Granulometry (morphology)

Let B be a structuring element in a Euclidean space or grid E, and consider the family ${\displaystyle \{B_{k}\}}$, ${\displaystyle k=0,1,\ldots }$, given by:

${\displaystyle B_{k}=\underbrace {B\oplus \ldots \oplus B} _{k{\mbox{ times}}}}$,

where ${\displaystyle \oplus }$ denotes morphological dilation. By convention, ${\displaystyle B_{0}}$ is the set containing only the origin of E, and ${\displaystyle B_{1}=B}$.

Let X be a set (i.e., a binary image in mathematical morphology), and consider the series of sets ${\displaystyle \{\gamma _{k}(X)\}}$, ${\displaystyle k=0,1,\ldots }$, given by:

${\displaystyle \gamma _{k}(X)=X\circ B_{k}}$,

where ${\displaystyle \circ }$ denotes the morphological opening.

The granulometry function ${\displaystyle G_{k}(X)}$ is the cardinality (i.e., area or volume, in continuous Euclidean space, or number of elements, in grids) of the image ${\displaystyle \gamma _{k}(X)}$:

${\displaystyle G_{k}(X)=|\gamma _{k}(X)|}$.

The pattern spectrum or size distribution of X is the collection of sets ${\displaystyle \{PS_{k}(X)\}}$, ${\displaystyle k=0,1,\ldots }$, given by:

${\displaystyle PS_{k}(X)=G_{k}(X)-G_{k+1}(X)}$.

The parameter k is referred to as size, and the component k of the pattern spectrum ${\displaystyle PS_{k}(X)}$ provides a rough estimate for the amount of grains of size k in the image X. Peaks of ${\displaystyle PS_{k}(X)}$ 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 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 ${\displaystyle \Psi _{k}(X)}$ 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., ${\displaystyle \Psi _{k}(X)\subseteq X}$,
2. Increasingness: The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., ${\displaystyle X\subseteq Y\Rightarrow \Psi _{k}(X)\subseteq \Psi _{k}(Y)}$,
3. "Stability": The result of passing through two sieves is determined by the sieve with smallest hole size. I.e., ${\displaystyle \Psi _{k}\Psi _{m}(X)=\Psi _{m}\Psi _{k}(X)=\Psi _{\min(k,m)}(X)}$.

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

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

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

• Particle-size distribution
• Grain size

• Random Sets and Integral Geometry, by Georges Matheron, Wiley 1975, ISBN 0-471-57621-2.
• Image Analysis and Mathematical Morphology by Jean Serra, ISBN 0-12-637240-3 (1982)
• Image Segmentation By Local Morphological Granulometries, Dougherty, ER, Kraus, EJ, and Pelz, JB., Geoscience and Remote Sensing Symposium, 1989. IGARSS'89, doi:10.1109/IGARSS.1989.576052 (1989)
• An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
• Morphological Image Analysis; Principles and Applications by Pierre Soille, ISBN 3-540-65671-5 (1999)