Granulometry (morphology)

Template:Granulometry 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 Template:Em in mathematical morphology.

Granulometry generated by a structuring element

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

$B_k=\underset{k\text{ times}}{\underbrace{B\oplus \dots \oplus B}}$,

where $\oplus$ 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 $\{{\gamma }_k(X)\}$, $k=0,1,\dots$, given by:

${\gamma }_k(X)=X\circ B_k$,

where $\circ$ 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 ${\gamma }_k(X)$:

$G_k(X)=|{\gamma }_k(X)|$.

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

$P{S}_k(X)=G_k(X)-G_{k+1}(X)$.

The parameter k is referred to as size, and the component k of the pattern spectrum $P{S}_k(X)$ 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 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 ${\mathrm{\Psi }}_k(X)$ 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., ${\mathrm{\Psi }}_k(X)\subseteq X$,
2. Increasingness: The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., $X\subseteq Y\Rightarrow {\mathrm{\Psi }}_k(X)\subseteq {\mathrm{\Psi }}_k(Y)$,
3. "Stability": The result of passing through two sieves is determined by the sieve with the smallest hole size. I.e., ${\mathrm{\Psi }}_k{\mathrm{\Psi }}_m(X)={\mathrm{\Psi }}_m{\mathrm{\Psi }}_k(X)={\mathrm{\Psi }}_{\min (k,m)}(X)$.

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

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

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

See also

• Particle-size distribution
• Grain size
• Optical granulometry

References

• Random Sets and Integral Geometry, by Georges Matheron, Wiley 1975, Template:ISBN.
• Image Analysis and Mathematical Morphology by Jean Serra, Template:ISBN (1982)
• Image Segmentation By Local Morphological Granulometries, Dougherty, ER, Kraus, EJ, and Pelz, JB., Geoscience and Remote Sensing Symposium, 1989. IGARSS'89, Template:Doi (1989)
• An Introduction to Morphological Image Processing by Edward R. Dougherty, Template:ISBN (1992)
• Morphological Image Analysis; Principles and Applications by Pierre Soille, Template:ISBN (1999)