# Granulometry (morphology)

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 *size* in mathematical morphology.

Granulometry | |
---|---|

Basic concepts | |

Particle size · Grain sizeSize distribution · Morphology | |

Methods and techniques | |

Mesh scale · Optical granulometrySieve analysis · Soil gradation | |

Related concepts | |

Granulation · Granular materialMineral dust · Pattern recognitionDynamic light scattering | |

- merge with Optical granulometry

## Granulometry generated by a structuring element

Let *B* be a structuring element in a Euclidean space or grid *E*, and consider the family
,
, given by:

- ,

where
denotes morphological dilation. By convention,
is the set containing only the origin of *E*, and
.

Let *X* be a set (i.e., a binary image in mathematical morphology), and consider the series of sets
,
, given by:

- ,

where denotes the morphological opening.

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

- .

The **pattern spectrum** or **size distribution** of *X* is the collection of sets
,
, given by:

- .

The parameter *k* is referred to as *size*, and the component *k* of the pattern spectrum
provides a rough estimate for the amount of grains of size *k* in the image *X*. Peaks of
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
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:

*Anti-extensivity*: Each sieve reduces the amount of grains, i.e., ,*Increasingness*: The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., ,- "
*Stability*": The result of passing through two sieves is determined by the sieve with smallest hole size. I.e., .

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

In the above case (granulometry generated by a structuring element), .

Another example of granulometry-generating family is when , where is a set of linear structuring elements with different directions.

## See also

## References

*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)