with missing values are easily handled [83].

#### 1.5. Theoretical Classical FCM Algorithm and Its Relationship with KBFWCM Algorithm

#### 1.6. Structure of the Paper

## 2. Related Work

#### 2.1. Criterion Function for BFWCOM Algorithm

#### 2.2. Clustering Prototype and Bias Field Update

#### 2.3. Calculation of Typicality of Data

## 3. Kernel-Induced Robust Bias Correction Fuzzy Weighted C-Order Mean Clustering Algorithm

#### 3.1. Definition and Theorem of Kernel Function

#### 3.1.1. Definition of Kernel Function

- 1
- Definiton of feature space: Suppose the pattern $x\in {\Re}^{p}$ maps the input space ${\Re}^{p}$ to a new space by mapping $\Phi $, that is, $\Phi :x\in \chi \subseteq {\Re}^{p}\mapsto \Phi \left(\mathbf{x}\right)\in F\subseteq {\Re}^{H}(p\ll H)$. Then, F is called the feature space.
- 2
- Definition of kernel function: Let X be a non-empty set, H be a inner product space, $\Phi $ be the mapping from X to H, if the function $\mathrm{K}:X\times X\to R$ satisfies the condition: for $\forall x,y\in X$, $K\left(x,y\right)=\u2329\Phi \left(x\right),\Phi \left(y\right)\u232a$. Then, K is the kernel function.
- 3
- Definition of positive semi-definite kernel function: if the continuous symmetric function $K\left(x,y\right)\in {L}_{\infty}\left(X\times Y\right)$ satisfies the following formula:$${\int}_{X\times Y}K\left(x,y\right)f\left(x\right)f\left(y\right)dxdy\ge 0,\forall f\in {L}_{2}\left(X\right)$$Then, the function $K\left(x,y\right)$ is a positive semi-definite kernel function. In the discrete case, for any positive integer n, sample set ${x}_{1},{x}_{2},\cdots ,{x}_{N}\in {\Re}^{p}$ and constant ${a}_{1},{a}_{2},\cdots ,{a}_{N}\in \Re $, if the symmetric function $K\left(x,y\right)\in {L}_{\infty}\left(X\times Y\right)$ and satisfies the following formula:$$\sum _{i,j=1}^{n}{a}_{i}{a}_{j}K\left({x}_{i},{x}_{j}\right)\ge 0$$Then, $K\left(x,y\right)$ is a positive semi-definite kernel function.
- 4
- Definition of Mercer conditions: The necessary and sufficient condition for any symmetric function $K\left(x,y\right)$ that can be represented as the inner product form $K\left(x,y\right)=\u2329\Phi \left(x\right),\Phi \left(y\right)\u232a$ is positive semi-definite, that is,$${\int}_{X\times Y}K\left(x,y\right)f\left(x\right)f\left(y\right)\ge 0$$For the discrete case, the following matrix$$\left[\begin{array}{cccc}K\left({x}_{1},{x}_{1}\right)& K\left({x}_{1},{x}_{2}\right)& \cdots & K\left({x}_{1},{x}_{\mathrm{n}}\right)\\ K\left({x}_{2},{x}_{1}\right)& K\left({x}_{2},{x}_{2}\right)& \cdots & K\left({x}_{2},{x}_{\mathrm{n}}\right)\\ \cdots & \cdots & \cdots & \cdots \\ K\left({x}_{\mathrm{n}},{x}_{1}\right)& K\left({x}_{n},{x}_{2}\right)& \cdots & K\left({x}_{\mathrm{n}},{x}_{n}\right)\end{array}\right]$$The nonnegative condition is as follows:$${\int}_{X\times Y}K\left(x,y\right)f\left(x\right)f\left(y\right)\ge 0,\forall f\in {L}_{2}\left(x\right)$$

#### 3.1.2. The Fundamentals of Kernel Functions

#### 3.1.3. Gaussian Radial Basis Function Clustering Algorithm

#### 3.2. The Kernel-Induced Distance Based FCM (KFCM)

Algorithm 1 Kernel-based fuzzy c-means algorithm. |

Input: Input dataset $X={\left\{{x}_{i}\right\}}_{i=1}^{N}$, the number of clusters $c(2\le c\le n)$, the threshold $\epsilon $, the maximum number of iterations t, the fuzziness index m; |

