# 论文学习 Fuzzy c Means clustering with local information and kernel metric for image segmentation

Posted by Sun on March 25, 2020

Fuzzy C-Means Clustering With Local Information and Kernel Metric for Image Segmentation(基于使用本地信息和核度量的模糊C-均值聚类的图像分割)

13_IEEE TRANSACTIONS ON IMAGE PROCESSING

Abstract—In this paper, we present an improved fuzzy C-means (FCM) algorithm for image segmentation by introducing a tradeoff weighted fuzzy factor and a kernel metric. The tradeoff weighted fuzzy factor depends on the space distance of all neighboring pixels and their gray-level difference simultaneously. By using this factor, the new algorithm can accurately estimate the damping extent of neighboring pixels. In order to further enhance its robustness to noise and outliers, we introduce a kernel distance measure to its objective function. The new algorithm adaptively determines the kernel parameter by using a fast bandwidth selection rule based on the distance variance of all data points in the collection. Furthermore, the tradeoff weighted fuzzy factor and the kernel distance measure are both parameter free. Experimental results on synthetic and real images show that the new algorithm is effective and efficient, and is relatively independent of this type of noise. Index Terms—Fuzzy clustering, gray-level constraint, image segmentation, kernel metric, spatial constraint.

## Introduction

Fuzzy c-means (FCM) algorithm

improved FCM algorithms

have been proposed by incorporating local spatial information into original FCM objective function

Motivation1

尽管RFLICM算法通过使用局部变化率可以利用更多的局部上下文信息来估计邻域像素之间的关系，**但忽视空间约束对中心像素和邻域像素之间关系的影响仍然是不合理的。**????????

## Motivation

#### FLICM

FLICM 提出了一种新的模糊因子Gki作为在其目标函数中的模糊局部相似性测量，旨在保证噪声不敏感和图像细节保存????。其将数据集 [xi_N i=1（在灰级空间中）分区到 c 群集的目标函数在

然后，通过更新隶属度$\left\{u_{ki}\right\}$和聚类中心$\left\{v_{k}\right\}_{k=1}^{c}$，可以获得最小值

#### B. Motivation of Using Non-Euclidean Distance

Gaussian Radial basis function (GRBF) kernel is a commonly- used method.

## METHODOLOGY

#### A. General Framework of KWFLICM Algorithm

where Ni stands for the set of neighbors in a window around xi, wij is the trade-off weighted fuzzy factor of jth in a local window around xi, 1 − K (xi, vk) represents a non-Euclidean distance measure based on kernel method, (1 − uki)m is a penalty which can accelerate the iterative convergence to some extent. {vk}c k=1is the centers or prototypes of the clusters and the array {uki} represents a membership matrix which also must satisfy the Eq. (2).

wij是围绕xi的局部窗口中jth的权衡加权模糊因子, 1-K是基于核方法的非欧距离度量, 1-u是可以一定程度上加速迭代收敛的惩罚. vk是聚类中心. u代表也满足2式的隶属度矩阵

the proposed algorithm can be summarized as follows

1. init 类别个数c, 模糊化参数m, 窗口尺寸Ni, 停止条件ε

2. 随机机初始化模糊聚类原型(prototypes)??

3. 设置循环计数器 b=0

4. 计算权衡加权模糊因子wij和修改的距离度量Dik^2???

5. 使用公式12更新分割矩阵

6. 使用公式13更新聚类中心

7.  if max Vnew− Vold < ε then stop, otherwise, set b = b+ 1 and go to step 4.

where V = [v1, v2, . . . , vc] are the vectors of the cluster prototypes.

#### C. Non-Euclidean Distance Based on Kernel Metric

objective function in KWFLICM is