Granulometry (morphology)

Granulometry generated by a structuring element

Let B be a structuring element in an Euclidean space or grid E, and consider the family { B k } , k = 0 , 1 , … , given by:

where ⊕ denotes morphological dilation. By convention, B 0 is the set containing only the origin of E, and B 1 = B .

Let X be a set (i.e., a binary image in mathematical morphology), and consider the series of sets { γ k ( X ) } , k = 0 , 1 , … , given by:

where ∘ denotes the morphological opening.

The granulometry function G k ( X ) is the cardinality (i.e., area or volume, in continuous Euclidean space, or number of elements, in grids) of the image γ k ( X ) :

The pattern spectrum or size distribution of X is the collection of sets { P S k ( X ) } , k = 0 , 1 , … , given by:

The parameter k is referred to as size, and the component k of the pattern spectrum P S k ( X ) provides a rough estimate for the amount of grains of size k in the image X. Peaks of P S k ( X ) 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 Ψ k ( X ) 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., Ψ k ( X ) ⊆ X ,
2. Increasingness: The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., X ⊆ Y ⇒ Ψ k ( X ) ⊆ Ψ k ( Y ) ,
3. "Stability": The result of passing through two sieves is determined by the sieve with smallest hole size. I.e., Ψ k Ψ m ( X ) = Ψ m Ψ k ( X ) = Ψ max ( k , m ) ( X ) .

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

In the above case (granulometry generated by a structuring element), Ψ k ( X ) = γ k ( X ) = X ∘ B k .

Another example of granulometry-generating family is when Ψ k ( X ) = ⋃ i = 1 N X ∘ ( B ( i ) ) k , where { B ( i ) } is a set of linear structuring elements with different directions.