Output: c clusters of X,U,V. |

Begin |

Fix $c,{t}_{max},m>1,\epsilon >0$, for some positive constant; |

Initialize the memberships ${\mu}_{ki}^{0}$; |

For $t=1,2,\dots ,{t}_{max}$ do: |

Update all prototypes ${\nu}_{ki}^{t}$ with Equation (13); |

Update all memberships ${\mu}_{ki}^{t}$ with Equation (12); |

Compute ${E}^{t}={max}_{ki}\left|{\mu}_{ki}^{t}-{\mu}_{ki}^{t-1}\right|$, |

If ${E}^{t}>\epsilon $, then $t\leftarrow t+1$ and goto Step 4 else stop |

If $t\ge {t}_{max}$, then stop |

end for; |

return c clusters of X,U,V; |

end |

#### 3.3. Kernel-Induced Distance Based KBFWCM with Dpatial Constraints

Algorithm 2 KBFWCM algorithm. | |

Input: Input Image $X={\left\{{x}_{i}\right\}}_{i=1}^{N}$, the number of clusters $c(2\le c\le N)$, the threshold $\epsilon $, the fuzziness index m, the parameter $\alpha $, the parameter $\beta $, the window size w; | |

Output: c clusters of X,U,V. | |

1: | Begin |

2: | Fix $X={\left\{{x}_{i}\right\}}_{i=1}^{N}$,$C\left(1<c<N\right)$,$m\in (1,\infty )$,$\epsilon >0$; |

3: | Initialize the center ${V}^{\left[0\right]}\in {R}_{pc}$; |

4: | Initialize U with random numbers and normalize with constraint (2); |

5: | Initialize all ${\beta}^{{}^{\prime}}s$ with ${1}^{\prime}s$; |

6: | Compute the mean or median filtered image $\overline{X}$ and set the iterative index k = 1; |

7: | For t = 1 to N do: |

8: | Update the ${U}^{\left[t\right]}$ with (23) and constraint (2); |

9: | for c = 1 to C do: (for each cluster); |

10: | For d = 1 to p do: (for each attribute) |

11: | for k = 1 to n do: (for each data item) |

12: | ${e}_{cdk}:=|{x}_{kd}-{v}_{cd}|$ (calculate the residual) |

13: | end for (for each data item) |

14: | sort residual; mark each residual with the number of sequences $\chi $ in the ordered sequence; |

15: | for k = 1 to n do: (for each residual); |

16: | calculate the characteristics of ${\beta}_{ckd}$ using formulas (4) and (5) or uniform weighting |

17: | end for; (for each residual) |

18: | end for; (for each attribute) |

19: | Update the prototypes for t-th iteration ${V}^{\left[t\right]}$ using (24); |

20: | Update the ${\eta}_{k}$ using (25); |

21: | end for; (for each cluster) |

22: | for k = 1 to n do: (for each data item) |

23: | calculate the characteristics of using formula (6) |

24: | end for; (for each data item) |

25: | if $\u2225{\mathbf{V}}_{new}-{\mathbf{V}}_{old}\u2225<\epsilon $ then goto 27 |

26: | end for |

27: | return c clusters of X,U,V; |

28: | end |

## 4. Algorithm Experiment

#### 4.1. IRIS Data Used for Comparison of Classification Effectiveness

#### 4.1.1. Classification Error Rate (CER)

#### 4.1.2. Deviation Degree (DD) Calculation

#### 4.2. Make Comparisons Between New Algorithm and Other Algorithms in Image Segmentation

#### 4.3. Comparison of Experimental Results Using Brain MR and Lena Images

#### 4.3.1. Denoising Experimental Results of MR Image

#### 4.3.2. Denoising Experimental Results of Lena’s Head Images

## 5. Conclusions

- (1)
- The local spatial information of the image and the typicality of the data item are used to improve the segmentation results and the kernel method is used to reduce the computation.
- (2)
- KBFWCM uses the membership filtering algorithm to exploit the local spatial constraint. Since noise points will have low compatibility in all clusters, the membership functions obtained by this algorithm more approach to the concept of typicality, making their impact on clustering negligible. Therefore, this algorithm is naturally more immune to noise.
- (3)
- This algorithm has additional advantages in calculation. It is a natural mechanism, assigning ’fuzzy labels’ to data in each iteration. Therefore, it can be used for more complex pattern recognition.
- (4)
- Experimental results show that the proposed KBFWCM can provide better segmentation results without adjusting parameters for different grayscale.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The though origin of the Kernel-based Robust Bias-correction Fuzzy Weighted C-ordered-means Clustering Algorithm (KBFWCM) algorithm.

