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Concept
Mathematical morphology
Summary
Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to s, but it can be employed as well on graphs, surface meshes, solids, and many other spatial structures.
Topological and geometrical continuous-space concepts such as size, shape, convexity, connectivity, and geodesic distance, were introduced by MM on both continuous and discrete spaces. MM is also the foundation of morphological , which consists of a set of operators that transform images according to the above characterizations.
The basic morphological operators are erosion, dilation, opening and closing.
MM was originally developed for s, and was later extended to grayscale functions and images. The subsequent generalization to complete lattices is widely accepted today as MM's theoretical foundation.
Mathematical Morphology was developed in 1964 by the collaborative work of Georges Matheron and Jean Serra, at the École des Mines de Paris, France. Matheron supervised the PhD thesis of Serra, devoted to the quantification of mineral characteristics from thin cross sections, and this work resulted in a novel practical approach, as well as theoretical advancements in integral geometry and topology.
In 1968, the Centre de Morphologie Mathématique was founded by the École des Mines de Paris in Fontainebleau, France, led by Matheron and Serra.
During the rest of the 1960s and most of the 1970s, MM dealt essentially with s, treated as sets, and generated a large number of binary operators and techniques: Hit-or-miss transform, dilation, erosion, opening, closing, granulometry, thinning, skeletonization, ultimate erosion, conditional bisector, and others. A random approach was also developed, based on novel image models. Most of the work in that period was developed in Fontainebleau.
From the mid-1970s to mid-1980s, MM was generalized to grayscale functions and s as well. Besides extending the main concepts (such as dilation, erosion, etc.
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Closing (morphology)
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.
Erosion (morphology)
Erosion (usually represented by ⊖) is one of two fundamental operations (the other being dilation) in from which all other morphological operations are based. It was originally defined for s, later being extended to grayscale images, and subsequently to complete lattices. The erosion operation usually uses a structuring element for probing and reducing the shapes contained in the input image. In binary morphology, an image is viewed as a subset of a Euclidean space or the integer grid , for some dimension d.
Dilation (morphology)
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.