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.

Let B be a structuring element in a Euclidean space or grid E, and consider the family

denotes morphological dilation.

is the set containing only the origin of E, and

Let X be a set (i.e., a binary image in mathematical morphology), and consider the series of sets

denotes the morphological opening.

The granulometry function

is the cardinality (i.e., area or volume, 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.

indicate relatively large quantities of grains of the corresponding sizes.

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

that returns the subset of elements in X with sizes that are smaller or equal to k. This family of operators satisfies 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),

is a set of linear structuring elements with different directions.