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Let X be a set (i.e., a binary image 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 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. Peaks of 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 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 satisfy the following properties:
Anti-extensivity: Each sieve reduces the amount of grains, i.e., ,
Increasingness: The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., ,
"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.