Granulometry (morphology)

Sieving axioms

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 ${\displaystyle {\mathrm{\Psi }}_k(X)}$ 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:

1. Anti-extensivity: Each sieve reduces the amount of grains, i.e., ${\displaystyle {\mathrm{\Psi }}_k(X)\subseteq X}$,
2. Increasingness: The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., ${\displaystyle X\subseteq Y\Rightarrow {\mathrm{\Psi }}_k(X)\subseteq {\mathrm{\Psi }}_k(Y)}$,
3. "Stability": The result of passing through two sieves is determined by the sieve with the smallest hole size. I.e., ${\displaystyle {\mathrm{\Psi }}_k{\mathrm{\Psi }}_m(X)={\mathrm{\Psi }}_m{\mathrm{\Psi }}_k(X)={\mathrm{\Psi }}_{min(k,m)}(X)}$.

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

In the above case (granulometry generated by a structuring element), ${\displaystyle {\mathrm{\Psi }}_k(X)={\gamma }_k(X)=X\circ B_k}$.

Another example of granulometry-generating family is when ${\displaystyle {\mathrm{\Psi }}_k(X)=\bigcup _{i=1}^{N}X\circ (B^{(i)}{)}_k}$, where ${\displaystyle \{B^{(i)}\}}$ is a set of linear structuring elements with different directions.