基于矩阵偏序关系的形态学算子

doi:10.12263/DZXB.20220007

摘要/Abstract

摘要：

已有形态偏序由于忽略了图像像素的局部相关性导致运算结果产生噪声叠加、拓扑失真等问题，本文在矩阵Frobenius范数的基础上定义了一种新的偏序关系，从理论上证明了该序满足的自反性、传递性和反对称性.在此基础上，本文提出了基于新序的形态学膨胀、腐蚀算子和相关的梯度运算算子.为验证新序和算子的有效性，与已有的算法进行对比实验，结果表明，新序和相应的算子在保证颜色分量相关性的同时，利用矩阵运算保证了像素的局部相关性，在抑制噪声和边缘保持方面均优于现有的算法.同时，通过结果对比和理论分析发现，新的形态算子对结构元素尺寸的包容性强，克服了形态算子在应用时难以选择合适结构元素大小的问题.这种新序是多通道图像处理的基础，可以扩宽形态学理论的应用范围.

关键词: 数学形态学, 矩阵的Frobenius范数, 偏序关系, 像素局部相关性, 噪声抑制, 拓扑保持

Abstract:

The existing morphological partial orderings neglect the local correlation of image pixels, which leads to problems such as noise superposition and topological distortion in the calculation results. This paper defines a new partial ordering relation based on the matrix Frobenius norm, which theoretically proves the reflexivity, transitivity and antisymmetry of the ordering. On this basis, this paper proposes the morphological dilation, erosion operators and the related gradient operators based on the new ordering. In order to verify the effectiveness of the new ordering and operators, those new operators are compared with the existing algorithms. The experimental results show that the new ordering and the corresponding operators ensure the correlation of the color components while using matrix operation to ensure the local correlation of the pixels. It outperforms the existing algotirhms in terms of noise suppression and edge preservation. At the same time, through comparison of the results and theoretical analysis, it is found that the new morphological operators have a strong tolerance for the size of structural element, which overcomes the difficulty of selecting suitable size when the morphological operators are applied. This new ordering is the basis of multi-channel image processing and it can broaden the application range of morphological.

Key words: mathematical morphology, Frobenius norm of matrix, partial ordering relation, pixel local correlation, noise suppression, topology maintenance

中图分类号:

TP391.4

王娜, 王俊平, 朱俊辉. 基于矩阵偏序关系的形态学算子[J]. 电子学报, 2023, 51(1): 213-221.

Na WANG, Jun-ping WANG, Jun-hui ZHU . Morphological Operators Based on Matrix Partial Ordering Relation[J]. Acta Electronica Sinica, 2023, 51(1): 213-221.

图/表 5

图1 经典图像局部像素排序结果对比

图2 灰度图像在经典形态算子、Loewner序形态算子、本偏序形态算子下的膨胀、腐蚀结果

图3 街道图像在基于R-ordering的形态算子、基于模糊序的形态算子、基于本偏序的形态算子下的膨胀、膨胀梯度结果

图4 帽子图像在基于R-ordering的形态算子、基于模糊序的形态算子、基于本偏序的形态算子下的腐蚀、腐蚀梯度结果

图5 三种形态算子在膨胀运算、腐蚀运算、开运算、闭运算下FSIMc值随结构元素尺寸的平均变化

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