Английская Википедия:Granulometry (morphology)
Шаблон: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 Шаблон: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 <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 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 (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 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 <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 satisfies 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 the smallest hole size. I.e., <math>\Psi_k\Psi_m(X)=\Psi_m\Psi_k(X)=\Psi_{\min(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.
See also
References
- Random Sets and Integral Geometry, by Georges Matheron, Wiley 1975, Шаблон:ISBN.
- Image Analysis and Mathematical Morphology by Jean Serra, Шаблон:ISBN (1982)
- Image Segmentation By Local Morphological Granulometries, Dougherty, ER, Kraus, EJ, and Pelz, JB., Geoscience and Remote Sensing Symposium, 1989. IGARSS'89, Шаблон:Doi (1989)
- An Introduction to Morphological Image Processing by Edward R. Dougherty, Шаблон:ISBN (1992)
- Morphological Image Analysis; Principles and Applications by Pierre Soille, Шаблон:ISBN (1999)
