提出灰度图像堆栈算子新框架，可由二值图像算子扩展而来。
On the representation of stack operators by mathematical morphology
- 基于截面运算的可交换性，定义灰度图像堆栈算子
- 堆栈算子继承特征集算子的格结构性质
- 适合设计具有数学形态学基础的图像处理算法
本文引入灰度图像堆栈算子类，其满足：(a) 将二值图像映射为二值图像；(b) 在平均意义下与截面运算可交换。等价地，堆栈算子是集合算子的1-Lipschitz扩张，通过在图像截面上应用特征集合算子并求和得到。它们推广了堆栈滤波器，后者对应的特征集合算子为单调递增。主要结果表明，堆栈算子继承特征集合算子的格性质。本文聚焦平移不变且局部定义的堆栈算子，通过推导其特征函数、核表示与基表示证明该结论。研究成果对图像算子设计有启示：解决某些灰度图像处理问题时，只需设计二值图像上的变换算子，再通过堆栈算子扩展即可。未来工作包括堆栈算子的机器学习及可解图像处理问题的刻画。
原文摘要 · Abstract (English)
This paper introduces the class of grey-scale image stack operators as those that (a) map binary-images into binary-images and (b) commute on average with cross-sectioning. Equivalently, stack operators are 1-Lipchitz extensions of set operators which can be represented by applying a characteristic set operator to the cross-sections of the image and adding. In particular, they are a generalisation of stack filters, for which the characteristic set operators are increasing. Our main result is that stack operators inherit lattice properties of the characteristic set operators. We focus on the case of translation-invariant and locally defined stack operators and show the main result by deducing the characteristic function, kernel, and basis representation of stack operators. The results of this paper have implications on the design of image operators, since imply that to solve some grey-scale image processing problems it is enough to design an operator for performing the desired transformation on binary images, and then considering its extension given by a stack operator. We leave many topics for future research regarding the machine learning of stack operators and the characterisation of the image processing problems that can be solved by them.
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