User:Renatokeshet/Morphological skeleton

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 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, can be characterized by MM on both continuous and discrete spaces. MM is also the foundation of morphological image processing, which consists of a set of operators that transform images according to the above characterizations.

MM was originally developed for binary images, and was later extended to grayscale functions and images. The subsequent generalization to complete lattices is widely accepted today as MM's theoretical foundation.

History
Mathematical Morphology was born in 1964 from 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, lead by Matheron and Serra.

During the rest of the 1960's and most of the 1970's, 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. A random approach was also developed, based on novel image models. Most of the work in that period was developed in Fontainebleau.

From mid-1970's to mid-1980's, MM was generalized to grayscale functions and images as well. Besides extending the main concepts (such as dilation, erosion, etc...) to functions, this generalization yielded new operators, such as morphological gradients, top-hat transform and the Watershed (MM's main segmentation approach).

In the 1980's and 1990's, MM gained a wider recognition, as research centers in several countries began to adopt and investigate the method. MM started to be applied to a large number of imaging problems and applications.

In 1986, Jean 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.

The 1990's and 2000's also saw further theoretical advancements, including the concepts of connections and levelings.

In 1993, the first International Symposium on Mathematical Morphology (ISMM) took place in Barcelona, Spain. Since then, ISMMs are organized every 2-3 years, each time in a different part of the world: Fontainebleau, France (1994); Atlanta, USA (1996); Amsterdam, Netherlands (1998); Palo Alto, CA, USA (2000); Sydney, Australia (2002); Paris, France (2004); Rio de Janeiro, Brazil (2007); and Groningen, Netherlands (2009).

Binary morphology
In binary morphology, an image is viewed as a subset of an Euclidean space $$\mathbb{R}^d$$ or the integer grid $$\mathbb{Z}^d$$, for some dimension d.

Structuring element
The basic idea in binary morphology is to probe an image with a simple, pre-defined shape, drawing conclusions on how this shape fits or misses the shapes in the image. This simple "probe" is called structuring element, and is itself a binary image (i.e., a subset of the space or grid).

Here are some examples of widely used structuring elements (denoted by B):


 * Let $$E=\mathbb{R}^2$$; B is an open disk of radius r, centered at the origin.
 * Let $$E=\mathbb{Z}^2$$; 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 $$E=\mathbb{Z}^2$$; B is the "cross" given by: B={(-1,0), (0,-1), (0,0), (0,1), (1,0)}.

Basic operators
The basic operations are shift-invariant (translation invariant) operators strongly related to Minkowski addition.

Let E be a Euclidean space or an integer grid, and A a binary image in E.

Other operators and tools

 * Hit-or-miss transform
 * Morphological skeleton
 * Filtering by reconstruction
 * Ultimate erosions and conditional bisectors
 * Granulometry
 * Geodesic distance functions

Grayscale morphology


In grayscale morphology, images are functions mapping a Euclidean space or grid E into $$\mathbb{R}\cup\{\infty,-\infty\}$$, where $$\mathbb{R}$$ is the set of reals, $$\infty$$ is an element larger than any real number, and $$-\infty$$ is an element smaller than any real number.

Grayscale structuring elements are also functions of the same format, called "structuring functions".

Other operators and tools

 * Morphological Gradients
 * Top-hat transform
 * Watershed algorithm

By combining these operators one can obtain algorithms for many image processing tasks, such as feature detection, image segmentation, image sharpening, image filtering, and classification.

Mathematical morphology on complete lattices
Complete lattices are partially ordered sets, where every subset has an infimum and a supremum. In particular, it contains a least element and a greatest element (also denoted "universe").

Adjunctions (Dilation and Erosion)
Let $$(L,\leq)$$ be a complete lattice, with infimum and minimum symbolized by $$\wedge$$ and $$\vee$$, respectively. Its universe and least element are symbolized by U and $$\emptyset$$, respectively. Moreover, let $$\{ X_{i} \}$$ be a collection of elements from L.

Particular cases
Binary morphology is a particular case of lattice morphology, where L is the power set of E (Euclidean space or grid), that is, L is the set of all subsets of E, and $$\leq$$ is the set inclusion. In this case, the infimum is set intersection, and the supremum is set union.

Similarly, grayscale morphology is another particular case, where L is the set of functions mapping E into $$\mathbb{R}\cup\{\infty,-\infty\}$$, and $$\leq$$, $$\vee$$, and $$\wedge$$, are the point-wise order, supremum, and infimum, respectively. That is, is f and g are functions in L, then $$f\leq g$$ if and only if $$f(x)\leq g(x),\forall x\in E$$; the infimum $$f\wedge g$$ is given by $$(f\wedge g)(x)=f(x)\wedge g(x)$$; and the supremum $$f\vee g$$ is given by $$(f\vee g)(x)=f(x)\vee g(x)$$.