In recent decades, image segmentation has aroused great interest of many researchers, and has become an important part of machine learning, pattern recognition, and computer vision. Among many methods of image segmentation, fuzzy C‐means (FCM) algorithm is undoubtedly a milestone in unsupervised method. With the further study of FCM, various different kinds of FCM algorithms are put forward to deal with the specific problems in image segmentation. Because there exist uncertainties in different regions of the image and similarity in the same region, reducing the uncertainty is still the main problem in image segmentation. Considering that Pythagorean fuzzy set (PFS) is a powerful tool to deal with uncertainty, in this paper, we use PFS to describe the uncertainty of image segmentation, including introducing fuzzification and defuzzification process and Pythagorean fuzzy element to describe the membership degree of pixel, combine the neighborhood information with weights and Pythagorean fuzzy distance, and propose Pythagorean fuzzy C‐means (PFCM) algorithm. Finally, we apply PFCM algorithm in image segmentation, such as different size images and Berkeley Segmentation Data Set to illustrate the effectiveness and applicability of our proposed algorithm. Meanwhile, we do comparison analysis between PFCM, fully convolution network and Deep‐image‐Prior networks, these results show that our proposed PFCM has good intuition and effectiveness.

中文翻译：

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毕达哥拉斯模糊C均值算法用于图像分割

近几十年来，图像分割引起了许多研究人员的极大兴趣，并且已经成为机器学习，模式识别和计算机视觉的重要组成部分。在许多图像分割方法中，模糊C均值（FCM）算法无疑是无监督方法中的一个里程碑。随着对FCM的深入研究，提出了各种不同的FCM算法来解决图像分割中的具体问题。由于图像的不同区域存在不确定性，而同一区域存在相似性，因此减少不确定性仍然是图像分割的主要问题。考虑到毕达哥拉斯模糊集（PFS）是处理不确定性的强大工具，在本文中，我们使用PFS来描述图像分割的不确定性，包括介绍模糊化和反模糊化过程以及勾股线模糊元素以描述像素的隶属度，将邻域信息与权重和勾股线模糊距离相结合，并提出勾股线模糊C均值（PFCM）算法。最后，我们将PFCM算法应用于图像分割中，例如不同尺寸的图像和Berkeley分割数据集，以说明该算法的有效性和适用性。同时，我们对PFCM，全卷积网络和Deep-image-Prior网络进行了比较分析，这些结果表明我们提出的PFCM具有良好的直觉性和有效性。并提出毕达哥拉斯模糊C均值（PFCM）算法。最后，我们将PFCM算法应用于图像分割中，例如不同尺寸的图像和Berkeley分割数据集，以说明该算法的有效性和适用性。同时，我们对PFCM，全卷积网络和Deep-image-Prior网络进行了比较分析，这些结果表明我们提出的PFCM具有良好的直觉性和有效性。并提出毕达哥拉斯模糊C均值（PFCM）算法。最后，我们将PFCM算法应用于图像分割中，例如不同尺寸的图像和Berkeley分割数据集，以说明该算法的有效性和适用性。同时，我们对PFCM，全卷积网络和Deep-image-Prior网络进行了比较分析，这些结果表明我们提出的PFCM具有良好的直觉性和有效性。