**Figure 2.**The piecewise linear ordered weighted averaging (PLOWA) and S-ordered weighted average (SOWA) weighting functions for n = 100 and ${p}_{a}=0.2$, ${p}_{c}=0.5$, ${p}_{l}=0.2$.

**Figure 3.**Comparison of segmentation results on the synthetic image from denoising base on Gaussian. (

**a**) Ground truth image; (

**b**) The image added Gaussian (zero mean and 20 variance); (

**c**) KFCM result; (

**d**) KFCMS1 result; (

**e**) KFCMS2 result; (

**f**) KGFCMS; (

**g**) FLICM result; (

**h**) ARKFCM result; (

**i**) FRFCM result;(

**j**) KBFWCM result.

**Figure 4.**Comparison of segmentation results on the synthetic image from denoising base on salt & pepper. (

**a**) Ground truth image; (

**b**) The image added salt & pepper (zero mean and $20\%$ variance); (

**c**) KFCM result; (

**d**) KFCMS1 result; (

**e**) KFCMS2 result; (

**f**) KGFCMS; (

**g**) FLICM result; (

**h**) ARKFCM result; (

**i**) FRFCM result;(

**j**) KBFWCM result.

**Figure 5.**Comparison of segmentation results on the brain MR image from denoising base on salt & pepper. (

**a**) Ground truth image; (

**b**) The image added salt & pepper (zero mean and $20\%$ variance); (

**c**) KFCM result; (

**d**) KFCMS1 result; (

**e**) KFCMS2 result; (

**f**) KGFCMS; (

**g**) FLICM result; (

**h**) ARKFCM result; (

**g**) FLICM result; (

**h**) ARKFCM result; (

**i**) FRFCM result;(

**j**) KBFWCM result.

**Figure 6.**Comparison of segmentation results on the brain MR image from denoising base on Gaussian. (

**a**) Ground truth image; (

**b**) The image added Gaussian (zero mean and 20 variance); (

**c**) KFCM result; (

**d**) KFCMS1 result; (

**e**) KFCMS2 result; (

**f**) KGFCMS; (

**g**) FLICM result; (

**h**) ARKFCM result; (

**i**) FRFCM result; (

**j**) KBFWCM result.

**Figure 7.**Comparison of segmentation results on the Lena image from denoising base on Gaussian. (

**a**) Ground truth image; (

**b**) The image added Gaussian (zero mean and 20 variance); (

**c**) KFCM result; (

**d**) KFCMS1 result; (

**e**) KFCMS2 result; (

**f**) KGFCMS; (

**g**) FLICM result; (

**h**) ARKFCM result; (

**i**) FRFCM result; (

**j**) KBFWCM result.

**Table 1.**Theoretical Formulations of Classical fuxxy c-means (FCM) Algorithm and some of its improved versions.

Algorithm Name | Abbreviation | Objective Function | Time |
---|---|---|---|

Fuzzy C-Means [37] | FCM | ${J}_{\mathrm{m}}={\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{k=1}^{n}}{\mathrm{u}}_{ik}^{\mathrm{m}}{\u2225{x}_{k}-{v}_{i}\u2225}^{2}$ | 1984 |

FCM with spatial constraints [49] | FCMS | ${J}_{\mathrm{m}}={\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{j=1}^{n}}{\mathrm{u}}_{ij}^{m}{\u2225{x}_{j}-{v}_{i}\u2225}^{2}+\frac{\alpha}{{N}_{R}}{\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{j=1}^{n}}{u}_{ij}^{m}{\displaystyle \sum _{r\in {N}_{i}}}{\u2225{x}_{r}-{v}_{i}\u2225}^{2}$ | 2002 |

