In mathematical morphology

Mathematical morphology

Mathematical morphology is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions...

, granulometry is an approach to compute a size distribution of grains in binary images, using a series of morphological opening

Opening (morphology)

In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:A\circ B = \oplus B, \, where \ominus and \oplus denote erosion and dilation, respectively....

 operations. It was introduced by Georges Matheron

Georges Matheron

Georges François Paul Marie Matheron was a French mathematician and geologist, known as the founder of geostatistics and a co-founder of mathematical morphology. In 1968 he created the Centre de Géostatistique et de Morphologie Mathématique at the Paris School of Mines in Fontainebleau...

 in the 1960s, and is the basis for the characterization of the concept of size in mathematical morphology.

Granulometry generated by a structuring element

Let B be a structuring element

Structuring element

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...

 in an Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 or grid

Lattice graph

The terms lattice graph, mesh graph, or grid graph refer to a number of categories of graphs whose drawing corresponds to some grid/mesh/lattice, i.e., its vertices correspond to the nodes of the mesh and its edges correspond to the ties between the nodes.-Square grid graph:A common type of a...

 E, and consider the family , , given by:
,

where denotes morphological dilation

Dilation (morphology)

Dilation is one of the basic operations in mathematical morphology. Originally developed for binary images, it has been expanded first to grayscale images, and then to complete lattices...

. By convention, is the set containing only the origin of E, and .

Let X be a set (i.e., a binary image

Binary image

A binary image is a digital image that has only two possible values for each pixel. Typically the two colors used for a binary image are black and white though any two colors can be used. The color used for the object in the image is the foreground color while the rest of the image is the...

 in mathematical morphology), and consider the series of sets , , given by:
,

where denotes the morphological opening.

The granulometry function is the cardinality (i.e., area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 or volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

, in continuous Euclidean space, or number of elements, in grids) of the image :
.

The pattern spectrum or size distribution of X is the collection of sets , , given by:
.

The parameter k is referred to as size, and the component k of the pattern spectrum provides a rough estimate for the amount of grains of size k in the image X. Peaks of indicate relatively large quantities of grains of the corresponding sizes.

Sieving axioms

The above common method is a particular case of the more general approach derived by Matheron.

The French

France

The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

 mathematician was inspired by sieving as a means of characterizing size. In sieving, a granular sample is worked through a series of sieve

Sieve

A sieve, or sifter, separates wanted elements from unwanted material using a woven screen such as a mesh or net. However, in cooking, especially with flour, a sifter is used to aerate the substance, among other things. A strainer is a type of sieve typically used to separate a solid from a liquid...

s with decreasing hole sizes. As a consequence, the different grain

GRAIN

GRAIN is a small international non-profit organisation that works to support small farmers and social movements in their struggles for community-controlled and biodiversity-based food systems. Our support takes the form of independent research and analysis, networking at local, regional and...

s in the sample are separated according to their sizes.

The operation of passing a sample through a sieve of certain hole size "k" can be mathematically described as an operator that returns the subset of elements in X with sizes that are smaller or equal to k. This family of operators satisfy the following properties:

1. Anti-extensivity: Each sieve reduces the amount of grains, i.e., ,
2. Increasingness: The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., ,
3. "Stability": The result of passing through two sieves is determined by the sieve with smallest hole size. I.e., .

A granulometry-generating family of operators should satisfy the above three axioms.

In the above case (granulometry generated by a structuring element), .

Another example of granulometry-generating family is when , where is a set of linear structuring elements with different directions.