User:Renatokeshet/mm

In mathematical morphology, reconstruction is an operation that...

Mathematical definition

Let X and Y be subsets of an Euclidean space $\mathbb {R} ^{d}$ or the integer grid $\mathbb {Z} ^{d}$, for some dimension d, such that $Y\subseteq X$. Also, let B be a structuring element.

The reconstruction of X from Y is given by:

$R\{X,Y\}=\lim _{n\rightarrow \infty }(\delta _{X})^{n}(Y)$,

where

$(\delta _{X})^{n}=\underbrace {\delta _{X}\delta _{X}\ldots \delta _{X}} _{n{\mbox{ times}}}$,

and $\delta _{X}(Y)$ denotes the conditional dilation of Y inside X:

$\delta _{X}(Y)=(Y\oplus B)\cap X$.

The symbol $\oplus$ denotes morphological dilation.

A structuring element is a simple, pre-defined shape, represented as a binary image, used to probe another binary image, in morphological operations such as erosion, dilation, opening, and closing.

Let $C$ and $D$ be two structuring elements satisfying $C\cap D=\emptyset$. The pair (C,D) is sometimes called composite structuring element. The hit-or-miss transform of a given image A by B=(C,D) is given by:

$A\odot B=(A\ominus C)\cap (A^{c}\ominus D)$,

where $A^{c}$ is the set complement of A.

That is, a point x in E belongs to the hit-or-miss transform output if C translated to x fits in A, and D translated to x misses A (fits the background of A).

Some applications

• Pattern detection. By definition, the hit-or-miss transform indicates the positions where a certain pattern (characterized by the composite structuring element B) occurs in the input image.

• Thinning. Let $E=Z^{2}$, and consider the eight composite structuring elements, composed by:

$C_{1}=\{(0,0),(-1,-1),(0,-1),(1,-1)\}$ and $D_{1}=\{(-1,1),(0,1),(1,1)\}$

$C_{2}=\{(-1,0),(0,0),(-1,-1),(0,-1),\}$ and $D_{2}=\{(0,1),(1,1),(1,0)\}$

and the three rotations of each by $90^{o}$, $180^{o}$, and $270^{o}$. The corresponding composite structuring elements are denoted $B_{1},\ldots ,B_{8}$. For any i between 1 and 8, and any binary image X, define

$X\otimes B_{i}=X\setminus (X\odot B_{i})$,

where $\setminus$ denotes the set-theoretical difference.

The thinning of an image A is obtained by cyclically iterating until convergence:

$A\otimes B_{1}\otimes B_{2}\otimes \ldots \otimes B_{8}\otimes B_{1}\otimes B_{2}\otimes \ldots$.

• Pruning.

• Computing the Euler number.

Bibliography

• An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)