Enhanced FCM [54] | EnFCM | ${J}_{\mathrm{m}}={\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{j=1}^{l}}{\mathrm{u}}_{ij}^{\mathrm{m}}{\u2225{\xi}_{j}-{v}_{i}\u2225}^{2},{\xi}_{j}=\frac{\alpha}{1+\alpha}\left({x}_{j}+\frac{\alpha}{{N}_{R}}{\displaystyle \sum _{r\in {N}_{j}}}{x}_{r}\right).$ | 2003 |

Two variants of FCMS [53] | FCMS1 FCMS2 | $\begin{array}{l}{J}_{\mathrm{m}}={\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{j=1}^{n}}{\mathrm{u}}_{ij}^{m}{\u2225{x}_{j}-{v}_{i}\u2225}^{2}+\frac{\alpha}{{N}_{R}}{\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{j=1}^{n}}{u}_{ij}^{m}{\u2225{\overline{x}}_{j}-{v}_{i}\u2225}^{2}\hfill \end{array}$ | 2004 |

Spatial FCM [52] | SFCM | $\begin{array}{l}J={\displaystyle \sum _{j=1}^{N}}{\displaystyle \sum _{i=1}^{c}}{\left({{u}^{\prime}}_{ij}\right)}^{m}{\u2225{x}_{j}-{v}_{i}\u2225}^{2}\hfill \\ Where{h}_{ij}={\displaystyle \sum _{k\in NB\left({x}_{j}\right)}}{u}_{ik},{u}_{ij}^{\prime}=\frac{{u}_{ij}^{p}{h}_{ij}^{q}}{{\sum}_{k=1}^{c}{u}_{kj}^{p}{h}_{kj}^{q}}.\hfill \end{array}$ | 2006 |

Fuzzy Local Information C-Means [64] | FLICM | $\begin{array}{c}{J}_{\mathrm{m}}={\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{j=1}^{n}}\left({\mathrm{u}}_{ij}^{m}{\u2225{x}_{j}-{v}_{i}\u2225}^{2}+{G}_{ij}\right)\hfill \\ Where,{G}_{ij}=\frac{\alpha}{{N}_{R}}{\displaystyle \sum _{r\in {N}_{j},j\ne r}}\frac{1}{1+{d}_{jr}}{\left(1-{u}_{ij}\right)}^{m}{\u2225{x}_{r}-{v}_{i}\u2225}^{2}\hfill \end{array}$ | 2012 |

Two Kernel variants of GIFPFCM [60] | KGFCMS1 KGFCMS2 | $\begin{array}{c}{J}_{m}=2{\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{j=1}^{n}}{u}_{ij}^{m}\left(1-K\left({x}_{j},{v}_{i}\right)\right)+{\displaystyle \sum _{j=1}^{n}}{a}_{j}{\displaystyle \sum _{i=1}^{c}}{u}_{ij}\left(1-{u}_{ij}^{m-1}\right)\hfill \\ +2\beta {\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{j=1}^{n}}{u}_{ij}^{m}\left(1-K\left({\overline{x}}_{j},{v}_{i}\right)\right)\hfill \end{array}$ | 2013 |

New Weighted Fuzzy C-Means [68] | NWFCM | $\begin{array}{c}{J}_{\mathrm{m}}={\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{k=1}^{n}}{\mathrm{u}}_{ik}^{\mathrm{m}}{\u2225{x}_{j}-{M}_{ij}\u2225}^{2}Where,\hfill \\ {M}_{ij}={\displaystyle \sum _{r=1,r\ne j}^{n}}{\u2225{x}_{j}-{x}_{r}\u2225}^{-1}{u}_{ir}{x}_{r}/\phantom{{\displaystyle \sum _{r=1,r\ne j}^{n}}{\u2225{x}_{j}-{x}_{r}\u2225}^{-1}{u}_{ir}{x}_{r}\left({\displaystyle \sum _{r=1,r\ne j}^{n}}{\u2225{x}_{j}-{x}_{r}\u2225}^{-1}{u}_{ir}\right)}\phantom{\rule{0.0pt}{0ex}}\left({\displaystyle \sum _{r=1,r\ne j}^{n}}{\u2225{x}_{j}-{x}_{r}\u2225}^{-1}{u}_{ir}\right)\hfill \end{array}$ | 2014 |

