Expansion, corrosion, opening and closing operations-morphology in Digital Image Processing

Expansion, corrosion, opening, and closing operations are the most fundamental transformations of mathematical morphology.
This article focuses on the morphology of binary images.
Expansion: expands the boundary of each component connected to a binary image by one pixel (filling the edge or holes inside 0 pixels );
Corrosion: remove the boundary points of the 1-pixel join component of the binary image to narrow down the layer (backbone information can be extracted, Burr removed, and isolated 0 pixels can be removed );
Open: First corrosion and then expansion can remove isolated points outside the target
Closed: expands first and then corrode. the holes in the target can be removed.
The following references: Application of Mathematical Morphology in image processing

Binary morphology
       In mathematical morphology, morphological transformation of binary images is a processing process for collections. The essence of morphological operators is to express the interaction between objects or shapes and structural elements. The shape of structural elements determines the shape information of signals extracted by such operations. Morphological image processing is to move a structural element in the image, and then combine the structural element with the following binary image, and perform other set operations.
       The basic morphological operations are corrosion and expansion.
       In morphology, structural elements are the most important and basic concepts. The role of structural elements in morphological transformation is equivalent to the "filter window" in signal processing ". B (X) is used to represent structural elements. The corrosion and expansion of each point X in Workspace e are defined:
       
       The result of corrosion on E with B (X) is a set composed of all vertices of E after B is translated. The result of B (X)'s expansion on E is a set of non-empty vertices after the Structure Element B is translated. The process of first corrosion and then expansion is called an open operation. It can eliminate small objects, separate objects in the fine, and slide the boundaries of large objects. The process of first expansion and then corrosion is called closed operation. It has the function of filling small holes in the body, connecting adjacent objects and smoothing boundaries.
       It can be seen that binary morphology expansion and corrosion can be converted into a set of logical operations. The algorithm is simple, suitable for parallel processing, and easy to implement by hardware, it is suitable for image segmentation, refinement, skeleton extraction, edge extraction, and Shape Analysis of binary images. However, in different applications, the selection of structure elements and their corresponding processing algorithms are different. Different structure elements and processing algorithms must be designed for different target images. The size and shape of the structure element are suitable or not, which will directly affect the morphological calculation result of the image. Therefore, many scholars have put forward a series of improved algorithms based on their actual application. For example, the edge detection algorithm proposed by Liang Yong using multidimensional morphological elements not only has good edge locating capability, but also has good Noise Smoothing capability. The design method proposed by Xu Chao to construct a quasi-circular structure element or a sequence structure element to generate quasi-circular structure elements is combined for skeleton extraction, which can greatly reduce the computational workload of morphological operations, it can also meet the compatibility of scale, translation and rotation, and is suitable for analyzing and describing the shape.

The main applications of mathematical morphology in image processing include edge detection, image segmentation, morphological skeleton extraction, and noise filtering.
Methods for selecting structure elements: Multi-structure elements and genetic algorithms.