Mathematical morphology

MM is most commonly applied to digital images, 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.

The basic morphological operators are erosion, dilation, opening and closing.

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.

During the rest of the 1960s and most of the 1970s, MM dealt essentially with binary images, 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.

to functions, this generalization yielded new operators, such as morphological gradients, top-hat transform and the Watershed (MM's main segmentation approach).

In the 1980s and 1990s, MM gained a wider recognition, as research centers in several countries began to adopt and investigate the method.

In 1986, Serra further generalized MM, this time to a theoretical framework based on complete lattices.

This generalization brought flexibility to the theory, enabling its application to a much larger number of structures, including color images, video, graphs, meshes, etc.

At the same time, Matheron and Serra also formulated a theory for morphological filtering, based on the new lattice framework.

In 1993, the first International Symposium on Mathematical Morphology (ISMM) took place in Barcelona, Spain.

Since then, ISMMs are organized every 2–3 years: Fontainebleau, France (1994); Atlanta, USA (1996); Amsterdam, Netherlands (1998); Palo Alto, CA, USA (2000); Sydney, Australia (2002); Paris, France (2005); Rio de Janeiro, Brazil (2007); Groningen, Netherlands (2009); Intra (Verbania), Italy (2011); Uppsala, Sweden (2013); Reykjavík, Iceland (2015); Fontainebleau, France (2017); and Saarbrücken, Germany (2019).

[1] In binary morphology, an image is viewed as a subset of a Euclidean space

This simple "probe" is called the structuring element, and is itself a binary image (i.e., a subset of the space or grid).

Let E be a Euclidean space or an integer grid, and A a binary image in E. The erosion of the binary image A by the structuring element B is defined by where Bz is the translation of B by the vector z, i.e.,

Example application: Assume we have received a fax of a dark photocopy.

Erosion process will allow thicker lines to get skinny and detect the hole inside the letter "o".

, which means that it is the locus of translations of the structuring element B inside the image A.

In the case of the square of side 10, and a disc of radius 2 as the structuring element, the opening is a square of side 10 with rounded corners, where the corner radius is 2.

Example application: Let's assume someone has written a note on a non-soaking paper and that the writing looks as if it is growing tiny hairy roots all over.

Opening essentially removes the outer tiny "hairline" leaks and restores the text.

The above means that the closing is the complement of the locus of translations of the symmetric of the structuring element outside the image A.

Here are some properties of the basic binary morphological operators (dilation, erosion, opening and closing): In grayscale morphology, images are functions mapping a Euclidean space or grid E into

In this case, the dilation and erosion are greatly simplified, and given respectively by In the bounded, discrete case (E is a grid and B is bounded), the supremum and infimum operators can be replaced by the maximum and minimum.

Thus, dilation and erosion are particular cases of order statistics filters, with dilation returning the maximum value within a moving window (the symmetric of the structuring function support B), and the erosion returning the minimum value within the moving window B.

In the case of flat structuring element, the morphological operators depend only on the relative ordering of pixel values, regardless their numerical values, and therefore are especially suited to the processing of binary images and grayscale images whose light transfer function is not known.

Along this line one should also look into Continuous Morphology[2] Complete lattices are partially ordered sets, where every subset has an infimum and a supremum.

Similarly, grayscale morphology is another particular case, where L is the set of functions mapping E into

A shape (in blue) and its morphological dilation (in green) and erosion (in yellow) by a diamond-shaped structuring element.
The erosion of the dark-blue square by a disk, resulting in the light-blue square.
The dilation of the dark-blue square by a disk, resulting in the light-blue square with rounded corners.
The opening of the dark-blue square by a disk, resulting in the light-blue square with round corners.
The closing of the dark-blue shape (union of two squares) by a disk, resulting in the union of the dark-blue shape and the light-blue areas.
Watershed of the gradient of the cardiac image