Granulometry (morphology)

In mathematical morphology, granulometry is an approach to compute a size distribution of grains in binary images, using a series of morphological opening operations. It was introduced by Georges Matheron 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 in a Euclidean space or grid "E", and consider the family formula_1, formula_2, given by:
where formula_4 denotes morphological dilation. By convention, formula_5 is the set containing only the origin of "E", and formula_6.
Let "X" be a set (i.e., a binary image in mathematical morphology), and consider the series of sets formula_7, formula_2, given by:
where formula_10 denotes the morphological opening.
The "granulometry function" formula_11 is the cardinality (i.e., area or volume, in continuous Euclidean space, or number of elements, in grids) of the image formula_12:
The pattern spectrum or size distribution of "X" is the collection of sets formula_14, formula_2, given by:
The parameter "k" is referred to as "size", and the component "k" of the pattern spectrum formula_17 provides a rough estimate for the amount of grains of size "k" in the image "X". Peaks of formula_17 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 mathematician was inspired by sieving as a means of characterizing "size". In sieving, a granular sample is worked through a series of sieves with decreasing hole sizes. As a consequence, the different grains 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 formula_19 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:
A granulometry-generating family of operators should satisfy the above three axioms.
In the above case (granulometry generated by a structuring element), formula_23.
Another example of granulometry-generating family is when formula_24, where formula_25 is a set of linear structuring elements with different directions.