granulometry (morphology)

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

:$B\_k=\backslash underbrace\{B\backslash oplus\backslash ldots\backslash oplus\; B\}\_\{k\backslash mbox\{\; times\}\}$,

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

:$\backslash gamma\_k(X)=X\backslash circ\; B\_k$,

where $\backslash circ$ 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 $\backslash gamma\_k(X)$:

:$G\_k(X)=|\backslash gamma\_k(X)|$.

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

:$PS\_k(X)\; =\; G\_\{k\}(X)-G\_\{k+1\}(X)$.

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

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

In the above case (granulometry generated by a structuring element), $\backslash Psi\_k(X)=\backslash gamma\_k(X)=X\backslash circ\; B\_k$.

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

• Particle-size distribution
• Grain size
• Optical granulometry

• 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)

Category:Mathematical morphology