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Concept# Structuring element

Summary

In mathematical morphology, a structuring element is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image. It is typically used in morphological operations, such as dilation, erosion, opening, and closing, as well as the hit-or-miss transform.
According to Georges Matheron, knowledge about an object (e.g., an image) depends on the manner in which we probe (observe) it. In particular, the choice of a certain structuring element for a particular morphological operation influences the information one can obtain. There are two main characteristics that are directly related to structuring elements:
Shape. For example, the structuring element can be a "ball" or a line; convex or a ring, etc. By choosing a particular structuring element, one sets a way of differentiating some objects (or parts of objects) from others, according to their shape or spatial orientation.
Size. For example, one structuring element can be a square or a square. Setting the size of the structuring element is similar to setting the observation scale, and setting the criterion to differentiate image objects or features according to size.
Structuring elements are particular cases of binary images, usually being small and simple. In mathematical morphology, s are subsets of a Euclidean space Rd or the integer grid Zd, for some dimension d. Here are some examples of widely used structuring elements (denoted by B):
Let E=R2; B is an open disk of radius r, centered at the origin.
Let E=Z2; B is a 3x3 square, that is, B={(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)}.
Let E=Z2; B is the "cross" given by: B={(-1,0),(0,-1),(0,0),(0,1),(1,0)}.
In the discrete case, a structuring element can also be represented as a set of pixels on a grid, assuming the values 1 (if the pixel belongs to the structuring element) or 0 (otherwise).
When used by a hit-or-miss transform, usually the structuring element is a composite of two disjoint sets (two simple structuring elements), one associated to the foreground, and one associated to the background of the image to be probed.

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Dilation (usually represented by ⊕) is one of the basic operations in mathematical morphology. Originally developed for , it has been expanded first to grayscale images, and then to complete lattices. The dilation operation usually uses a structuring element for probing and expanding the shapes contained in the input image. In binary morphology, dilation is a shift-invariant (translation invariant) operator, equivalent to Minkowski addition. A binary image is viewed in mathematical morphology as a subset of a Euclidean space Rd or the integer grid Zd, for some dimension d.

In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B: where and denote erosion and dilation, respectively. Together with closing, the opening serves in computer vision and as a basic workhorse of morphological noise removal. Opening removes small objects from the foreground (usually taken as the bright pixels) of an image, placing them in the background, while closing removes small holes in the foreground, changing small islands of background into foreground.

In mathematical morphology, the closing of a set () A by a structuring element B is the erosion of the dilation of that set, where and denote the dilation and erosion, respectively. In , closing is, together with opening, the basic workhorse of morphological noise removal. Opening removes small objects, while closing removes small holes. It is idempotent, that is, . It is increasing, that is, if , then . It is extensive, i.e., . It is translation invariant.