Adaptively Regularized Kernel-based FCM [69] | ARKFCM | $\begin{array}{c}{J}_{ARKFCM}=2\left[{\displaystyle \sum _{i=1}^{N}}{\displaystyle \sum _{j=1}^{c}}{u}_{ij}^{m}\left(1-K\left({x}_{i},{v}_{j}\right)\right)\right.\hfill \\ \left.+{\displaystyle \sum _{i=1}^{N}}{\displaystyle \sum _{j=1}^{c}}{\phi}_{i}{u}_{ij}^{m}\left(1-K\left({\overline{x}}_{i},{v}_{j}\right)\right)\right]\hfill \end{array}$ | 2016 |

Fuzzy C-Ordered-Means [70] | FCOM | $\begin{array}{c}J(\mathbf{U},\mathbf{V})={\displaystyle \sum _{i=1}^{c}}{\displaystyle \sum _{k=1}^{N}}{\beta}_{ik}{\left({u}_{ik}\right)}^{m}\phantom{\rule{0.222222em}{0ex}}\mathcal{D}({\mathbf{x}}_{k},{\mathbf{v}}_{i}),\hfill \\ Where,\mathcal{D}({\mathbf{x}}_{k},{\mathbf{v}}_{i})={\u2225{\mathbf{x}}_{k}-{\mathbf{v}}_{i}\u2225}_{\mathbf{A}}^{2}={({\mathbf{x}}_{k}-{\mathbf{v}}_{i})}^{T}\mathbf{A}({\mathbf{x}}_{k}-{\mathbf{v}}_{i}).\hfill \end{array}$ | 2016 |

Fuzzy weighted C-ordered-means [73] | FWCOM | $\begin{array}{c}J\left(U,V,Z\right)={\displaystyle \sum _{c=1}^{C}}{\displaystyle \sum _{i=1}^{X}}{\beta}_{ci}{\left({u}_{ci}\right)}^{m}{\displaystyle \sum _{d=1}^{D}}{\left({z}_{cd}\right)}^{\varphi}{\left({x}_{id}-{v}_{cd}\right)}^{2},\hfill \\ Where,\forall c\in C,{\displaystyle \sum _{d=1}^{D}}{z}_{cd}=1,\forall k\in X,{\displaystyle \sum _{c=1}^{C}}{\beta}_{ck}{u}_{ck}={f}_{k}.\hfill \end{array}$ | 2017 |

Reconstruction and membership Filtering FCM [66] | FRFCM | ${J}_{\mathrm{m}}={\displaystyle \sum _{l=1}^{q}}{\displaystyle \sum _{k=1}^{c}}{\gamma}_{l}{u}_{kl}^{m}{\u2225{\zeta}_{l}-{v}_{k}\u2225}^{2},{\displaystyle \sum _{l=1}^{q}}{\gamma}_{l}=N$ | 2018 |

Algorithm | Misclassification Numbers | CER | Clustering Center | DD |
---|---|---|---|---|

FCM | 16 | 10.7% | $\begin{array}{c}{v}_{1}=(5.0040,3.4141,1.4828,0.2535)\hfill \\ {v}_{2}=(5.8887,2.7610,4.3636,1.3972)\hfill \\ {v}_{3}=(6.7748,3.0523,5.6465,2.0534)\hfill \end{array}$ | 0.081 |

FCMS | 10 | 6.7% | $\begin{array}{c}{v}_{1}=(4.9940,3.3649,1.4833,0.2537)\hfill \\ {v}_{2}=\left(5.7787,2.7851,4.2636,1.3882\right)\hfill \\ {v}_{3}=(6.6648,3.0573,5.6485,2.0564)\hfill \end{array}$ | 0.063 |

KFCMS | 9 | 6% | $\begin{array}{c}{v}_{1}=(4.9834,3.3781,1.5033,0.2641)\hfill \\ {v}_{2}=\left(5.6772,2.8001,4.3023,1.3872\right)\hfill \\ {v}_{3}=(6.7101,3.1001,5.7023,2.1032)\hfill \end{array}$ | 0.057 |

FWCOM | 8 | 5.3% | $\begin{array}{c}{v}_{1}=\left(4.9940,3.3649,1.4833,0.2537\right)\hfill \\ {v}_{2}=\left(5.7787,2.7851,4.2636,1.3882\right)\hfill \\ {v}_{3}=(6.6648,3.0573,5.6485,2.0564)\hfill \end{array}$ | 0.049 |

