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In [[mathematical morphology]], '''granulometry''' is an approach to compute a size distribution of grains in binary images, using a series of [[opening (morphology)|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 generated by a structuring element ==

Let ''B'' be a [[structuring element]] in an [[Euclidean space]] or [[Lattice graph|grid]] ''E'', and consider the family <math>\{B_k\}</math>, <math>k=0,1,\ldots</math>, given by:

:<math>B_k=\underbrace{B\oplus\ldots\oplus B}_{k\mbox{ times}}</math>,

where <math>\oplus</math> denotes [[dilation (morphology)|morphological dilation]]. By convention, <math>B_0</math> is the set containing only the origin of ''E'', and <math>B_1=B</math>.

Let ''X'' be a [[Set (mathematics)|set]] (i.e., a [[binary image]] in mathematical morphology), and consider the series of sets <math>\{\gamma_k(X)\}</math>, <math>k=0,1,\ldots</math>, given by:

:<math>\gamma_k(X)=X\circ B_k</math>,

where <math>\circ</math> denotes the morphological opening.

The ''granulometry function'' <math>G_k(X)</math> is the [[cardinality]] (i.e., [[area]] or [[volume]], in continuous Euclidean space, or number of elements, in grids) of the image <math>\gamma_k(X)</math>:

:<math>G_k(X)=|\gamma_k(X)|</math>.

The '''pattern spectrum''' or '''size distribution''' of ''X'' is the collection of sets <math>\{PS_k(X)\}</math>, <math>k=0,1,\ldots</math>, given by:

:<math>PS_k(X) = G_{k}(X)-G_{k+1}(X)</math>.

The parameter ''k'' is referred to as ''size'', and the component ''k'' of the pattern spectrum <math>PS_k(X)</math> provides a rough estimate for the amount of grains of size ''k'' in the image ''X''. Peaks of <math>PS_k(X)</math> 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 [[France|French]] mathematician was inspired by [[sieving]] as a means of characterizing ''size''. In sieving, a [[granular]] sample is worked through a series of [[sieve]]s with decreasing hole sizes. As a consequence, the different [[grain]]s 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 <math>\Psi_k(X)</math> 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., <math>\Psi_k(X)\subseteq X</math>,
# ''Increasingness'': The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., <math>X\subseteq Y\Rightarrow\Psi_k(X)\subseteq\Psi_k(Y)</math>,
# "''Stability''": The result of passing through two sieves is determined by the sieve with smallest hole size. I.e., <math>\Psi_k\Psi_m(X)=\Psi_m\Psi_k(X)=\Psi_{\max(k,m)}(X)</math>.

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

In the above case (granulometry generated by a structuring element), <math>\Psi_k(X)=\gamma_k(X)=X\circ B_k</math>.

Another example of granulometry-generating family is when <math>\Psi_k(X)=\bigcup_{i=1}^{N} X\circ (B^{(i)})_k</math>, where <math>\{B^{(i)}\}</math> is a set of linear structuring elements with different directions.

== 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)

[[Category:Mathematical morphology]]