ARKFCM | 7 | 4.7% | $\begin{array}{c}{v}_{1}=\left(4.9840,3.3549,1.4533,0.2497\right)\hfill \\ {v}_{2}=(5.7817,2.8051,4.2539,1.3082)\hfill \\ {v}_{3}=(6.3948,3.0583,5.3485,2.0821)\hfill \end{array}$ | 0.046 |

FRFCM | 5 | 3.3% | $\begin{array}{c}{v}_{1}=\left(4.9941,3.3650,1.4863,0.2577\right)\hfill \\ {v}_{2}=(5.7767,2.7853,4.2615,1.2782)\hfill \\ {v}_{3}=(6.4648,3.0893,5.3485,2.1564)\hfill \end{array}$ | 0.037 |

KBFWCM | 5 | 3.3% | $\begin{array}{c}{v}_{1}=\left(4.9969,3.3649,1.4623,0.2158\right)\hfill \\ {v}_{2}=(5.7723,2.7852,4.1582,1.2782)\hfill \\ {v}_{3}=(6.4490,2.9253,5.4026,2.0110)\hfill \end{array}$ | 0.035 |

Algorithm | ${\mathit{\sigma}}_{\mathit{g}}$ = 10 | ${\mathit{\sigma}}_{\mathit{g}}$ = 20 | ${\mathit{\sigma}}_{\mathit{g}}$ = 25 | Salt & Pepper (5%) | Salt & Pepper (10%) | Salt & Pepper (20%) |
---|---|---|---|---|---|---|

KFCM | 85.94 | 86.16 | 85.34 | 92.71 | 93.41 | 95.35 |

KFCMS1 | 86.07 | 86.17 | 87.05 | 93.68 | 94.56 | 95.78 |

KFCMS2 | 89.15 | 87.40 | 86.44 | 94.68 | 95.85 | 94.89 |

KGFCMS | 71.50 | 94.74 | 94.90 | 98.84 | 96.97 | 99.32 |

FLICM | 96.64 | 96.64 | 96.64 | 97.34 | 96.44 | 97.28 |

ARKFCM | 93.17 | 93.17 | 93.17 | 94.23 | 95.84 | 97.84 |

FRFCM | 97.84 | 98.78 | 96.97 | 95.96 | 96.94 | 97.92 |

KBFWCM | 98.91 | 98.93 | 97.86 | 98.35 | 97.77 | 97.98 |

Denosing Method | MSE | MSE $\left(\%\right)$ | PSNR/db | SNR/db |
---|---|---|---|---|

Image with noise | 640.5461 | 100 | 20.0653 | 13.4480 |

KFCM | 138.6119 | 21.64 | 26.7128 | 21.0557 |

KFCMS1 | 179.6752 | 28.05 | 26.1314 | 20.0553 |

KFCMS2 | 182.4737 | 28.49 | 25.5188 | 19.8616 |

KGFCMS | 105.4836 | 16.47 | 26.0125 | 20.6052 |

FLICM | 107.1739 | 16.73 | 26.1670 | 20.5099 |

ARKFCM | 91.4922 | 14.28 | 28.8986 | 22.2414 |

FRRCM | 67.5066 | 10.54 | 29.5425 | 23.8853 |

KBFWCM | 54.9023 | 8.53 | 31.5349 | 24.6808 |

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**MDPI and ACS Style**

Zhang, W.; Guo, X.; Huang, T.; Liu, J.; Chen, J.
Kernel-Based Robust Bias-Correction Fuzzy Weighted C-Ordered-Means Clustering Algorithm. *Symmetry* **2019**, *11*, 753.
https://doi.org/10.3390/sym11060753

**AMA Style**

Zhang W, Guo X, Huang T, Liu J, Chen J.
Kernel-Based Robust Bias-Correction Fuzzy Weighted C-Ordered-Means Clustering Algorithm. *Symmetry*. 2019; 11(6):753.
https://doi.org/10.3390/sym11060753

**Chicago/Turabian Style**

Zhang, Wenyuan, Xijuan Guo, Tianyu Huang, Jiale Liu, and Jun Chen.
2019. "Kernel-Based Robust Bias-Correction Fuzzy Weighted C-Ordered-Means Clustering Algorithm" *Symmetry* 11, no. 6: 753.
https://doi.org/10.3390/sym11060753